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#### UNIT TEST 6

11th Standard

Reg.No. :
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Chemistry

USING BLUE OR BLACK PEN ONLY

Time : 01:00:00 Hrs
Total Marks : 75

PART - A

15 x 1 = 15
1. Gases deviate from ideal behavior at high pressure. Which of the following statement(s) is correct for non-ideality?

(a)

at high pressure the collision between the gas molecule become enormous

(b)

at high pressure the gas molecules move only in one direction

(c)

at high pressure, the volume of gas become insignificant

(d)

at high pressure the intermolecular interactions become significant

2. Rate of diffusion of a gas is

(a)

directly proportional to its density

(b)

directly proportional to its molecular weight

(c)

directly proportional to its square root of its molecular weight

(d)

inversely proportional to the square root of its molecular weight

3. Which of the following is the correct expression for the equation of state of van der Waals gas?

(a)

$\left( P+\frac { a }{ { n }^{ 2 }{ V }^{ 2 } } \right) (V-nb)=nRT$

(b)

$\left( P+\frac { na }{ { n }^{ 2 }{ V }^{ 2 } } \right) (V-nb)=nRT$

(c)

$\left( P+\frac { { an }^{ 2 } }{ { V }^{ 2 } } \right) (V-nb)=nRT$

(d)

$\left( \frac { P+{ n }^{ 2 }{ a }^{ 2 } }{ { V }^{ 2 } } \right) (V-ab)=nRT$

4. When an ideal gas undergoes unrestrained expansion, no cooling occurs because the molecules

(a)

are above inversion temperature

(b)

exert no attractive forces on each other

(c)

do work equal to the loss in kinetic energy

(d)

collide without loss of energy

5. Equal weights of methane and oxygen are mixed in an empty container at 298 K. The fraction of total pressure exerted by oxygen is

(a)

$\frac { 1 }{ 3 }$

(b)

$\frac { 1 }{ 2 }$

(c)

$\frac { 2 }{ 3 }$

(d)

$\frac { 1 }{ 3 } \times 273\times 298$

6. The temperatures at which real gases obey the ideal gas laws over a wide range of pressure is called

(a)

Critical temperature

(b)

Boyle temperature

(c)

Inversion temperature

(d)

Reduced temperature

7. A bottle of ammonia and a bottle of HCI connected through a long tube are opened simultaneously at both ends. The white ammonium chloride ring first formed will be

(a)

At the center of the tube

(b)

Near the hydrogen chloride bottle

(c)

Near the ammonia bottle

(d)

Throughout the length of the tube

8. The value of universal gas constant depends upon

(a)

Temperature of the gas

(b)

Volume of the gas

(c)

Number of moles of the gas

(d)

units of Pressure and volume.

9. The value of the gas constant R is

(a)

0.082 dm2 atm.

(b)

0.987 cal mol-1K-1

(c)

8.3 J mol-1 K-1

(d)

8 erg mol-1 K-1

10. Use of hot air balloon in sports at meteorological observation is an application of

(a)

Boyle's law

(b)

Newton's law

(c)

Kelvin's law

(d)

Brown's law

11. The table indicates the value of van der Waals constant 'a' in (dm3)2 atm. mol-2.

 Gas O2 N2 NH3 CH4 a 1.360 1.390 4.170 2.253

The gas which can be most easily liquefied is

(a)

O2

(b)

N2

(c)

NH3

(d)

CH4

12. Consider the following statements
i) Atmospheric pressure is less at the top of a mountain than at sea level
ii) Gases are much more compressible than solids or liquids
iii) When the atmospheric pressure increases the height of the mercury column rises.
Select the correct statement

(a)

I and II

(b)

II and III

(c)

I and III

(d)

I, II and III

13. Compressibility factor for CO2 at 400 K and 71.0 bar is 0.8697. The molar volume of CO2 under these conditions is

(a)

22.04 dm3

(b)

2.24 dm3

(c)

0.41 dm3

(d)

19.5 dm3

14. If temperature and volume of an ideal gas is increased to twice its values, the initial pressure P becomes

(a)

4P

(b)

2P

(c)

P

(d)

3P

15. At identical temperature and pressure, the rate of diffusion of hydrogen gas is 3$\sqrt { 3 }$ times that of a hydrocarbon having molecular formula CnH2n-2. What is the value of n ?

(a)

8

(b)

4

(c)

3

(d)

1

16. PART - B

Answer six questions.  Question No. 24 is compulsory.  Answer any five from the remaining.

6 x 2 = 12
17. Suppose there is a tiny sticky area on the wall of a container of gas. Molecules hitting this area stick there permanently. Is the pressure greater or less than on the ordinary area of walls?

18. Which of the following gases would you expect to deviate from ideal behavior under conditions of low-temperature F2, CI2, or Br2? Explain.

19. Distinguish between diffusion and effusion.

20. Aerosol cans carry clear warning of heating of the can. Why?

21. Why do astronauts have to wear protective suits when they are on the surface of moon?

22. When ammonia combines with HCI, NH4CI is formed as white dense fumes. Why do more fumes appear near HCI?

23. Write a short note on the consequence of Boyle's law. (or) Give the relationship between pressure and density

24. When a real gas is converted from its initial to final state by adiabatic expansion, it is not possible to calculate its volume using Boyle's law. Why?

25. Calculate the pressure exerted by 2 moles of sulphur hexafluoride in a steel vessel of volume 6 dm3 at 70°C assuming it is an ideal gas.

26. PART - C

Answer six question.     Question No. 32 compulsory.    Answer any 5 form the remaining

6 x 3 = 18
27. State Boyle's law.

28. A balloon filled with air at room temperature and cooled to a much lower temperature can be used as a model for Charle's law.

29. Name two items that can serve as a model for Gay Lusaac' law and explain.

30. Give the mathematical expression that relates gas volume and moles. Describe in words what the mathematical expression means.

31. What are ideal gases? In what way real gases differ from ideal gases.

32. Give suitable explanation for the following facts about gases.
(a) Gases don't settle at the bottom of a container.
(b) Gases diffuse through all the space available to them
(c) Explain with an increase in temperature.

33. Suggest why there is no hydrogen (H2) in our atmosphere. Why does the moon have no atmosphere?

34. When the driver of an automobile applies brake, the passengers are pushed toward the front of the car but a helium balloon is pushed toward back of the car. Upon forward acceleration the passengers are pushed toward the front of the car. Why?

35. In the below figure, let us find the missing parameters [volume in (b) and pressure in (c)]
P1 = 1 atm, P2 = 2 atm, P3 = ? atm
V1 = 1dm3, V2 =? dm3, V3 = 0.25 dm3
T1 = 298 K, T2 = 298 K, T3 = 298 K.

36. PART - D

5 x 5 = 25
1. Write the Van der Waals equation for a real gas. Explain the correction term for pressure and volume.

2. Derive the values of vanderwaals equation constants in terms of critical constants.

1. A sample of gas at 15°C at 1 atm. has a volume of 2.58 dm3. When the temperature is raised to 38°C at 1 atm does the volume of the gas increase? If so, calculate the final volume.

2. Derive the ideal gas equation by combining the empirical gas laws.

1. In an experiment of verification of Charle's law, the following are the set of readings taken by a student.

Experiment Volume (L) Temperature (0C)
1 1.54 20
2 1.65 40
3 1.95 100
4 2.07 120

What is the average value of the constant of proportionality?

2. A neon-di-oxygen mixture contains 70.6 g of di-oxygen and 167.5 g of neon. If the pressure of the mixture of gases in the cylinder is 25 bar, what is the partial pressure of di-oxygen and neon in the mixture? (Atomic mass of Ne=20u)

1. Solve

Effect of temperature on volume of the gas to verify Charles law All the container a, b and c have same pressure of 1 atm. If T1, T2, and T3 are, respectively, at 200, 300 and 100 K, and V1 = 0.3 dm3, calculate V2 and V3.

2. Use the graphs to answer the question below.

i) Which of the graphs is the best representation of pressure and temperature (measured I kelvin) for one mole of ideal gas
ii) Which of the graphs is the best representation of pressure and volume of one mole of an ideal gas.
iii) Which of the above graph represents PV vs P of one mole of an ideal gas?

1. In an experimant of verification of Charle's law, the following are the set of readings taken by a student

 Experiment Volume (L) Temperature (°C) 1 1.54 20 2 1.65 40 3 1.95 100 4 2.07 120

What is the average value of the constant of proportionality?

2. Which of following flasks has higher pressure
(a) 5.00 L containing 4.15 g of Helium at 298 K
(b) 10.0 L containing 56.2 g Argon at 303 K


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# How do you differentiate f(x)=(sinx+x)(x+e^x) using the product rule?

