text:
When I try to invoke CURL i get the following message

error

Error: Could not retrieve catalog from remote server: Error 500 on SERVER: Server Error: Invalid relationship: Exec[install app] { require => Package[curl] }, because Package[curl] doesn't seem to be in the catalog
Warning: Not using cache on failed catalog
Could not retrieve catalog; skipping run

My Manifest

class pwc_testmobility {
$ensure = present exec{'install app': command=> "curl -X GET -k -sS -u user:pass https://mdm1-stg.com/api/v2/labels/label_summary?adminDeviceSpaceId=1", require => [Package['curl']], path => '/etc/puppetlabs/code/environments/production/modules/mobile/manifests', }  Am using the common phase, new to puppet and coding, so any help would be appreciated . I am trying to deploy a mobile using API which is done via curl edit retag close merge delete 2 Answers Sort by » oldest newest most voted It seems like you have not declare a Package Resource (for curl).You declared a relationship (with require), but you didn't declare the resource you are referring to on that dependency. Add the following to your code and check again (you might also need to share a bit more of that code so that we understand the context). package { 'curl': ensure => present, }  more Comments Thanks Stivesso, I will post the cord here in a while ( 2017-09-19 06:02:33 -0600 )edit TLDR: Append package { 'curl': ensure => installed, } somewhere in the class/module/environment where it can be called by this class. First, let's break this down into a readable format: class pwctestmobility {$ensure = present

exec{'install app':
command=> "curl -X GET -k -sS -u user:pass https://mdm1-ukstg.pwcinternal.com/ap...summary?adminDeviceSpaceId=1",
require => [Package['curl']],
path => '/etc/puppetlabs/code/environments/production/modules/mobile/manifests',
}
}

The error indicated that require line. Do you have a package resource declaration for curl? Should look something like:

package { 'curl':
ensure => installed,
}

Also, while unrelated to the error given, the exec resource "path =>" declaration is the search path used for command execution, (similar to the \$PATH environment variable) on the node. So unless the curl executable exists at '/etc/puppetlabs/code/environments/production/modules/mobile/manifests' on each node you plan to push this to it's probably not going to work.

Generally, you need to either declare the absolute path of the executable (ie /usr/bin/curl or similar) or declare the path(s) that it may exist such as:

 path => '/usr/bin'

OR

path => ['/usr/bin', '/usr/sbin'],

You might want to do some more research on those resources:

more

thanks a lot, I will test this and update back, a basic question, I understand CURL will be installed by default in PE or should we do that

( 2017-09-19 06:05:29 -0600 )edit


text:
Package of problems

# Name that graph Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

## Solutions

Can you find an equation for this parabola?

• In how many ways can you choose to show this equation?
• Which format did you initially choose and why?

Given that the diagram shows a parabola, I know that I am looking for a quadratic equation.

There are three forms in which I could choose to write a quadratic equation:

$$$y=ax^2 + bx + c\label{eq:1}$$$ $$$y=a(x-d)(x-e)\label{eq:2}$$$ $$$y=a(x-f)^2+g\label{eq:3}$$$

In order to write the equation in form $\eqref{eq:1}$ I need to know three pieces of information (because there are three unknowns). Any three different coordinate points on the parabola would work as this would enable me to write three equations in $a, b, c$ which I could then solve simultaneously.

To write the equation directly in form $\eqref{eq:2}$ (fully factorised form) I also need to know three pieces of information. This time however I specifically need to know the location of the $x$-intercepts (assuming that there are any), which will give me the values of $d$ and $e$. Given another point on the parabola I can then identify the value of $a$.

Form $\eqref{eq:3}$ (completed square form) will require a different three pieces of information. The location of the minimum or maximum point (the vertex) of the parabola, $(f, g)$, plus one additional point on the parabola to identify the value of $a$.

I can now consider the specific case of the red parabola presented above. Attempting to identify the equation of this parabola in each of the three forms described will allow me to compare efficiency.

For this particular parabola form $\eqref{eq:2}$, factorised form, is probably the most efficient form to aim for. The $x$-intercepts are clear from the graph and there are several other integer coordinates visible that can be used to identify the scaling factor, $a$.

What is the same and what is different about your approach to finding the equation of the two parabolas below, compared to the example above?

First I will consider the blue parabola.

For this parabola form $\eqref{eq:3}$, completed square form, is the more efficient form. The coordinates of the minimum point can be read directly from the graph and an integer $y$-intercept makes for a more straight-forward calculation to confirm the scaling factor, $a$. It is worth noting that the method of translating the parabola discussed in form $\eqref{eq:2}$ is also an efficient approach here.

Now I will consider the purple parabola.

For this parabola form $\eqref{eq:2}$ and form $\eqref{eq:3}$ are equivalent and more efficient that the approach used in form $\eqref{eq:1}$.

• What properties do the three parabolas have in common?

The three parabolas have the following properties in common:

• A vertical line of symmetry passing through the maximum or minimum point.

• Real roots - two of the parabolas cross the $x$-axis in two places, one touches the $x$-axis.

• A $y$-intercept that can be read from the diagram.

• At least three points that pass through integer coordinates that can be read from the graph.

• Is there an approach to finding the equation that works for all three of these parabolas, and indeed for any parabola you could be given?

• Which approach to finding the equation was the most efficient? Is it the same approach for each example?

In general all of the approaches discussed here rely on the ability to read information directly from the graph.