Dec 26, 2015

Apply the product rule to find:

$\frac{d}{\mathrm{dx}} f \left(x\right) = \left(\cos x + 1\right) \left(x + {e}^{x}\right) + \left(\sin x + x\right) \left(1 + {e}^{x}\right)$

#### Explanation:

The product rule tells us that:

$\frac{d}{\mathrm{dx}} u \left(x\right) v \left(x\right) = u ' \left(x\right) v \left(x\right) + u \left(x\right) v ' \left(x\right)$

So with $u \left(x\right) = \left(\sin x + x\right)$ and $v \left(x\right) = \left(x + {e}^{x}\right)$, we find:

$\frac{d}{\mathrm{dx}} f \left(x\right) = \left(\cos x + 1\right) \left(x + {e}^{x}\right) + \left(\sin x + x\right) \left(1 + {e}^{x}\right)$

format: plain


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# Codes and Expansions (CodEx) Seminar

## Chris Manon (University of Kentucky)Spaces of Eigenvalues

For an $m \times n$ matrix $\Phi$ with columns $\phi_1, \ldots, \phi_n$ and a subset $A \subset [n]$ of the index set of the columns, let $\lambda_A(\Phi)$ be the spectrum of the Hermitian matrix $\sum_{i \in A} \phi_i\phi_i^*.$ For a set of such sets $S = \{A_1, \ldots, A_k\}$ we get a map $\lambda_S: M_{m \times n}(\mathbb{C}) \to (\mathbb{R}^m)^{|S|}$. We call the image $P_S$ of $\lambda_S$ a space of eigenvalues. In this talk I'll explain how a few familiar objects from representation theory and geometry appear as $P_S$ for some set $S$, give a general condition implying that $P_S$ is a convex polyhedral cone, and give a general condition implying that $P_S$ is not convex. This is joint work with Alex Fink and Milena Hering.


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# Tag Archives: Plasmons

## Acoustic Plasmon

In regards to the posts I’ve made about plasmons in the past (see here and here for instance), it seems like the plasmon in a metal will always exist at a finite energy at $q=0$ due to the long-ranged nature of the Coulomb interaction. Back in 1956, D. Pines published a paper, where in collaboration with P. Nozieres, he proposed a method by which an acoustic plasmon could indeed exist.

The idea is actually quite simple from a conceptual standpoint, so a cartoony description should suffice in describing how this is possible. The first important ingredient in realizing an acoustic plasmon is two types of charge carriers. Pines, in his original paper, chose $s$-electrons and $d$-electrons from two separate bands to illustrate his point. However, electrons from one band and holes from another could also suffice. The second important ingredient in realizing the acoustic plasmon is that the masses of the two types of carriers must be very different (which is why Pines chose light $s$-electrons and heavy $d$-electrons).

Screening of heavy charge carrier by light charge carrier

So why are these two features necessary? Well, simply put, the light charge carriers can screen the heavy charge carriers, effectively reducing the range of the Coulomb interaction (see image above). Such a phenomenon is very familiar to all of us who study solids. If, for instance, the interaction between the ions on the lattice sites in a simple 3D monatomic solid were not screened by the electrons, the longitudinal acoustic phonon would necessarily be gapped because of the Coulomb interaction (forgetting, for the moment, about what the lack of charge neutrality would do to the solid!). In some sense, therefore, the longitudinal acoustic phonon is indeed such an acoustic plasmon. The ion acoustic wave in a classical plasma is similarly a manifestation of an acoustic plasmon.

This isn’t necessarily the kind of acoustic plasmon that has been so elusive to solid-state physicists, though. The original proposal and the subsequent search was conducted on systems where light electrons (or holes) would screen heavy electrons (or holes). Indeed, it was suspected that Landau damping into the particle-hole continuum was preventing the acoustic plasmon from being an observable excitation in a solid. However, there have been a few papers suggesting that the acoustic plasmon has indeed been observed at solid surfaces. Here is one paper from 2007 claiming that an acoustic plasmon exists on the surface of beryllium and here is another showing a similar phenomenon on the surface of gold.

To my knowledge, it is still an open question as to whether such a plasmon can exist in the bulk of a 3D solid. This has not stopped researchers from suggesting that electron-acoustic plasmon coupling could lead to the formation of Cooper pairs and superconductvity in the cuprates. Varma has suggested that a good place to look would be in mixed-valence compounds, where $f$-electron masses can get very heavy.

On the experimental side, the search continues…

A helpful picture: If one imagines light electrons and heavy holes in a compensated semimetal for instance, the in-phase motion of the electrons and holes would result in an acoustic plasmon while the out-of-phase motion would result in the gapped plasmon.

## Interactions, Collective Excitations and a Few Examples

Most researchers in our field (and many outside our field that study, e.g. ant colonies, traffic, fish schools, etc.) are acutely aware of the relationship between the microscopic interactions between constituent particles and the incipient collective modes. These can be as mundane as phonons in a solid that arise because of interactions between atoms in the lattice or magnons in an anti-ferromagnet that arise due to spin-spin interactions.

From a theoretical point of view, collective modes can be derived by examining the interparticle interactions. An example is the random phase approximation for an electron gas, which yields the plasmon dispersion (here are some of my own notes on this for those who are interested). In experiment, one usually takes the opposite view where inter-particle interations can be inferred from the collective modes. For instance, the force constants in a solid can often be deduced by studying the phonon spectrum, and the exchange interaction can be backed out by examining the magnon dispersions.

In more exotic states of matter, these collective excitations can get a little bizarre. In a two-band superconductor, for instance, it was shown by Leggett that the two superfluids can oscillate out-of-phase resulting in a novel collective mode, first observed in MgB2 (pdf!) by Blumberg and co-workers. Furthermore, in 2H-NbSe2, there have been claims of an observed Higgs-like excitation which is made visible to Raman spectroscopy through its interaction with the charge density wave amplitude mode (see here and here for instance).

As I mentioned in the post about neutron scattering in the cuprates, a spin resonance mode is often observed below the superconducting transition temperature in unconventional superconductors. This mode has been observed in the cuprate, iron-based and heavy fermion superconducting families (see e.g. here for CeCoIn5), and is not (at least to me!) well-understood. In another rather stunning example, no less than four sub-gap collective modes, which are likely of electronic origin, show up below ~40K in SmB6 (see image below), which is in a class of materials known as Kondo insulators.

Lastly, in a material class that we are actually thought to understand quite well, Peierls-type quasi-1D charge density wave materials, there is a collective mode that shows up in the far-infrared region that (to my knowledge) has so far eluded theoretical understanding. In this paper on blue bronze, they assume that the mode, which shows up at ~8 cm$^{-1}$ in the energy loss function, is a pinned phase mode, but this assignment is likely incorrect in light of the fact that later microwave measurements demonstrated that the phase mode actually exists at a much lower energy scale (see Fig. 9). This example serves to show that even in material classes we think we understand quite well, there are often lurking unanswered questions.

In materials that we don’t understand very well such as the Kondo insulators and the unconventional superconductors mentioned above, it is therefore imperative to map out the collective modes, as they can yield critical insights into the interactions between constituent particles or couplings between different order parameters. To truly understand what is going on these materials, every peak needs to be identified (especially the ones that show up below Tc!), quantified and understood satisfactorily.

As Lestor Freamon says in The Wire:

All the pieces matter.

## Plasma Frequency, Screening Response Time and the Independent Electron Approximation

The plasma frequency in the study of solids arises in many different contexts. One of the most illuminating ways to look at the plasma frequency is as a measure of the screening response time in solids. I’ve discussed this previously in reference to the screening of longitudinal phonons in semiconductors, but I think it is worth repeating and expanding upon.

What I mean by “screening response time” is that in any solid, when one applies a perturbing electric field, the electrons take a certain amount of time to screen this field. This time can usually be estimated by using the relation:

$t_p = \frac{2\pi}{\omega_p}$

Now, suppose I introduce a time-varying electric field perturbation into the solid that has angular frequency $\omega$. The question then arises, will the electrons in the solid be able to respond fast enough to be able to screen this field? Well, for frequencies $\omega < \omega_p,$ the corresponding perturbation variation time is $t = 2\pi/\omega > t_p$. This means the the perturbation variation time is longer than the time it takes for the electrons in the solid to screen the perturbation. So the electrons have no problem screening this field. However, if $\omega > \omega_p$ and $t < t_p$, the electronic plasma in the solid will not have enough time to screen out the time-varying electric field.

This screening time interpretation of the plasma frequency is what leads to what is called the plasma edge in the reflectivity spectra in solids. Seen below is the reflectivity spectrum for aluminum (taken from Mark Fox’s book Optical Properties of Solids):

One can see that below the plasma edge at ~15eV, the reflectivity is almost perfect, resulting in the shiny and reflective quality of aluminum metal in the visible range. However, above $\hbar\omega$=15eV, the reflectivity suddenly drops and light is able to pass through the solid virtually unimpeded as the electrons can no longer respond to the quickly varying electric field.

Now that one can see the effect of the screening time on an external electric field such as light, the question naturally arises as to how the electrons screen the electric field generated from other electrons in the solid. It turns out that much of what I have discussed above also works for the electrons in the solid self-consistently. Therefore, it turns out that the electrons near the Fermi energy also have their electric fields, by and large, screened out in a similar manner. The distance over which the electric field falls by $1/e$ is usually called the Thomas-Fermi screening length, which for most metals is about half a Bohr radius. That the Thomas-Fermi approximation works well is because one effectively assumes that $\omega_p \rightarrow \infty$, which is not a bad approximation for the low-energy effects in solids considering that the plasma frequency is often 10s of eV.

Ultimately, the fact that the low-energy electrons near the Fermi energy are well-screened by other electrons self-consistently permits one to use the independent electron approximation — the foundation upon which band theory is built. Therefore, in many instances that the independent electron approximation is used to describe physical phenomena in solids, it should be kept in mind the hidden role the plasmon actually plays in allowing these ideas to work.

Naively, from my discussion above, it would seem like the independent electron approximation would then break down in a band insulator. However, this is not necessarily so. There are two things to note in this regard: (i) there exists an “interband plasmon” at high energies that plays essentially the same role that a free-carrier plasmon does in a metal for energies $E_g << E < \hbar\omega_p$ and (ii) whether the kinetic or Coulomb energy dominates will determine the low energy phenomenology. An image below is taken from this paper on lithium fluoride, which is a band insulator with a band gap of about 5eV and exhibits a plasmon at ~22eV:

The interband plasmon ultimately contributes to the background dielectric function, $\epsilon$, which reduces the Coulomb energy between the electrons in the form:

$V_{eff} = \frac{e^2}{\epsilon_0 \epsilon r}$

For example, this is the Coulomb interaction felt between an electron and hole when an exciton is formed (with opposite sign), as can be seen for LiF in the above image.

Now, the kinetic energy can be approximated by the band width, $W$, which effectively gives the amount of “wavefunction overlap” between the neighboring orbitals. Now, if $W >> V_{eff}$, then the independent electron approximation remains a good approximation. In this limit, one can get a band insulator, that is adequately described using the independent electron approximation. In the opposite limit, however, often one gets what is called a Mott insulator. Because d- and f-electrons tend to be closely bound to the atomic site, there is usually less wavefunction overlap between the electrons, leading to a small band width. This is why Mott insulators tend to occur in materials that have d- and f-electrons near the Fermi energy

Most studies on strongly correlated electron systems tend to concentrate on low-energy phenomenology.  While this is no doubt important, in light of this post, I think it may be worth looking up from time to time as well.

## Plasmons of a Luttinger Liquid

There’s a quite remarkable experiment in a paper by the Wang group at Berkeley that recently came out in Nature Photonics demonstrating the existence of peculiar plasmons in carbon nanotubes. These are significant because this may constitute the first observations of plasmons in a Luttinger liquid. To observe these plasmons, the group used scattering-scanning near-field optical microscopy (s-SNOM).

The plasmons are novel in that they appear to have a quantized propagation velocity that depends only on the number of conducting channels. Also, the ratios for the propagation velocities appear to be in the form $1:\sqrt{2}:\sqrt{3}:\sqrt{4}$ for one, two, three and four nanotubes respectively. (Each nanotube has one conducting channel).

The plasmons are extremely well-confined spatially ($\lambda_p/\lambda \sim 1/100$, where $\lambda_p$ and $\lambda$ are the plasmon wavelength and the wavelength of free-space light respectively) and also have a quality factor of $\sim$20. This means that there may be important applications in store for these kinds of plasmons as well, though I find the result from a more fundamental perspective quite intriguing.

## Screening, Plasmons and LO-TO Splitting: One Last Time

I hope you are not by now fed up with my posts on this topic, but there is a great paper by Mooradian and Wright, which I’ve actually linked before in a different context, that is worth tackling. In this paper, they discuss the concept of plasmon-longitudinal optical (LO) phonon coupling.

To my mind, there is a significant aspect of the data which they do not explicitly address in their paper. Of course, I’m referring to degeneracy, LO-TO splitting, and screening. An image of their data is shown below (click to enlarge):

A quick run-down of the experiment: they are using Raman scattering on several different samples of GaAs with different doping levels. The carrier density can be read off in the image above.

It can be seen that for lower doped samples of GaAs, that there is strong LO-TO splitting. This is because of the long-ranged nature of the Coulomb interaction, as detailed here. As the carrier concentration is increased beyond the plasmon-phonon mixing region, the LO-TO splitting starts to disappear. This observation is noteworthy because there exists a “critical carrier density”, beyond which the LO and TO phonons are degenerate.

One can think of this in the following way: the plasmon energy is a measure of how quickly the free carriers can respond to an electric field. Therefore, for the highly doped GaAs samples, where the plasmon is at a significantly higher energy than the phonons, the free carriers can quickly screen the Coulomb field set up by the polar lattice. The electric field that is set up by the phonons can hence be approximated by a screened electric field (of the Thomas-Fermi kind) in this limit, and the Coulomb interaction is hence no longer long-ranged.

While the points I have made above will be quite obvious to many of you, I still find the data and its implications from a historical perspective quite profound.

Aside: I was heartened by Sarang’s post on the concept of emergence and upward heritability. One tends to think harder about one’s stance when there is an opposing view. He made some extremely important points regarding this topic, though I have to admit that I still lean towards Wilczek-ian concepts at present.

## Plasmons, the Coulomb Interaction and a Gap

In a famous 1962 paper entitled Plasmons, Gauge Invariance and Mass (pdf!), P.W. Anderson described the relationship between the gap in the plasmon spectrum and the idea of spontaneous symmetry breaking. It is an interesting historical note that Higgs cites Anderson’s paper in his landmark paper concerning the Higgs mechanism.

While there are many different formulations, including Anderson’s, of why the plasmon is gapped at zero momentum in a 3D solid, they all rely on one crucial element: the long-range nature of the Coulomb interaction (i.e. the electrons are charged particles). Of these formulations, I prefer one “cartoon-y” explanation which captures the essential physics well.

Before continuing, let me stress that it is quite unusual for a fluid medium (such as the electrons in a metal) to possess no zero frequency excitations at long wavelengths. For instance,  the dispersion relation for surface gravity waves on water (pdf!) is:

$\omega^2(k)=gk \tanh kh$.

Now, in 3D and in the long-wavelength limit, the plasmon sets up opposite charges on the surfaces of the solid as pictured below:

The long-wavelength plasmon therefore sets up the same electric field as in a capacitor. The electric field for a capacitor is $\textbf{E} = \frac{\sigma\hat{x}}{\epsilon_0}$. This expression is surprisingly independent of the distance separating the surfaces of the solid. Therefore, it takes a finite amount of energy to set up this electric field, even in the limit of infinite distance. This finite energy results in the gapping of the plasmon.

This cartoon can be extended further to 2D and 1D solids. In the 2D case, the electric field for the 1D “lines of charge” bounding the solid falls off like $\textbf{E}\sim\frac{1}{\textbf{r}}$. Therefore, in the infinite distance limit, it takes no energy to create this electric field and the plasmon is not gapped at $\textbf{q}=0$. Similarly, for the 1D case, the electric field from the points bounding the solid falls of as $\frac{1}{\textbf{r}^2}$, and the plasmon is again gapless.

This reasoning can be applied further to the phenomenon known as LO-TO splitting in a polar solid. Here, the longitudinal optical phonon (LO) and the transverse optical phonon (TO) branches are non-degenerate down to the very lowest (but non-zero!) momenta. Group theory predicts these modes to be degenerate at $\textbf{q}=0$ for the zincblende crystal structure of typical semiconducting compounds. Below is the phonon dispersion for GaAs demonstrating this phenomenon:

Again, the splitting occurs due to the long-ranged nature of the Coulomb interaction. In this case, however, it is the polar ionic degree of freedom that sets up the electric field as opposed to the electronic degrees of freedom. Using the same reasoning as above, one would predict that the LO-TO splitting would disappear in the 2D limit, and a quick check in the literature suggests this to be the case as reported in this paper about mono-layer Boron Nitride.

I very much appreciate toy models such as this that give one enough physical intuition to be able to predict the outcome of an experiment. It has its (very obvious!) limitations, but is valuable nonetheless.


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# Why is the square root of 24 irrational?

Wiki User

2009-03-12 20:06:32

If it were rational, then the square root of 6 would be also since sqrt 24 is sqrt(4x6)=2xsqrt(6) and here is a proof the sqrt(6) is irrational So let's assume by contradiction that the square root of 6 is rational. By definition, that means there are two integers a and b with no common divisors where: a/b = square root of 6. So let's multiply both sides by themselves:

(a/b)(a/b) = (square root of 6)(square root of 6)

a2/b2 = 6

a2 = 6b2

But this last statement means the RHS (right hand side) is even, because it is a product of integers and one of those integers (at least) is even. So a2 must be even. But any odd number times itself is odd, so if a2 is even, then a is even. Since a is even, there is some integer c that is half of a, or in other words:

2c = a.

Now let's replace a with 2c:

a2 = 6b2

(2c)2 = (2)(3)b2

2c2 = 3b2

But now we can argue the same thing for b, because the LHS is even, so the RHS must be even and that means b is even. Now this is the contradiction: if a is even and b is even, then they have a common divisor (2). Then our initial assumption must be false, so the square root of 6 cannot be rational.

Wiki User

2009-03-12 20:06:32
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Mastodon on mathstodon.xyz is sporadically using a lot of CPU, causing intermittent outages. I have no idea why.

@christianp I don't know about this instance, but there have been at least one other that was doing crypto mining in the browser.

A Mastodon instance for maths people. The kind of people who make $\pi z^2 \times a$ jokes.

Use $ and $ for inline LaTeX, and $ and $ for display mode.


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# Newbie Question

This topic is 2764 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

## Recommended Posts

Anyways, I started work on a money planner program (in C++) and ran into a few (probably) common newbie problems.

First off, can cin read anything else besides an intiger? I want it to read doubles, by the way.

Secondly, can you make a user-made function return a double? If so, how is this done? I am making a function that calculates total gas money needed and then outputs a total which is then given to another function which adds up all the money totals (gas price, extra money for emergency gas and such, trip money (to buy souvenirs, food and stuff like that) and more things later on) and then finally gives its output to cout.

While on that last topic, can I round doubles to the nearest tenth? I don't want an unreadable and extremely long number thrown at the user. I was thinking of making another function to do this, but is there an easier way?