It is possible to use three simultaneous equations to find the equation of each parabola in form $\eqref{eq:1}$. However, this is a laborious approach and consequently there are plenty of opportunities for “silly” arithmetical errors! This is certainly not the most efficient approach!

Form $\eqref{eq:2}$, factorised form, is a very efficient approach for the red and purple parabolas. Although it may seem that this approach cannot work for the blue parabola (and indeed any parabola which doesn’t cross the $x$-axis), it may be possible to consider a translation of the parabola and to obtain the equation from there, as demonstrated in the approach to form $\eqref{eq:2}$ (for the blue parabola). This is still more efficient than attempting the original approach for form $\eqref{eq:1}$.

Form $\eqref{eq:3}$, completed square form, is a very efficient approach for all three parabolas as the coordinates of the minimum or maximum point (the vertex) of each parabola can be read directly from the graph.

format: plain


text:
# satisfy the Euler-Lagrange equation

Two circles of unit radius, each normal to the line through their centers are a distance d apart. A soap film is formed between themas shown below; energetic considerations require the filem to assume a shape of minimum surface area.

a) show that for any value of d less than a critical value $d_{c}$, there are two surfaces satisfying the appropriate Euler-Lagrange equation, while for d > $d_{c}$ there is no surface. evaluate $d_{c}$ for a typical d < $d_{c}$, sketch the 2 surfaces. which has the lesser area?

b) show that in a certain range $d_{o}$ < d < $d_{c}$, the minimum surface as given by the Euler-Lagrange equation has an area larger than 2$\pi$, so hat the surface is "metastable"; that is, the surface is stable against small perturbations but not against arbitrary perturbation. evaluate $d_{o}$.

c) what happens when d is in creased beyond $d_{c}$?

Here is an drawing of the surface.

Here is what i have so far.

-
What have you tried? –  Matthew Conroy Mar 29 '13 at 22:27
ok i edited the question with what i have been able to do so far. –  Bigb Mar 29 '13 at 23:29
Please make the effort to write your development as MathJax. Desciphering handwriting is enough of my job to not wanting to do it in my hobby here. –  vonbrand Mar 30 '13 at 0:22
haha i understand i will type it up when i get a chance im about to have to get offline for the night. –  Bigb Mar 30 '13 at 0:38

The free energy of the film is to be minimized by surface area minimization. If we assume that d is along x-axis due to axial-symmetry minimal surface is a surface of revolution given by $$A[y]=2\pi \int_{x_1}^{x_2} y \sqrt{1+y'^2}\ dx$$ Correspending Euler Lagrange equation $$\frac{\partial g}{\partial y}-\frac{d}{dx}\bigg(\frac{\partial g}{\partial y'}\bigg)=0$$ where ommiting constants $$\frac{\partial g}{\partial y}=\sqrt{1+y'^2}$$ $$\frac{d}{dx}\bigg(\frac{\partial g}{\partial y'}\bigg)=\frac{d}{dx}\bigg(\frac{y\ y'}{\sqrt{1+y'^2}}\bigg)=\frac{y'^2}{\sqrt{1+y'^2}}+\frac{y''}{\sqrt{1+y'^2}}-\frac{y\ y'^2\ y''}{\big(\sqrt{1+y'^2}\big)^3}$$ If you replace the partials and collect the terms you have $$\frac{1}{\sqrt{1+y'^2}}-\frac{y\ y''}{\big(\sqrt{1+y'^2}\big)^3}=0$$ Further simplification by multiplying $y'$ and collecting terms $$\frac{y'}{\sqrt{1+y'^2}}-\frac{y\ y'\ y''}{\big(\sqrt{1+y'^2}\big)^3}=\frac{d}{dx}\bigg(\frac{y}{\sqrt{1+y'^2}}\bigg)=0$$ The minimization problem reduces to find a solution to a first-order differential equation solution. $$\frac{y}{\sqrt{1+y'^2}}=\alpha$$ which can be recast as $$\frac{dy}{dx}=\sqrt{\frac{y^2}{\alpha^2}-1}$$ and separation of variables $$\int dx=\int \frac{dy}{\sqrt{\frac{y^2}{\alpha^2}-1}}$$ Necessary substitution as $y=\alpha\ \cosh t$ and $$\int dx=\alpha \int dt\Rightarrow x+\beta=\alpha\ t$$ and by back-substitution $$y=\alpha \cosh \frac{x+\beta}{\alpha}$$

The boundary conditions must be integrated to the solution. To be able to use symmetry I assume origin lies in the middle of length d. The curve must pass through points $(\frac{-d}{2},1)$ and $(\frac{d}{2},1)$. Due to symmetry we can say that $\beta=0$. The following condition must be satisfied $$1=\alpha \cosh \frac{d}{2\alpha}$$ I guess from this point you can analyze required points.

-
@vonbrand Thanks for correction –  Occupy Gezi Mar 30 '13 at 23:23


text:
1. ## Linear space question

say you have the set of all functions defined at 1 with f(1)=0 or a homogeneous differential equation y''+ay'+by=0, these are both linear spaces because they satisfy closure under addition/multiplication but why isn't it closed if you change 0 to a number c?