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cin can take any input you want. I suggest using cin.getline or cin.readline (I haven't coded in C++ in years now, so I might forget some names here and there.

Once you have your string, you can convert it to any data type you want.

A function can return any data type.

eg:
double ReturnTotal(const double input)
{
double Total = //whatever code;
}

You can use cout to format the output of your double.

http://www.cplusplus.com/reference/iostream/manipulators/setprecision/

cout << setprecision (2) << yourdoublevariable;

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cin can read doubles, as well as chars, strings, ints, and floats. All you need to do is cin >> myVariable; and have myVariable be the desired type (in this case just declare it as a double).

User made functions can return doubles. When you are writing the function, just use double as the return type:
double myFunction(){  //Some code  return myDoubleVariable;}

Should give you an idea about some of that, at the very least the decimal part:
Quote:
 A special "magic formula" for controlling how many decimal places are printed: cout.setf(ios::fixed); // specifies fixed point notation cout.setf(ios::showpoint); // so that decimal point will always be shown cout.precision(2); // sets floating point types to print to // 2 decimal places (or use your desired number)

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Quote:
 Original post by Daaarkcin can take any input you want. I suggest using cin.getline or cin.readline (I haven't coded in C++ in years now, so I might forget some names here and there.Once you have your string, you can convert it to any data type you want.A function can return any data type.eg:double ReturnTotal(const double input){ double Total = //whatever code; return Total;}You can use cout to format the output of your double.http://www.cplusplus.com/reference/iostream/manipulators/setprecision/cout << setprecision (2) << yourdoublevariable;

Thanks. Does cin.getline and cin.readline read strings too? I get errors when I try to read strings with just cin. Maybe I forgot to #include "string" (or "string.h", I'm not sure).

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What kind of errors? Post the compiler messages.

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Quote:
 Original post by DragonsouljWhat kind of errors? Post the compiler messages.

It was a while ago, so I forget now. I probably initialized the string as a integer.

Edit: Your link was broken. There was a quote at the end.

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Either way, that last link has some info on cin and cout as well as getline.

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Quote:
 Original post by SomeGuy49Thanks. Does cin.getline and cin.readline read strings too? I get errors when I try to read strings with just cin. Maybe I forgot to #include "string" (or "string.h", I'm not sure).

#include <string> <-- like that

Yes, getline reads strings. But you must #include <string> for getline to recognize strings.

With getline you capture the input into a string, and then you can do some sanity checking on the values to make sure you have good input before converting it into a double. But after reading your next posts, that might be a little advanced for you.

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Alright, this is the program, in all its glory. Is there any errors? It ran perfectly in my computer. Is there ways to make it simpler?

#include <iostream>
#include <string>
using namespace std;

double gascalc(double length, double mpg, double price)
{
double gasmoney;
//The following calculates money needed for gas.
gasmoney = length / mpg * price;
return gasmoney;
}

double maincalc(double gas, double extra, double spend)
{
double moneytotal;
moneytotal = gas + extra + spend;
return moneytotal;
}

int main()
{
cout << "(=)This is a money calculator for Canadian road trips(=)" << endl;
cout << "How long is the road trip (miles)?" << endl;
double milength;
cin >> milength;
cout << "What is the average (hwy/city average) MPG of the car?" << endl;
double MPG;
cin >> MPG;
cout << "What is the average Canadian gas price (in USD)?" << endl;
double gasprice;
cin >> gasprice;
cout << "How much money do you want to spend on things there (food included)?" << endl;
double spendmoney;
cin >> spendmoney;
//The following is a simple equation that finds how much money you should take
//as extra in case you want to buy more things than you originally thought.
double extramoney = gasprice / 2 + spendmoney * 1.2;
double gasmoney = gascalc(milength, MPG, gasprice);
double finaltotal = maincalc(gasmoney, extramoney, spendmoney);
cout << "It will cost: " << finaltotal << endl;
return 0;
}

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Looks good from here, but you will want to use the source tags next time, it will make it easier to read, as it preserves the indentation of the code.

The only thing I would do differently is declare the variables at the top of the function, and then space out sections of the code with blank lines just to break it up a bit.

You could eliminate a lot of lines just by skipping the variable assignments.

double moneytotal;
moneytotal = gas + extra + spend;
return moneytotal;

can be replaced with

return (gas + extra + spend);

#include <iostream>#include <string>using namespace std;double gascalc(double length, double mpg, double price){	return length / mpg * price;}double maincalc(double gas, double extra, double spend){	return gas + extra + spend;}int main(){	double milength;	double MPG;	double gasprice;	double spendmoney;	cout << "(=)This is a money calculator for Canadian road trips(=)" << endl;	cout << "How long is the road trip (miles)?" << endl;	cin >> milength;	cout << "What is the average (hwy/city average) MPG of the car?" << endl;	cin >> MPG;	cout << "What is the average Canadian gas price (in USD)?" << endl;	cin >> gasprice;	cout << "How much money do you want to spend on things there (food included)?" << endl;	cin >> spendmoney;	//The following is a simple equation that finds how much money you should take	//as extra in case you want to buy more things than you originally thought.	double extramoney = gasprice / 2 + spendmoney * 1.2;	double gasmoney = gascalc(milength, MPG, gasprice);	double finaltotal = maincalc(gasmoney, extramoney, spendmoney);	cout << "It will cost: " << finaltotal << endl;		return 0;}


text:
# Find relevant digit groupings

Recently, my reputation was 25,121. I noticed that each digit grouping (i.e. the numbers separated by commas) was a perfect square.

Your challenge is, given a non-negative integer N and a unary boolean Black Box Function f : Z*B , yield a truthy value if each value of f applied to the digit groupings of N is truthy, and falsey otherwise.

One can find the digit groupings by splitting the number into groups of 3, starting from the right side. The leftmost group may have 1, 2, or 3 digits. Some examples:

12398123  -> 12,398,123    (3 digit groupings)
10        -> 10            (1 digit grouping)
23045     -> 23,045        (2 digit groupings)
100000001 -> 100,000,001   (3 digit groupings)
1337      -> 1,337         (2 digit groupings)
0         -> 0             (1 digit grouping)

• This function can map to either booleans (e.g. true and false), 1s and 0s, or any truthy/falsey value. Please specify which format(s) are supported by your answer.
• You may take an integer as input, or an integer string (i.e. a string composed of digits).
• You may write a program or a function.
• When passing the digital groups to the function f, you should trim all unnecessary leading zeroes. E.g., f, when applied to N = 123,000 should be executed as f(123) and f(0).

## Test cases

Function notation is n -> f(n), e.g., n -> n == 0. All operators assume integer arithmetic. (E.g., sqrt(3) == 1)

function f
integer N
boolean result

n -> n == n
1230192
true

n -> n != n
42
false

n -> n > 400
420000
false

n -> n > 0
0
false

n -> n -> 0
1
true

n -> sqrt(n) ** 2 == n
25121
true

n -> sqrt(n) ** 2 == n
4101
false

n -> mod(n, 2) == 0
2902414
true

n -> n % 10 > max(digits(n / 10))
10239120
false

n -> n % 10 > max(digits(n / 10))
123456789
true

• If we are unable to take functions as arguments, are we allowed to assume that the function is defined as a variable and we reference that in our program? – caird coinheringaahing Mar 5 '18 at 19:19
• @cairdcoinheringaahing Please read the reference for Black box functions, specifically the references at the end of that post. To summarize, yes, you can, even if your language is capable of taking functions as arguments (afaict) – Conor O'Brien Mar 5 '18 at 19:23
• Can the input be negative? Zero? You talk about integers, but all the examples are positive. I'd also suggest including test cases where a grouping of 000 needs to be handled. – xnor Mar 5 '18 at 21:57
• @ConorO'Brien In that case you should add (n -> n > 0 applied to 0) to the test cases because most answers fail on it. – Asone Tuhid Mar 7 '18 at 8:53
• @EriktheOutgolfer They are [0]. – Conor O'Brien Mar 7 '18 at 14:35

# Jelly, 5 bytes

bȷÇ€Ạ

Try it online!

The command-line argument is the number. The line above the line this function resides in is the main line of the rest of the program, that is, the code that gets called for each of the groups. Be careful not to refer to the line bȷÇ€Ạ is in! The example used here is the 5th test case.

• I'm pretty sure that this is a failure when applying n -> n != 0 to 0 – Asone Tuhid Mar 7 '18 at 9:01
• @AsoneTuhid It's not; The number is 0, and its digit group list is [0], so then Ṇ gets mapped to each element (the single 0 here), turning the list into [1] and, since all of the elements of this list are truthy, 1 is returned. Note that if I had the digit group list be [] instead the result wouldn't change, as all of the elements of [] are truthy (vacuous truth). However, the result can be different for different programs, and the rules aren't exactly clear on this (asked OP). – Erik the Outgolfer Mar 7 '18 at 13:45
• Sorry then, I barely understand Jelly. Nice solution. – Asone Tuhid Mar 7 '18 at 13:47

## Brachylog, 8 bytes

ḃ₁₀₀₀↰₁ᵐ

Try it online!

The blackbox function goes on the second line (or the "Footer" on TIO) and the integer is read from STDIN. Prints true. or false. accordingly.

ḃ₁₀₀₀      Compute the base-1000 digits of the input.
↰₁ᵐ   Map the blackbox predicate over each digit. We don't care about the
result of the map, but the predicate must succeed for each digit,
otherwise the entire map fails.