2. ## Re: Linear space question

Originally Posted by mathnerd15
say you have the set of all functions defined at 1 with f(1)=0 or a homogeneous differential equation y''+ay'+by=0, these are both linear spaces because they satisfy closure under addition/multiplication but why isn't it closed if you change 0 to a number c?
suppose f(1)=c, and g(1)=c

(f+g)(1) = 2c

other than c=0, there is no number such that 2c=c so closure under addition is immediately violated

3. ## Re: Linear space question

I see the sum must be 0 but how exactly does this follow from addition closure which says that there is a unique sum x+y, isn't 2c unique? is there a deeper insight for differential equations and a relation to one to one injectiveness?

4. ## Re: Linear space question

Originally Posted by mathnerd15
I see the sum must be 0 since by definition (f+g)(1)=f(1)=g(1)=0 is there a deeper insight for differential equations and a relation to one to one injectiveness?
Yes. The differential equations of the form some left hand side = 0, are known as the homogeneous solutions to that differential equation. Because the right hand side is zero if you have two solutions to that equation then their sum is also a solution and in general the homogeneous solutions form a space and finding various bases for that space is important.

Some of the harder core math guys on here can expand on this I'm sure.

I remember that we'd use a series of solutions to the homogeneous equation to solve for boundary conditions using Fourier analysis.

5. ## Re: Linear space question

Originally Posted by mathnerd15
I see the sum must be 0 but how exactly does this follow from addition closure which says that there is a unique sum x+y, isn't 2c unique?
this isn't what closure says

closure says if you have a set of elements with an addition operation then if you add two elements of that set the sum is also in that set.

so if my set of functions if specified as

$$f_i(1)=c$$

then if closure is to be satisfied

$$(f_i+f_j)(1)=c\;\;\forall i,j$$

which is violated if $$c \neq 0$$

6. ## Re: Linear space question

Thanks very much!!! Apostol defines addition closure as- For every pair of elements x and y in V there corresponds a unique elemnt in V called the sum of x and y, denoted by x+y.
Yes that is so beautiful that the solutions to homogeneous equations form a space and superposition can be used. there is a really nice Braun Springer text on DE

7. ## Re: Linear space question

Originally Posted by mathnerd15
Thanks very much!!! Apostol defines addition closure as- For every pair of elements x and y in V there corresponds a unique elemnt in V called the sum of x and y, denoted by x+y.
Yes that is so beautiful that the solutions to homogeneous equations form a space and superposition can be used. there is a really nice Braun Springer text on DE
ok, this is still my form of closure as it specifies that (x+y) is in fact in V. It just goes one step further and says that the + operation is bijective.

8. ## Re: Linear space question

I wonder why Apostol adds bijection...there's a really nice online latex editor- I'm trying to get the type on this site to load with mathjax


text:
Math and Arithmetic
Numbers
Irrational Numbers

# Can 2 irrational numbers add together to form a rational?

###### Wiki User

Yes. For example:

a = 10 - pi

b = pi

Both are irrational; the sum a + b is 10.

🙏
0
🤨
0
😮
0
😂
0

## Related Questions

Integers are whole numbers. They are the counting numbers, 0 and the corresponding negative numbers. Rational numbers are numbers that can be expressed as a ratio of two integers (the second one being non-zero). Irrational numbers are numbers that are not rational numbers. Rational and irrational number together form the set of real numbers.

No. Irrational and rational numbers can be non-negative.

Rational numbers can be represented in the form x/y but irrational numbers cannot.

They are not rational, that is, they cannot be expressed as a ratio of two integers.Their decimal equivalent is infinitely long and non-recurring.Together with rational numbers, they form the set of real numbers,Rational numbers are countably infinite, irrational numbers are uncountably infinite.As a result, there are more irrational numbers between 0 and 1 than there are rational numbers - in total!

Real numbers can be rational or irrational because they both form the number line.

Rational numbers are numbers that can be written as the division of two integers where the divisor is not zero. Irrational numbers are numbers that are not rational.Irrational numbers, therefore, are numbers that can notbe written as the division of two integers where the divisor is not zero.

Rational numbers are sometimes represented in tabular form, e.g. for proofs relating to different infinities. Irrational and real numbers are not because there is no way to do it.

-8.0 is a rational number. Irrational numbers are things such as numbers that can't be expressed as a fraction in the a/b form. Such numbers that are irrational include pi, square roots of negative numbers, etc.

A rational number is one which can be expressed as a ratio of two integers in the form p/q where q > 0. An irrational number is one which cannot be expressed in such a form.

4.72 is a rational number because if so required it can be expressed as fraction in the form of 118/25 in its lowest terms.

rational and irrational numbers are two types of real Numbers. all real numbers which are terminating and non terminating but repeating comes in the category of rational numbers. all real numbers which are non terminating and non recurring comes in the category of irrational numbers. rational numbers are expressed in the p/q form where p and q are both integers and q is not equal to 0.the opposite the case is with irrational numbers. they are not expressed in the p/q form

It is a rational number because it can be expressed as a fraction in the form of 11/30 whereas irrational numbers can't be expressed as fractions

A rational number is a number that can be expressed as a ratio of two integers in the form A/B where B>0. An irrational number is a real number that is not rational.

The rational numbers form a field. In particular, the sum or difference of two rational numbers is rational. (This is easy to check directly). Suppose now that a + b = c, with a rational and c rational. Since b = c - a, it would have to be rational too. Thus you can't ever have a rational plus an irrational equalling a rational.

Yes. All rational numbers can be changed into fraction form, while all numbers that can't are irrational.

Rational numbers can be expressed in the form p/q (where q is not equal to zero). Irrational numbers cannot be expressed in this form. For example, the square root of 2 cannot be expressed as p/q.