# APL (Dyalog), 16 13 bytes

3 bytes saved thanks to @Adám

∧/⎕¨1e3⊥⍣¯1⊢⎕

Try it online!

How?

1e3⊥⍣¯1⊢⎕ - input the number and encode in base 1000

⎕¨ - input the function and apply on each

∧/ - reduce with logical and

• Save 3 bytes by going explicit: Try it online! – Adám Mar 6 '18 at 11:33
• @Adám thanks! I liked the alpha-alpha though... – Uriel Mar 6 '18 at 13:33
• correct me if i'm wrong but i think this is a failure – Asone Tuhid Mar 7 '18 at 8:45

# Python 2, 46 bytes

g=lambda f,n,k=1000:f(n%k)and(n<k or g(f,n/k))

Try it online!

• 44 bytes by using * in place of and. – Bubbler Mar 14 '18 at 7:17

f#n=f(mod n 1000)&&(n<1||f#div n 1000)

The blackbox function must return True or False.

Try it online!

Edit: -2 -4 bytes thanks to @ovs.

# C (gcc), 6658 48 bytes

-10 bytes thanks to @Neil!

t=1000;g(f,i)int f();{i=f(i%t)&(i<t||g(f,i/t));}

Try it online!

## Wolfram Language (Mathematica), 30 bytes

And@@#/@#2~IntegerDigits~1000&

Try it online!

## JavaScript (ES6), 40 36 bytes

f=>g=i=>f(i%1e3)&(i<1e3||g(i/1e3|0))

Takes the function and value by currying and returns 0 or 1. Edit: Saved 4 bytes thanks to @Shaggy.

• 1000 -> 1e3 to save a couple of bytes. And could you replace && with & for another byte? – Shaggy Mar 6 '18 at 15:32
• @Shaggy Yes, I think that's safe enough. Same goes for betseg's answer? – Neil Mar 6 '18 at 16:29
• fails for function_name(n=>n>0)(0) (returns true) – Asone Tuhid Mar 7 '18 at 8:38
• @AsoneTuhid Thanks, fixed. – Neil Mar 7 '18 at 10:43

# Pyth, 9 bytes

.AyMjQ^T3

Try it online! (uses the third test case)

Assumes the black-box function is named y. You can declare such a function using L (argument: b), as shown on TIO. I will implement all the test cases later, if I have time.

# Stax, 8 bytes

Vk|Eym|A

Stax programs do not have function calls or arguments, so we store a block in the Y register that consumes and produces a single value. This can be done before the program code.

{...}Yd     store a block in the Y register that executes ...
Vk|E        get "digits" of input using base 1000
ym      map "digits" to array using y as mapping function
|A    all elements are truthy?

Here's an example using the perfect square function.

# Clean, 54 bytes

import StdEnv
$n=if(n<1)True($(n/1000))&&f(n rem 1000)

Try it online!

Defines the function $:: Int -> Bool, expecting a function f :: Int -> Bool to be defined elsewhere. # Java (OpenJDK 9), 94 bytes f->n->{int r=0,l=n.length();for(;l>0;l-=3)r+=f.test(n.substring(l<4?0:l-3,l))?0:1;return r<1;}  Try it online! # Common Lisp, 73 72 bytes (defun g(x f)(and(funcall f(mod x 1000))(or(< x 1e3)(g(floor x 1e3)f))))  Try it online! • This is my first Lisp answer, so it's probably terrible. – Dennis Mar 10 '18 at 18:36 # 05AB1E, 8 bytes ₄вεI.V}P  Try it online! Explanation ₄в # convert first input to base-1000 ε } # apply to each element I.V # execute second input as code P # product of the resulting list  Takes the number as the first line of input and the function as the second. Outputs 1 for truthy and 0 for falsy. • This doesn't execute the code on each element, but on the whole list. – Erik the Outgolfer Mar 6 '18 at 11:48 • @EriktheOutgolfer: With 05AB1E's automatic vectorization it will in most cases. I just realized that it won't work for Qand Ê though. I'll revert back to the 8-byte version. – Emigna Mar 6 '18 at 11:50 • Still, it's not .V that vectorizes, it doesn't even take the list as an argument itself to begin with. – Erik the Outgolfer Mar 6 '18 at 11:53 • @EriktheOutgolfer: I've never said that .V vectorizes. In the example in my link it's È. – Emigna Mar 6 '18 at 11:55 • Actually Q and Ê would work with vectorization unlike I previously stated, but using automatic vectorization would make these commands map on the whole list which feels outside the spirit of the challenge so we need ε. – Emigna Mar 6 '18 at 12:11 # Excel VBA, 79 bytes An anonymous VBE immediate window function that takes input, n as type integer from range [A1], and the name of a publically defined VBA function from range [B1]. t=1:n=[A1]:While n:t=t*-Application.Run(""&[B1],n Mod 1E3):n=Int(n/1E3):Wend:?t  ### Usage example In a public module, the input function, in this case f() is defined. Public Function f(ByVal n As Integer) As Boolean Let f = (n Mod 2 = 0) End Function  The input variables are set. [A1]=2902414 '' Input Integer [B1]="f" '' input function  The immediate window function is then called. t=1:n=[A1]:While n:t=t*-Application.Run(""&[B1],n Mod 1E3):n=Int(n/1E3):Wend:?t 1 '' Function output (truthy)  # Ruby, 37 bytes g=->f,n{f[n%x=1000]&&(n<x||g[f,n/x])}  Try it online! A recursive lambda, taking function and integer and returning boolean. ## 36 bytes (only positive n) g=->f,n{n>0?f[n%k=1000]&&g[f,n/k]:1}  This version returns 1 for truthy, false for falsey. Unfortunately it can fail when n = 0 Try it online! • I think you have to count g= if it's recursive – Asone Tuhid Mar 5 '18 at 23:53 • @AsoneTuhid Oh, that makes sense. I'll add it in. – benj2240 Mar 6 '18 at 0:06 • Also, try this (-1 byte), it returns 1 as the truthy value – Asone Tuhid Mar 6 '18 at 0:11 • That wrinkled my brain a little... I had to tinker around a bit to convince myself it worked in all cases. Thanks! – benj2240 Mar 6 '18 at 19:29 • I was wrong, this version doesn't work for g[->n{n>0},0] (returns true). It only fails if the input is 0 but the question says "non-negative" so you should go back to 37. sorry – Asone Tuhid Mar 7 '18 at 8:37 # Appleseed, 51 bytes (lambda(n f)(all(map f(or(to-base 1000 n)(q(0))))))  Anonymous lambda function that takes a number and a function and returns a boolean value. Try it online! (lambda (n f) ; Function with parameters n and f (all ; Return true if all elements of this list are truthy: (map f ; Map the function f to each element of (or ; This list if it is nonempty: (to-base 1000 n) ; Convert n to a list of "digits" in base 1000 (q (0)) ; Or if that list is empty (when n=0), then use the list (0) instead ))))  # Add++, 15 bytes L,1000$bbbUª{f}

Try it online!

Requires a function f to be declared in the TIO header.

## How it works

D,f,@,0.5^i2^A=	; Declares a function 'f' to check if a perfect square
; E.g. 25 -> 1; 26 -> 0

L,		; Declare the main lambda function
; Example argument: 		[25121]
1000\$bb	; Convert to base 1000	STACK = [[25 121]]
bUª{f}	; Is 'f' true for all?	STACK = [1]

# Husk, 7 bytes

ΛmrC_3

Try it online! Takes N as a string and f as a function object. Given a number, f can return any truthy or falsy value.

# Japt, 9 bytes

Had to sacrifice 3 bytes and take input as a string in order to handle the n=>n>0 test case when n=0. I could get them back if it was permitted to pass the groupings to the function as strings and convert them to integers within the function, when necessary

ò3n)mn eV

Try it

The function, as allowed by consensus, is preassigned to variable V on the second line of the header - the empty first line is essential.

Here are all the test case functions rewritten for Japt.

n -> n == n                           X{X==X}
n -> n != n                           X{X!=X}
n -> n > 400                          X{X>400}
n -> n > 0                            X{X>0}
n -> n -> 0                           X{X{0}}
n -> sqrt(n) ** 2 == n                X{Xq v1}
n -> mod(n, 2) == 0                   X{X%2==0}
n -> n % 10 > max(digits(n / 10))     X{X%10>Xz10 ì rw}
`


text:
## Description

Unique Binary Search Trees

Given n, how many structurally unique BST's (binary search trees) that store values 1 … n?

Example:

Input: 3

Output: 5

Explanation: Given n = 3, there are a total of 5 unique BST's:

   1         3     3      2      1
\       /     /      / \      \
3     2     1      1   3      2
/     /       \                 \
2     1         2                 3

## Solution

n = 3为例，有数字1、2、3；设解为F(3)=f(1~3)

F(3)分解：

1. 如果以1root结点，则右边2~3F(2)=f(2~3)个解
2. 如果以2root，则左边有F(1)=f(1~1)、右边有F(1)=f(3~3)；总F(1)*F(1)
3. 如果以3root，则左边有F(2)=f(1~2)

$$F(n) = \sum_{i=0}^{n}F(i) * F(n-i-1) ; F(0) = 1;$$

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17  class Solution { public: int numTrees(int n) { int *dp = new int[n+1]{1, 1}; // {F(0), F(1)} for (int i = 2; i <= n; ++i) { dp[i] = 0; for (int k = 0; k < i; ++k) { dp[i] += (dp[k] * dp[i - k - 1]); } } int r = dp[n]; delete []dp; return r; } };


text:
11/19/19

09/11/19

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An item sells for $80 and is reduced to sell for $50. Find the markdown amount and the rate of markdown.