It is a non-integer. It can be a rational fraction (in decimal or rational form); it can be an irrational number (including transcendental numbers); it could be a complex number or a quaternion.

A rational number is one which can be expressed as a ratio of two integers, in the form p/q where q>0. However, there are even more numbers which cannot be so expressed. So we need irrational numbers.

Integers are whole numbers. Fractions, whether in decimal or divisional form, can be either rational or irrational.

Such numbers cannot be ordered in the manner suggested by the question because: For every whole number there are integers, rational numbers, natural numbers, irrational numbers and real numbers that are bigger. For every integer there are whole numbers, rational numbers, natural numbers, irrational numbers and real numbers that are bigger. For every rational number there are whole numbers, integers, natural numbers, irrational numbers and real numbers that are bigger. For every natural number there are whole numbers, integers, rational numbers, irrational numbers and real numbers that are bigger. For every irrational number there are whole numbers, integers, rational numbers, natural numbers and real numbers that are bigger. For every real number there are whole numbers, integers, rational numbers, natural numbers and irrational numbers that are bigger. Each of these kinds of numbers form an infinite sets but the size of the sets is not the same. Georg Cantor showed that the cardinality of whole numbers, integers, rational numbers and natural number is the same order of infinity: aleph-null. The cardinality of irrational numbers and real number is a bigger order of infinity: aleph-one.


text:
# Physics (Version 8.4)

### Rationale/Aims

Physics is a fundamental science that endeavours to explain all the natural phenomena that occur in the universe. Its power lies in the use of a comparatively small number of assumptions, models, laws and theories to explain a wide range of phenomena, from the incredibly small to the incredibly large.

### Structure of Physics

In Physics, students develop their understanding of the core concepts, models and theories that describe, explain and predict physical phenomena.

### Links to Foundation to Year 10

The Physics curriculum continues to develop student understanding and skills from across the three strands of the F-10 Australian Curriculum: Science. In the Science Understanding strand, the Physics curriculum draws on knowledge and understanding from across the four sub-strands of Biological, Physical, Chemical and Earth and Space Sciences.

### Representation of Cross-curriculum priorities

While the significance of the cross-curriculum priorities for Physics varies, there are opportunities for teachers to select contexts that incorporate the key concepts from each priority.

## Unit 4: Revolutions in modern physics

### Unit 4: Revolutions in modern physics Description

The development of quantum theory and the theory of relativity fundamentally changed our understanding of how nature operates and led to the development of a wide range of new technologies, including technologies that revolutionised the storage, processing and communication of information. In this unit, students examine observations of relative motion, light and matter that could not be explained by existing theories, and investigate how the shortcomings of existing theories led to the development of the special theory of relativity and the quantum theory of light and matter. Students evaluate the contribution of the quantum theory of light to the development of the quantum theory of the atom, and examine the Standard Model of particle physics and the Big Bang theory.

Contexts that could be investigated in this unit include technologies such as GPS navigation, lasers, modern electric lighting, medical imaging, nanotechnology, semiconductors, quantum computers and particle accelerators; and related areas of science such as space travel, the digital revolution and the greenhouse effect.

Through the investigation of appropriate contexts, students explore the ways in which these models and theories, and associated technologies, have developed over time and through interactions with social, economic, cultural and ethical considerations. They investigate the ways in which science contributes to contemporary debate about local, regional and international issues, including evaluation of risk and action for sustainability, and recognise the limitations of science to provide definitive answers in different contexts.

Through investigation, students apply their understanding of relativity, black body radiation, wave/particle duality, and the quantum theory of the atom, to make and/or explain observations of a range of phenomena such as atomic emission and absorption spectra, the photoelectric effect, lasers, and Earth’s energy balance. They continue to develop skills in planning, conducting and interpreting the results of investigations, in synthesising evidence to support conclusions, and in recognising and defining the realm of validity of physical theories and models.

### Unit 4: Revolutions in modern physics Learning Outcomes

By the end of this unit, students:

• understand the consequences for space and time of the equivalence principle for inertial frames of reference
• understand how the quantum theory of light and matter explains blackbody radiation, the photoelectric effect, and atomic emission and absorption spectra
• understand how the Standard Model explains the nature of and interaction between the fundamental particles that form the building blocks of matter
• understand how models and theories have developed over time, and the ways in which physical science knowledge and associated technologies interact with social, economic, cultural and ethical considerations
• use science inquiry skills to design, conduct, analyse and evaluate investigations into frames of reference, diffraction, black body and atomic emission spectra, the photoelectric effect, and photonic devices, and to communicate methods and findings
• use algebraic and graphical models to solve problems and make predictions related to the theory and applications of special relativity, quantum theory and the Standard Model
• evaluate the experimental evidence that supports the theory of relativity, wave-particle duality, the Bohr model of the atom, the Standard Model, and the Big Bang theory
• communicate physics understanding using qualitative and quantitative representations in appropriate modes and genres.