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#### What is the entry of it?

Bought merchandise on account 1000 and paid freight of 500

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#### How do I add common stocks to the cash balance on a balance sheet?

I am currently working on my Accounting homework and I cannot figure out this answer. I have everything figured out except for the correct cash balance on the balance sheet. A side not says that... more

08/20/19

#### why a sole trader might want to become a partnership

why a sole trader might want to become a partnership- Benefits Disadvantages Conclusion

#### Which Version of Microsoft Word created a given document?

I'm starting to get multiple MS Word documents sent to me, and I'd like to detect what version of MS word created, or last saved, each doc. Is there some tool for that? Or, can I determine such... more

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When Christopher Columbus was 41 yrs. old he sailed the ocean. His final expedition 10 years later ,which took 2 yrs.. He died 2 years after his final expedition ended. What year was Christopher born? more

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I am struggling to find what exactly is wrong about these journal entries? All the other examples I've looked up look the exact same. Please advise! Thanks.      Assume that you are the chief... more

## Still looking for help? Get the right answer, fast.

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Most questions answered within 4 hours.

#### OR

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text:
## Monday, July 27, 2009 ... //

### Don Page: Born Again

I am convinced that the first paper on the hep-th archive has a higher average value than the average paper. Physicists often know that they have found something pretty and they want others to see it. Because most of the people who submit papers to the arXiv are pretty bright, their selection is usually wise, too.

Unfortunately, on some days, this mechanism works in reverse. Don Page wrote two new papers and I will look at both of them. The first of them is called

Born Again.
The title is clever: it means that the paper is another attack on Max Born's interpretation of quantum mechanics. It also subtly invites you to join Page's church by becoming a born-again Christian - and it may stimulate your thinking about reincarnation. ;-)

Unfortunately, the title is the last clever part of the paper. Pretty much every paragraph of the paper contains fundamental misconceptions about the character of physical law and the very nature of rational thinking. The anthropic considerations are far from being the only misunderstood aspects of reality in this paper.

Parts of quantum theories

The first two paragraphs contain a summary of the wrong paper - he plans to show that Born's interpretation of quantum mechanical amplitudes can't apply to cosmology - and we will discuss the points in detail. The second section sketches how the author imagines the inner structure of a physical theory. In his viewpoint, it is made out of a whopping number of six qualitatively distinct components:
1. Kinematic variables
2. Dynamical laws
3. Boundary conditions
4. Specification of what has probabilities
5. Probability rules
6. Specification of what the probabilities mean
You should be shocked by such a huge number of components because in normal physics, a quantum mechanical theory is said to be composed out of two components only:
1. The logical, interpretational framework of quantum mechanics
2. The dynamical laws
How could he have possibly obtained six instead of two? You might think that it's just some philosophy because one can't empirically answer the question "How many parts does a physical theory have?". However, there is a reason why this philosophy matters: the confused, fragmented, redundant picture of physics that Don Page keeps in mind is pretty much the primary source of all of his confusions that are expanded later in the paper. All the aspects of physics simply don't fit together in his mind which is why he thinks that he can combine them, xerox them, modify them, and recombine them in arbitrary ways.

So why are there two components only and what do they mean? The first component if universal for all quantum mechanical theories: it contains the general postulates of quantum mechanics.

It tells us that the (squared) complex amplitudes determine the probabilities that can be measured by the repetitions of the same experiment, whenever it's possible; it tells us that Yes/No questions are given by projection operations; their expectation values determine the probabilities; other observables are given by Hermitian linear operators; the evolution is given by a Hermitian Hamiltonian or a unitary evolution operator (or S-matrix), and a few others. These rules are completely general for all quantum mechanical theories and I formulated them in such a way that a sensible physicist should accept them regardless of her favorite interpretation of quantum mechanics. However, for the sake of clarity, I will assume that the reader uses the Consistent Histories, just like I do, and the rules of this most sensible logical framework for quantum mechanics are fully included in the first category.

On the other hand, the Hermiticity or unitarity rules that I previously included in the same group are not independent assumptions: they may be viewed as constraints on the types of Hamiltonians or evolution operators that are allowed in the second category. So once you define a proper Hermitian Hamiltonian in the following paragraph, you may forget about them.

The second category contains the actual formula for the Hamiltonian (or the S-matrix). This part contains everything that is specific for a given quantum mechanical theory. It also automatically contains all the nontrivial mathematics underlying your quantum theory, all the information about the "kinematics" or sensible degrees of freedom, all the information about the degrees of freedom that like to behave classically (because the Hamiltonian is enough to calculate the rate of decoherence in any situation). In the case of quantum field theories, this second category knows everything about the spacetime dimensionality, the particle species, and their interactions. All these things can be deduced from the definition of the Hamiltonian.

So why did Mr Page obtain six separable components?

It may be useful to describe Page's four redundant points one by one. Let's identify his points (5) and (2) with the correct categories of a quantum theory, the logical framework and the dynamical laws. What about the other four?

First, it is not true that "kinematic variables" have to be the starting point for any quantum theory. And even when they exist, it's not true that they're independent from the dynamical laws, either. What do I mean? Well, in simple quantum mechanical models from the undergraduate textbooks, quantum mechanical theories are obtained by the "quantization" of a classical starting point. But that's not a general way how quantum mechanical theories arise.

Such a construction of a quantum mechanical theory is only meaningful if the theory has a classical limit and if this limit is useful to reconstruct the whole quantum theory, even outside the limit. But it's not true that every quantum mechanical theory must have such a classical limit. For example, the (2,0) superconformal quantum field theory in 5+1 dimensions has no dimensionless coupling that can be made small. So all the dynamical questions about this theory are inherently quantum mechanical. There is no classical Lagrangian - at least not a manifestly Lorentz-covariant one - whose quantization would generate this quantum theory. The (2,0) theory is surely not the only example.

What is actually needed for a quantum theory is a Hilbert space. But there's no freedom to choose an infinite-dimensional Hilbert space because all infinite-dimensional Hilbert spaces are unitarily equivalent to each other. The only general constraint is that there is a space of states, and this rule clearly belongs to the logical framework of quantum mechanics: it's a postulate. Claims that quantum mechanical theories have to have "classical degrees of freedom" whose quantization produces the quantum theory are artifacts of a blind acceptance of some features of simple textbook examples that are in no way general. There doesn't have to be any privileged or well-defined way to parameterize the Hilbert space by eigenvalues of a fixed collection of observables. A Hamiltonian (or an evolution operator) acting on a Hilbert space is enough.

The second point is even more important as a source of Page's confusion. Imagine that we deal with a theory that is obtained by the quantization of a classical theory. Page thinks that the definition of some kinematic variables is an independent piece of information that is needed to get the complete theory. But that's wrong, too. And this statement is actually wrong for all quantum mechanical theories.

Choose an arbitrary basis of the Hilbert space and define the dynamical laws using a Hamiltonian written as an infinite matrix. Together with the completely general quantum mechanical postulates, you already have everything you need to make any predictions or retrodictions. Isn't it obvious? You may evolve states and calculate the amplitudes i.e. probabilities for any projection operators. You must also have some good questions - good projection operators - but you're free to choose your favorite ones. However, you will be highly restricted by the logical framework of quantum mechanics - by the consistency of your histories, in our case - and decoherence will de facto forbid most questions that you could a priori ask.

But do we need to specifically define the number of dimensions, the fact that particles can have three-dimensional positions, and so on? Not at all. All these things are aspects that can be deduced from the Hamiltonian. For example, the most interesting Hamiltonians - especially those that are consistent with special relativity - enjoy a degree of locality that is often exact. But this locality can be derived from the Hamiltonian, too. Once you derive it, you also know the dimension of the spacetime in which the laws are local. Why? Local Hamiltonians have the property that you can write them (approximately or exactly) as sums of commuting smaller Hamiltonians associated with different, non-overlapping regions.

The Hamiltonian is enough to calculate the rate of decoherence in all situations, too.

The misunderstanding of the fact that the position is just another observable in quantum mechanics and that its "local properties" are emerging from the Hamiltonian, if you wish, and they can't be separately incorporated into the theory according to your own choices, is arguably the main source of Page's confusion about "regions" and his bizarre conclusion that the "regions" make Born's interpretation of the amplitudes invalid.

Summarizing the redundant points

There are only two parts of a quantum mechanical theory while Page proposed six of them. One of them, the (1) kinematic variables, shouldn't have been included because of the reasons we have discussed in detail.

Another one, the (3) boundary conditions, shouldn't have been included because the boundary conditions are either a part of the dynamical laws and allow us to predict something, or they're not a part of the theory at all.

For example, if the initial conditions are given by the Hartle-Hawking state or its generalization, this state is described by the same path integral that is also used for the evolution. It's just the same path integral applied to a different topology of spacetime (one that has no non-vanishing initial slice). It makes no sense to discuss it separately from the dynamical laws.

On the other hand, if the boundary conditions are left arbitrary by the physical theory, they're not a part of the physical theory. One always needs to know the boundary and especially initial conditions to get some answers. But the boundary conditions are not a part of the theory but a part of the question that we are answering using the theory. They are parts of the application of the physical theory but not parts of the theory itself. In the same way, we also didn't include the salary for the physicists as a separate category because while it may be a part of the solution to a problem, it is surely not a part of the theory. ;-)

A theory must define its set of concepts and the legitimate rules how and which questions can be formulated and how answers may look like (the set of allowed questions and answers may be determined by a complicated recipe that needs a lot of calculations!). But it doesn't and cannot really tell us which questions should be asked at a given moment.

The remaining two redundant points are (4) and (6) which are just some confused questions that should have been included into (5), the logical framework of quantum mechanics.

Page asks what has probabilities: any proposition (i.e. any projection operator) has probabilities. And he asks what the probabilities mean: well, they mean "N_k/N" where "N" is the number of repetitions of the same experiment and "N_k" is the number of the repetitions which have some property "k" - whose probability we're talking about.

If the same situation is repeated many times, they can be understood in the frequentist fashion. If the situations can't be repeated many times, probabilities become largely "empirically meaningless" and "unmeasurable" but they can still sometimes be predicted by complete theories. Every scientist or mathematician should know these elementary things and they shouldn't be claimed to be "on par" with the whole Hamiltonian or with the logical framework of quantum mechanics which are much more nontrivial beasts.

The second section ends up with another paragraph that I find very controversial, to put it mildly.

Page says that "a goal of science" is to produce complete theories labeled by "i" that predict probabilities of different observations "j", namely "P_j (i)". I don't think that a goal of science is to have many theories. A goal of science is to find out which hypothesis is actually correct, by eliminating all the other, wrong (falsified) ones and by refining the viable (temporarily correct) one(s): Page's "goal" is just an intermediate step in a typical scientific process. And it's questionable whether he should be using the word "theory" before science determines which of the mutually exclusive hypotheses (would-be theories) is actually right.

Are there problems with Born's rule?

By Born's rule, we will mean the rule that the probability of "j" is the expectation value of the projection operator "P_j". Both of these objects are associated with a given theory, "i", which is another index identifying these objects.

Page thinks that this rule fails in a large universe that may have many copies of the same observer. Why? Because he thinks that by not knowing his location, the observer either can't define the projection operator or can't trust the rule. Page is very vague about his precise statement. At any rate, it is completely manifest that the following paragraphs don't contain any evidence supporting his statement.

He correctly notices that mutually exclusive projection operators satisfy
Pj(L) Pk(L) = delta jk Pj(L)
i.e. that the product vanishes for two different projection operators and each of them squares to itself. But quite suddenly, he says that the product
Pj(L) Pk(M)
doesn't have to be zero for two different regions L,M. Well, indeed, it doesn't have to be zero. And what?

If you restrict the projection operators "P_j(L)" to act on the smaller Hilbert space associated with the region "L" only (which is enough to discuss all experiments in "L"), and similarly for "M", then the product is not even well-defined because the domain of the left operator doesn't coincide with the range of the right operator. In other words, the indices don't match: Page is summing over a pair of indices that are made equal even though they can't really be equal since their types differ.

This means that a consistency check has failed. Objects such as "delta_{jk}" are only well-defined if "j,k" are indices of the same type. But if "j,k" parameterize basis vectors of two Hilbert spaces describing two different regions "L,M", then no "delta_{jk}" exists because the bases of these two Hilbert spaces are really different. The Kronecker symbol is ill-defined and it cannot appear in any error-free calculation. If such a meaningless object appears, it doesn't prove an error in Born's basic postulates of quantum mechanics but rather an error in the paper where this inconsistent product appreared - e.g. in Page's paper.

If the observer has identical friends in other regions, there are several issues to be addressed. First, the very question "which of them he is" is largely unphysical for him. He is one of them, and as long as the other copies are distant and the laws are local, the other copies can't even influence any of his physical measurements (and therefore, they can't influence any of his correct predictions, either). Quantum mechanically, fully identical particles are really identical and it makes no sense to ask which is which, not even in principle: the only way to distinguish them is by a different eigenvalue of an operator (e.g. location).

Second, and this point is even more important, the possible existence of other copies of the observer shouldn't change anything about the observer's will to avoid elementary childish errors in his considerations. When he investigates the causal relationships between any pair of observations or maneuvers in his lab, he must always work with operators that act on the same Hilbert space - the Hilbert space describing his lab. Whenever he multiplies two operators that act on different spaces - e.g. Hilbert spaces associated with different regions - he knows that he has made a mistake.

The observer may try to confuse himself by saying that he may be the observer in region "1945" or "2009" or whatever (such a simple numerical identification of the copies is meaningless and impossible, anyway). But he must never get confused enough to apply Einstein's summation rule to a pair of indices of a different type.

This is such a trivial and sharp point that I am simply flabbergasted that someone can fail to get it - and create fog about it - for a few years. Let me repeat the trivial point again.

The possible existence of additional copies of an observer, whether they exist or not, is completely inconsequential for the correct predictions of probabilities of any outcome of any measurement in his "region" as long as these predictions are made correctly. The basic logical framework of physics combined with (at least approximate) locality guarantees that it is the case, and any hypothesis or a "system of reasoning" that denies locality at this very basic level is instantly ruled out. Why do we have to read this nonsense on hep-th for years?

The mathematical expressions that appear in the last paragraphs of the third section are uniformly wrong, too. Page tries to propose his own Born's rule where the probability is given by
Pj(i) = < I - ProductL (I - PjL) >.
Jesus Christ, this is formally some probability that "j" occurs at least in one region. That's why he needs the absurd product over the regions. But such a product is completely unphysical. One can't ever make a measurement that correlates all copies of an observer, especially not those who will be born in the future or spatially separated regions.

If an observer prepares a neutron and tries to find the probability that the outgoing electron's helicity is right-handed, either experimentally or theoretically, he is interested in the final state of particles in the same region where he defined the initial state, too. It doesn't matter a single bit whether there are similar or identical regions somewhere, how many of them there are, and what the other observers are doing. The local character of the laws of physics guarantees that.

No dynamics - or projection operators - from the other regions and copies can influence the observed result in a particular experiment in a particular region, whether it is "known" or "unknown". That's also why all these operators from wrong regions have to drop out of all the correct theories and calculations of the probabilities. It's a trivial consistency check on any calculation.

In the text above, I assumed that the observer had to define something about the initial conditions, for example that he had a neutron. The location where this neutron was assumed to exist automatically "marked" the correct region where the final probabilities are predicted, too. You could think that there could exist answers (e.g. about the probabilities of a spin, or the "probability of life") that don't depend on the question (including some knowledge about the initial conditions) but I assure you that you won't find any such scientific answers that have no questions, perhaps except for 42. ;-)

This statement should be obvious in the case of ordinary "laboratory experiments", such as the measurement of a spin of a decay product. The initial conditions are as important to be specified as the final state whose probability is to be calculated. The initial state is associated with the degrees of freedom in a particular region, and the dynamical laws can only relate this information to the predictions of observables in the same region. The information from other regions can never enter the discussion at all.

But even if you ask the "more grandiose" questions, such as "what is the probability of life?", you will find out that there's no science without questions. First of all, the exact answer to this question is known empirically: the answer is 100%. Life surely exists. Any viable theory must be compatible with the fact that life exists.

But that's it. If a hypothesis predicts that life in a given world emerges with a certain probability between 0% and 100%, you must be extremely careful whether such a prediction is scientifically justified, measurable, testable, and well-defined, and whether the figure changes the likelihood that the hypothesis is valid (and in which way).

Once again, the experimentally measured number is 100%. The life surely exists. But this result may have been an artifact of accidents in the history of the Cosmos and the Earth, not a direct consequence of the laws of physics. That's true but what is not true is that a greater probability that life exists somewhere is enough to locate more likely theories.

Such an opinion is just an artifact of a biased attitude of people who want to see that "things exist". But in our Universe and any other world, it is equally important that many other things don't exist. The anthropic and Boltzmann-brain charlatans are often tempted to say that a theory with a lot of stuff - many copies of everything etc. - is more likely because it produces a lot of copies of things that may evolve into life (or your brain) and other effects we observe, and they think that this feature increases the probability that these scenarios are valid.

What they're forgetting is that their frameworks, because of their having a lot of stuff, also include a huge amount of things that we don't observe, which lowers their likelihood to be correct. In fact, a typical theory with Boltzmann brains or supervast multiverses predicts many more things that aren't there, according to observations, than those that are there. And you know, a correct theory must predict not only that a chicken is born out of an egg at least somewhere in the multiverse. It must also predict that a car is not born out of this egg, unless it is a Kinder surprise egg. ;-)

In the discussion about the Standard Model vs Minimal Supersymmetric Standard Model likelihood, I've explained that one must be very careful about philosophical arguments that have a different character in both models because they could apparently "punish" one of the models hugely, relatively to the other model, but the resulting probabilities would depend on your choice of the "selection criteria". So these criteria are very dangerous and one should prefer experiments where both models predict statements of the same kind that may be compared (such as the question whether the W boson is heavier than a certain bound). In other words, one needs to choose "fair contests" to compare two models.

The same comment applies to the Boltzmann brain and anthropic considerations, but it becomes even more pressing. The anthropic and the Boltzmann brain people never resist the temptation to believe that a model with a lot of copies of something is more likely because it predicts a lot of copies of the good stuff. But they always forget that it predicts a lot of copies of the bad stuff, too.