### Unit 4: Revolutions in modern physics Content Descriptions

#### Science Inquiry Skills

Identify, research and construct questions for investigation; propose hypotheses; and predict possible outcomes (ACSPH114)

Design investigations, including the procedure to be followed, the materials required, and the type and amount of primary and/or secondary data to be collected; conduct risk assessments; and consider research ethics (ACSPH115)

Conduct investigations, including use of simulations and manipulation of spectral devices, safely, competently and methodically for the collection of valid and reliable data (ACSPH116)

Represent data in meaningful and useful ways, including using appropriate SI units, symbols and significant figures; organise and analyse data to identify trends, patterns and relationships; identify sources of uncertainty and techniques to minimise these uncertainties; utilise uncertainty and percentage uncertainty to determine the cumulative uncertainty resulting from calculations, and evaluate the impact of measurement uncertainty on experimental results; and select, synthesise and use evidence to make and justify conclusions (ACSPH117)

Interpret a range of scientific and media texts, and evaluate processes, claims and conclusions by considering the quality of available evidence; and use reasoning to construct scientific arguments (ACSPH118)

Select, construct and use appropriate representations, including text and graphic representations of empirical and theoretical relationships, simulations, simple reaction diagrams and atomic energy level diagrams, to communicate conceptual understanding, solve problems and make predictions (ACSPH119)

Select, use and interpret appropriate mathematical representations, including linear and non-linear graphs and algebraic relationships representing physical systems, to solve problems and make predictions (ACSPH120)

Communicate to specific audiences and for specific purposes using appropriate language, nomenclature, genres and modes, including scientific reports (ACSPH121)

#### Science as a Human Endeavour (Units 3 & 4)

ICT and other technologies have dramatically increased the size, accuracy and geographic and temporal scope of datasets with which scientists work (ACSPH122)

Models and theories are contested and refined or replaced when new evidence challenges them, or when a new model or theory has greater explanatory power (ACSPH123)

The acceptance of science understanding can be influenced by the social, economic and cultural context in which it is considered (ACSPH124)

People can use scientific knowledge to inform the monitoring, assessment and evaluation of risk (ACSPH125)

Science can be limited in its ability to provide definitive answers to public debate; there may be insufficient reliable data available, or interpretation of the data may be open to question (ACSPH126)

International collaboration is often required when investing in large-scale science projects or addressing issues for the Asia-Pacific region (ACSPH127)

Scientific knowledge can be used to develop and evaluate projected economic, social and environmental impacts and to design action for sustainability (ACSPH128)

#### Special relativity

Examples in context

Support materials only that illustrate some possible contexts for exploring Science as a Human Endeavour concepts in relation to Science Understanding content.

## Development of the special theory of relativity

Many scientists, including Albert Michelson, Hendrik Lorentz and Henri Poincaré, contributed to the development of the special theory of relativity. Lorentz’s Transformation and his ideas about the aether initially explained the Doppler effect. They were improved upon by the next generation of scientists developing theories about electromagnetic mass and ideas about inertial frames of reference and relative motion. Albert Einstein’s work on special relativity built upon the work of scientists such as Maxwell and Lorentz, while subsequent studies by Max Planck, Hermann Minkowski and others led to the development of relativistic theories of gravitation, mass-energy equivalence and quantum field theory. The Michelson-Morley and Fizeau experiments provided evidence for the special theory of relativity (ACSPH123).

## Ring laser gyroscopes and navigation

Ring laser gyroscopes (RLG) are inertial guidance systems that do not rely on signals from an external source but from instruments on board a moving object. RLGs use small differences in the time it takes light to travel around the ring in two directions, known as the Sagnac effect. RLGs have many advantages over other systems: they are highly accurate, have no moving parts, are compact and lightweight, and do not resist changes to their orientation (ACSPH122). RLGs are commonly used in aircraft for accurate navigation and have military applications in helicopters, ships, submarines and missiles.

## Nuclear reactors

Special relativity leads to the idea of mass-energy equivalence, which has been applied in nuclear fission reactors. Nuclear reactors are most commonly used for power generation, propulsion and scientific research. Research reactors have resulted in advances in areas such as medicine and materials testing and fabrication through provision of nuclear isotopes for industrial and medical applications. Although nuclear reactors provide a range of benefits, there is considerable public concern over safety and security issues (ACSPH124). Data from the nuclear industry indicates that nuclear power reactors pose an acceptable risk to public safety and that much has been done to limit that risk. However, other groups argue that such a risk is not acceptable, and, even if no accidents occur, storage of the radioactive waste produced from nuclear facilities remains a safety concern (ACSPH126).

Mathematical representations and relationships

$$\mathrm t\;=\frac{{\mathrm t}_\mathrm o}{\sqrt[{}]{\left(1-\frac{\mathrm v^2}{\mathrm c^2}\right)}}$$

$$\mathrm t\;=$$ time interval in the moving frame as measured by the observer in the proper frame, $${\mathrm t}_\mathrm o=$$ proper time interval (time interval for a clock at rest in the observer’s frame), $$\mathrm v\;=$$ relative speed of the two inertial frames, $$\mathrm c\;=$$ speed of light in a vacuum $$\left(3\;\times10^8\;\text{m }\text{s}^{-1}\right)$$

$$\mathrm l=\;{\mathrm l}_\mathrm o\sqrt[{}]{\left(1-\frac{\mathrm v^2}{\mathrm c^2}\right)}$$

$$\mathrm l=\;$$ length interval in the frame moving at velocity $$\mathrm v$$ with respect to the observer, $${\mathrm l}_\mathrm o=\;$$ lo= proper length (length in a frame at rest with respect to the observer), $$\mathrm c\;=$$ speed of light $$\left(3\;\times10^8\;\text{m }\text{s}^{-1}\right)$$

$${\mathrm p}_\mathrm v=\frac{\mathrm m\mathrm v}{\sqrt[{}]{\left(1-\frac{\mathrm v^2}{\mathrm c^2}\right)}}$$

$${\mathrm p}_\mathrm v=$$ relativistic momentum for an object moving with velocity, $$\mathrm v$$ with respect to the observer, $$\mathrm m$$ $$=$$ mass, $$\mathrm c\;=$$ speed of light

$$\left(3\;\times10^8\;\text{m }\text{s}^{-1}\right)$$

$$\operatorname\Delta\mathrm E=\mathrm\Delta\mathrm m\mathrm c^2$$

$$\mathrm\Delta\mathrm E=$$ change in energy, $$\;\mathrm\Delta\mathrm m=$$ change in mass, $$\;\mathrm c=\;$$ c= speed of light

$$\left(3\;\times10^8\;\text{m }\text{s}^{-1}\right)$$

Observations of objects travelling at very high speeds cannot be explained by Newtonian physics (for example, the dilated half-life of high-speed muons created in the upper atmosphere, and the momentum of high speed particles in particle accelerators) (ACSPH129)

Einstein’s special theory of relativity predicts significantly different results to those of Newtonian physics for velocities approaching the speed of light (ACSPH130)

The special theory of relativity is based on two postulates: that the speed of light in a vacuum is an absolute constant, and that all inertial reference frames are equivalent (ACSPH131)

Motion can only be measured relative to an observer; length and time are relative quantities that depend on the observer’s frame of reference (ACSPH132)

Relativistic momentum increases at high relative speed and prevents an object from reaching the speed of light (ACSPH133)

The concept of mass-energy equivalence emerged from the special theory of relativity and explains the source of the energy produced in nuclear reactions (ACSPH134)

#### Quantum theory

Examples in context

Support materials only that illustrate some possible contexts for exploring Science as a Human Endeavour concepts in relation to Science Understanding content.

## Development of the quantum model

Max Planck and Einstein were the first to describe light and energy as being quantised, a finding that led to light being described as spatially quantised photons of energy. The Bohr model of the atom was built on this quantised description of light energy and Rutherford’s nuclear model. The Bohr model was a quantum-based modification to Rutherford’s model and was rapidly accepted due to its ability to explain the emission lines of atomic hydrogen. Prior to Bohr’s model, the Rydberg formula describing the wavelengths of spectral lines of many chemical elements was known but could not be explained. A more elaborate quantum mechanical model of the atom, however, was required to explain other observations made about atoms. The quantum mechanical model of the atom uses quantum theory and describes electron orbitals that can be used to calculate the probability of finding an electron at a specific point (ACSPH123).

## Black body radiation and the greenhouse effect

All objects in the universe, including the sun and Earth, emit black body radiation. The natural temperature of Earth can be predicted using the Stefan-Boltzmann black body radiation equation which assumes there is a balance between incoming and outgoing radiation. The true temperature is significantly higher due to the absorption of emitted black body radiation from the surface by molecules in the atmosphere (the greenhouse effect). Models of Earth’s energy balance enable scientists to monitor changes in global temperature, assess the evidence for changes in climate due to the enhanced greenhouse effect, and evaluate the risk posed by anthropogenic climate change (ACSPH125). Further development of models of Earth’s energy dynamics and climate change enables scientists to more accurately predict the scenarios that will result in global warming, the time frames involved, and the likely impacts of these changes (ACSPH128).

Mathematical representations and relationships

$$\mathrm E=\mathrm h\mathrm f$$

$$\mathrm E\;=$$ energy of photon, $$\mathrm f\;=$$ frequency, $$\mathrm h\;=$$

$$\left(6.626\;\times\;10^{-34}\;\mathrm J\;\mathrm s\right)$$

$${\mathrm\lambda}_{\mathrm m\mathrm a\mathrm x}\;=\frac{\mathrm b}{\mathrm T}$$

$${\mathrm\lambda}_{\mathrm m\mathrm a\mathrm x}=\;$$ peak wavelength, $$\mathrm T\;=\;$$ absolute temperature, $$\mathrm b\;=$$ Wien’s displacement constant $$\text{(2.898 × }10^{-3}\;\text{m K)}$$

$${\mathrm E}_\mathrm k=\mathrm h\mathrm f-\;\mathrm W$$

$${\mathrm E}_\mathrm k=\;$$ Ek= kinetic energy of photoelectron, $$\mathrm h\mathrm f\;=$$ energy of incident photon, $$\;\mathrm W\;=\;$$ work function of the material

$$\mathrm\lambda=\frac{\mathrm h}{\mathrm p}$$

$$\mathrm\lambda=$$ wavelength associated with particle, $$\mathrm p\;=$$ momentum of particle, $$\mathrm h\;=$$ Planck’s constant $$\left(6.626\;\times\;10^{-34}\;\mathrm J\;\mathrm s\right)$$

$$\mathrm n\mathrm\lambda=2\mathrm\pi\mathrm r$$

$$\mathrm n\;=$$ an integer 1, 2, 3, 4... , $$\mathrm\lambda=$$wavelength of electron, $$\mathrm r\;=$$ orbital radius of electron

$$\mathrm m\mathrm v\mathrm r=\frac{\mathrm n\mathrm h}{2\mathrm\pi}$$

$$\mathrm m\;=$$ mass of electron, $$\mathrm v\;=$$ velocity of electron, $$\mathrm r\;=$$ orbital radius of electron, $$\mathrm n\;=$$ an integer 1, 2, 3, 4, etc., $$\mathrm h\;=$$ Planck’s constant $$\left(6.626\;\times\;10^{-34}\;\mathrm J\;\mathrm s\right)$$

$$\frac1{\mathrm\lambda}=\mathrm R\left(\frac1{\mathrm n_\mathrm f^2}-\frac1{\mathrm n_\mathrm i^2}\right)$$

$$\mathrm\lambda=$$ wavelength of spectral line, $${\mathrm n}_\mathrm i=\;$$ principal quantum number of initial electron state, $$\;{\mathrm n}_\mathrm f=\;$$ nf= principal quantum number of final electron state, $$\mathrm R\;=$$ Rydberg’s constant $$\text{(1.097 × }10^7\;\text{m}^{-1}\text{)}$$