In the Boltzmann brain case, the number of the bad (unobserved) things predicted by the crazy hypothesis is exponentially higher than the number of good (observed) things. If we're just a fluctuation, the predicted probability that our neighborhood is chaotic and manifestly follows no "grand set of rules" is at least "exp(S)" times greater than the probability that we would observe an apparent order in another experiment. Any kind of such a Boltzmann brain hypothesis is falsified by pretty much every single observation that we ever make.

The Boltzmann brain hypothesis is ludicrous and can't follow from any serious working theory created by mature physicists. A "fair contest" actually exists: simply use your theory to predict whether your next observation will be consistent with a long-term order of the Cosmos. The sane theories will predict "yes" while the Boltzmann brain theory will predict "almost certainly no": and it is therefore instantly falsified. But even many of the multiverse research directions fall into the same trap. Very huge bubbling multiverses can produce many planets where our life has a chance to appear. And their proponents think that a lot of "volume" or "Lebensraum" for intelligent beings is a good feature. But these models also produce a lot of celestial bodies where wrong particles live and wrong phenomena take place.

The latter objects reduce the probability that the scenario is valid, as long as you are at least a little bit balanced about the "positive" and "negative" observations and their influence on your priors. When you summarize these observations, it is very clear that you can't possibly obtain a more likely theory just by looking at theories with "more stuff" in them. The right theories (and vacua) should have the "right stuff", not "more stuff".

Fourth section

Let's return to Page's article. It gets even more crazy. He starts the section with
Probability Symmetry Principle (PSP): If the quantum state is an eigenstate of equal number of observations of two different observations, then the probabilities of these two observations are equal.
The repetition of the word "observation" suggests a typo. But let's analyze this statement as it appears above. Imagine that you measure the spin of the same one electron N times and you obtain "up" N1 times and "down" N2 times.

The assumption of the "principle" is that a quantum state is the eigenstate of N1 and N2. If either N1 or N2 is equal to zero, it can be because the probability to get either "up" or "down" is zero. But let's assume that both N1,N2 are nonzero.

If a quantum state is an eigenstate of N1, it has a predetermined number of repetitions when it will be spinning "up". How is it possible? One possible explanation is that each measurement is separately predicted to be "up" or "down" with the probability 100%. In that case, everything is known for certain and you shouldn't be talking about probabilities.

Probabilities only refer to the infinite repetition of the same situation. And in the (N+1)-th measurement and the following ones, you won't know whether the measurement will be like those "up" measurements, or your "down" measurements, so you can't possibly say anything about the probability.

On the other hand, the results of individual measurements may be undetermined by the quantum state, but the state is such that the total number of "up" results, N1, is determined. This can only be the case if there exist sharp correlations - i.e. entanglement - between the individual measurements in the ensemble of N repetitions. There are "quota" that tell you that if you get "up" twice, you must get "down" for the third time, and so on.

Such correlations between the individual experiments mean that these "repetitions" are not really independent, so you're not allowed to assume that these experiments are repetitions of the "same" experiment. They're not the same experiment because some of them are correlated with some events in the past while others have a correlation with other events in the future. That's why you're not allowed to talk about probabilities of particular things, either.

At any rate, the "Probability Symmetry Principle" is complete nonsense because it uses the word "probability" for some random ratios that don't satisfy the basic defining feature of a "probability", namely the repeated existence of the same situation. The statement is meaningless - and of course, whether such a meaningless statement would be true or false would be an entirely different question. If you give it the same meaning as Page does, it's wrong, as we will see.

The second principle offered in the section is the "Prior Rule Principle" that requires that the projection operators to calculate probabilities don't depend on the state. That's a bizarre statement because if you want to find out the appropriate mathematical expression for a projection operator that encodes a given property, you need to know the Hilbert space, its physics and the links between the observables and the real observations quite well. For example, the property that "an electron is found in the micrometer vicinity of a proton" is expressed by very differently looking operators in non-relativistic quantum mechanics and in the Standard Model.

So both of these "principles" are silly but what Page is doing with them is even more stupid. Indeed, he considers an entangled state "psi", being equal to "b12 ket12 + b21 ket21". The measurements in regions 1,2 generate results 1,2 or 2,1 - they're anticorrelated. Ordinary entanglement.

But the shocker comes when he uses his "Probability Symmetry Principle" to argue that all theories are always required to predict the same probability for both options allowed above, "12" and "21". Does he really want to claim that the absolute value of both "b12" and "b21" must be "1/sqrt(2)"? That would be a really bad joke. A basic postulate of quantum mechanics is the superposition principle that guarantees that all linear combinations of states are allowed. So if a principle forbids states unless they satisfy a nonlinear constraint (involving absolute values of the amplitudes), it's surely a wrong principle.

I've tried hard to determine how he got to his weird conclusions, and my verdict is that he misunderstands very basic things about the way how probabilities are calculated from wave functions and from density matrices. Quantum mechanics surely allows entangled states with different absolute values of the amplitudes. If a physicist doesn't know the wave function, but only knows that it is one of many orthogonal wave functions that have different probabilities, he calculates the result for each of them and takes the weighted average. That's what the density matrix calculus gives him, too. This is no rocket science.

You can see very easily and explicitly why the "Probability Symmetry Principle" is completely wrong. Imagine that a theory predicts that there are two planets that are separated from one another. Each planet contains one observer - otherwise identical to his counterpart. Both of them are going to measure the spin of an electron. Imagine that the theory predicts that the Universe prior to the measurements must be found in the pure state,
psi = 0.6 |12> + 0.8 i |21>
I only added the "i" to remind the readers that the phases can be arbitrary and that quantum amplitudes are complex. Now, this state clearly predicts that there is a 36% probability that the results "12" will be measured by the two observers but 64% probability that it will be "21".

If an observer knows that he is the first one, he has 36% to get "1". If he knows he is the second one, he has 64% to get "1". If he doesn't know which one he is, then indeed, he can't say what the probability is. If he has a good reason to think that he is a random one among the two, with a uniform probability distribution, his probability to get "1" is (36+64)/2 = 50%: he uses the "maximum ignorance" density matrix. But there's nothing preferred about this "uniform distribution".

If he had a dynamical theory that would show that his ancestors randomly chose the planet, or something like that, he would know that the 50%:50% expectation would be the most sensible one. But if he has no such a picture of his history (and such a picture can't even exist if the number of copies is infinite because there's no uniform distribution on infinite sets), the 50%:50% assumption is as good or as bad as any other one. He may also have a different theory that implies that his planet is the planet where the results "1" are pretty often. Dividing the odds equally among possibilities can't ever reduce your actual ignorance. You can't get any closer to the right length of the emperor's nose by averaging over the opinions of millions of people who haven't seen him.

Measurements and theories can sometimes strictly tell you what the right answer is: but if you don't have the sufficient information about your initial or current state, scientific theories simply can't tell you the "only correct" probabilities that you are obliged to assume and especially not the "only correct" probabilities that you are someone rather than someone else. If one is ignorant about his past or his identity, he is really ignorant and tricks and random answers such as "uniform probabilities" don't and can't make him any less ignorant!

At any rate, it's just not true that every theory predicts that the probability of his getting "1" must be 50%. In other words, it's not true that the dumbest egalitarian version of the anthropic principle must be a part of every theory in physics. In fact, it is a part of no sensible theory in physics, except for the theories where the uniform distribution can actually be derived (e.g. for microstates in thermal equilibrium according to statistical physics).

The rest of the paper is dedicated to replacing Born's rule with a new "principle" - one that completely denies the local character of physics and leads you to predict things according to various absurd considerations about the unphysical, unmeasurable copies of yourself that can't possibly influence you - and that can't possibly influence any properly calculated probability in your lab or planet, either.

The paper is just complete bullshit, and there are literally hundreds of such nonsensical papers in the arXiv already. Another one is the very next hep-th paper by the same author that predicts universal cosmic doomsday within 20 billion years.

This may have something to do with some unscientific religious craziness that the author believes. But I just can't understand how someone can seriously believe such things. What happens in 20 billion years will depend on the current state of the Universe and the physical laws that will evolve it.

The world is not obliged to follow some simple predetermined dogmas of evolution. The character of our world and the odds of one fate or another were decided in the past and they were not decided in the future, or according to the future. The future is whatever evolves from the past and the present. Up to the dynamical laws that connect it with the present, the future is free.

The world has no simple purpose. It has no goal of creating the nicest beings, or to kill them in the minimal possible time, or to allow them to have the minimal or maximum number of descendants. Nothing like that (unless a similar mechanism is actually demonstrated by some independent rational arguments and evidence). The world evolves as its pretty complex dynamical laws, combined with the known complex information about the initial or current state, demand. The result is complicated, too.

The basic fallacy is the same one as in the previous paper. Page thinks that predictions of outcomes in physics depend on the copies of yourself, including the copies that will live in the future bilions of years. But the reality is very different. It's not only true that the correctly calculated predictions don't have to depend on the other copies: in fact, they are not even allowed to depend on any features of the copies, or their very existence.

For spatially separated copies of yourself, this condition follows from locality. For the copies of yourself that will live in the future, the condition is even more strict and follows from causality. Your current evolution simply can't be affected to the fate of some possible humans who may exist in billions of years in the future. If someone uses any infinite products over his copies or some averaging over the future (that is unknown) to make any predictions about the world where we live is simply insane and violates basic rules of physics, science, and logical thinking.

And that's the memo.