Atomic phenomena and the interaction of light with matter indicate that states of matter and energy are quantised into discrete values (ACSPH135)

On the atomic level, electromagnetic radiation is emitted or absorbed in discrete packets called photons; the energy of a photon is proportional to its frequency; and the constant of proportionality, Planck’s constant, can be determined experimentally (for example, from the photoelectric effect or the threshold voltage of coloured LEDs) (ACSPH136)

A wide range of phenomena, including black body radiation and the photoelectric effect, are explained using the concept of light quanta (ACSPH137)

Atoms of an element emit and absorb specific wavelengths of light that are unique to that element; this is the basis of spectral analysis (ACSPH138)

The Bohr model of the hydrogen atom integrates light quanta and atomic energy states to explain the specific wavelengths in the hydrogen spectrum and in the spectra of other simple atoms; the Bohr model enables line spectra to be correlated with atomic energy-level diagrams (ACSPH139)

On the atomic level, energy and matter exhibit the characteristics of both waves and particles (for example, Young’s double slit experiment is explained with a wave model but produces the same interference pattern when one photon at a time is passed through the slits) (ACSPH140)

#### The Standard Model

Examples in context

Support materials only that illustrate some possible contexts for exploring Science as a Human Endeavour concepts in relation to Science Understanding content.

## Evidence for the Higgs boson particle

The Higgs boson particle was predicted in the early 1960s by the Standard Model of particle physics. Evidence for the Higgs boson particle would confirm the existence of the Higgs field and help to explain why fundamental particles have mass. Discovery of the particle would guide other theories and discoveries in this field, including validation of the Standard Model, and insights into cosmic inflation and the cosmological constant problem (ACSPH123). Production of the Higgs boson requires an extremely powerful particle accelerator. The Large Hadron Collider at CERN was built to test particle physics theories, and specifically to try to produce and detect the Higgs boson particle. Since the commencement of its operation, previously unobserved particles have been produced (ACSPH123) and most recently a new particle has been observed that is consistent with the theorised Higgs boson particle.

## Particle Accelerators

Particle accelerators propel charged particles to high speeds using a combination of electric and magnetic fields. High-energy particle accelerators are used in particle physics research to create and observe particles. These machines have gradually increased in size, complexity and in their ability to accelerate particles to higher speeds, thus increasing scientists’ ability to observe new particles (ACSPH122). More practical uses of particle accelerators include their use in production of radioisotopes for medical treatments and as synchrotron light sources. The construction of the Australian Synchrotron involved collaboration between Australian and New Zealand science organisations, state and federal governments and international organisations and committees including the International Science Advisory Committee and the International Machine Advisory Committee (ACSPH127).

## The Big Bang Theory

The Big Bang Theory describes the early development of the universe including the formation of subatomic particles from energy and the subsequent formation of atomic nuclei. There is a variety of evidence that supports the Big Bang Theory including Cosmic Background Radiation, the abundance of light elements and the red shift of light from galaxies that obey Hubble’s Law. Alternate theories exist including the Steady State theory, but the Big Bang Theory is the most widely accepted theory today (ACSPH123). There is opposition to this theory in both scientific and religious communities due to its inability to explain what came before the singularity and because it cannot explain all aspects of the universe as it exists today (ACSPH124).

The Standard Model is based on the premise that all matter in the universe is made up from elementary matter particles called quarks and leptons; quarks experience the strong force, leptons do not (ACSPH141)

The Standard Model explains three of the four fundamental forces (strong, weak and electromagnetic forces) in terms of an exchange of force-carrying particles called gauge bosons; each force is mediated by a different type of gauge boson (ACSPH142)

Reactions between particles can be represented by simple reaction diagrams (ACSPH143)

Lepton number and baryon number are examples of quantities that are conserved in all reactions between particles; conservation laws can be used to support or invalidate proposed reactions (ACSPH144)

Variations of reactions can be found by applying symmetry operations to known reactions. These include reversing the direction of the reaction diagram (time reversal symmetry) and replacing all particles with their antiparticles and vice versa (charge reversal symmetry). Energy and momentum must also be conserved for such a reaction to be possible. (ACSPH145)

High-energy particle accelerators are used to test theories of particle physics including the Standard Model (ACSPH146)

The Standard Model is used to describe the evolution of forces and the creation of matter in the Big Bang theory (ACSPH147)


text:
# Container Material Considerations for Collection and Storage of Atomic Absorption Spectroscopy Samples

Analysis by Atomic Absorption Spectroscopy requires collection, storage and at times transportation of samples. Containers can pose a contamination risk of Atomic Absorption Spectroscopy samples and the extent of contamination will be significant over extended storage periods.

Container materials in common use are made from

• Glass
• Quartz
• Plastics

In this article we shall cover the properties of such materials that make them desirable for collection and storage of AAS samples.

Container pre-treatment

It goes without saying that the container should be inert with least possibilities of leaching or adsorption. New containers require pre-treatment. Glass containers should be soaked overnight in 1:1 concentrated at $$HNO_3$$ or HCl prior to use. The pre-treatment minimizes the alkaline character of glassware. Plastics should be soaked in 1:1 nitric acid for a week followed by a week in HCl and finally a week in deionised water.

Used containers should be washed serially with a non-phosphate detergent, deionised water, acidified potassium dichromate and finally with deionised distilled water.

#### Container materials

Glass

Some solvents and acids gradually dissolve glass and get contaminated with elements such as Si, K, Na, Ca, Mg and Al.

Borosilicate glass

• Most commonly used
• Resistant to most acids except HF and boiling $$H_3PO_4$$
• Not suitable for heating or storage of hot alkaline solutions
• Should not be heated to temperatures above 500°C

Quartz

Quartz exists in opaque and transparent forms. Opaque form has higher trace element concentration and is not suitable for trace metal analysis. Containers made from transparent parts are used for sampling for trace metals AAS analysis.

• Borosilicate glass parts also attacked by HF and boiling $$H_3PO_4$$
• Suitable for strong alkaline solutions
• Can be heated up to 1100°C

Plastics

Plastic containers have their own advantages and disadvantages. However, availability of different chemically modified plastic materials do limit some of the disadvantages.

• Unbreakable
• Preferred for trace metal analysis due to greater inertness

• Heating not possible
• Cannot be used with organic solvents

The commonly used plastic container materials are:

• Polypropylene
• Polytetrafluoroethylene (PTFE or Teflon)
• Fluorinatedethylenepropylene (FEP)
• Perfluoroalkoxy (PFA)
• HDPE (high-density polyethylene)
• LDPE (low-density polyethylene)

PFA has some excellent desirable properties such as acid digestions upto 250°C using acids such as HF, HNO3, HCl, H2SO4 and H3PO4. It is also used for making microwave digestions vessels

LDPE and HDPE bottles are typically used for extended storage of solutions with upto 10% $$HNO_3.$$

The choice of container material will depend on a number of considerations such as:

• Required level of detection
• Inertness of container
• Ability to handle hot boiling acids
• Storage period prior to analysis.

As a good practice samples should be analysed at the earliest opportunity post preparation or collection.

## A simple way to understand and practice the clauses covered by ISO 17025:2005

I have been a part of an accredited laboratory for 10 years now and have successfully faced more than 12 audits based on the ISO…

## 125 Job Profiles for an Analytical Chemist!

The ever increasing consumer awareness and the demand for quality have made analytical chemistry and analytical chemist, an integral and essential part of all industries.…

## Characteristic properties of commonly used laboratory plastic materials

A laboratory chemist is always on the lookout for container materials which can be used for handling and dispensing of all types of solvents, chemicals,…

## How to test a hand sanitizer at home? 3 Easy Simple Tests!

How to test you Hand Sanitizer at home I am sure you are also using a hand sanitizer and sometimes wonder if it is really…

## Essentials of Good Laboratory Practices (GLP)

Quality Product testing is the backbone for promotion of global trade. A product tested and approved to meet defined specifications should qualify when tested in…

#### Responses

1. […] trace metals analysis in ppm and ppb levels for different elements. It is essential to work in a clean laboratory environment and use inert containers for collection and storage of samples so that errors due to cross contamination are […]

2. Hi there,

I was just wondering if anyone has information on the process and manufacturing of PMP lab plastics, as I am doing a leachability study on this plastic aswell as pyrex glassware for an Elemental impurities lab.

Thank you,

1. Hi Grace, I think you should contact the manufacturer like Brand for this information.

3. Hi there
Thanks for information on this site, it hep me to understand AAS. can you guide me about storage of metal standards.most of them contain HCl and some HF.

1. Hi, it is best to refer the documentation provided by the supplier and the MSDS for proper storage as the concentrations may vary.


text:
## LATEX-L@LISTSERV.UNI-HEIDELBERG.DE

 Options: Use Classic View Use Monospaced Font Show Text Part by Default Condense Mail Headers Topic: [<< First] [< Prev] [Next >] [Last >>]

 Sender: Mailing list for the LaTeX3 project <[log in to unmask]> Date: Tue, 9 Sep 2008 20:44:05 +0200 Reply-To: Mailing list for the LaTeX3 project <[log in to unmask]> MIME-Version: 1.0 Content-Transfer-Encoding: 7bit In-Reply-To: <[log in to unmask]> Content-Type: text/plain; charset=us-ascii From: Frank Mittelbach <[log in to unmask]> Parts/Attachments: text/plain (57 lines) I don't really think that it is necessary or even desirable to define all types of variant expansion forms (in all permutations) up front just for the sake of it. Ideally when I write \foo:ABC where ABC is some form of arg spec, eg "xoc" "noo" or ... then what should happen is that automagically the arguments should be get properly expanded and then the corresponding base function (ie \foo:nnn should be called passing the correctly expanded arguments). And that should work for any permutation that you can think of. Unfortunately, TeX can't autogenerate functions if they are missing because we are missing a callback function that gets called in case of "undefined csname". So either  - we provide all variants up front  - we do all our parsing ourselves (so that we can detect missing functions)    but that makes a lot of processing very slow but it would be an option  - get thatm missing functionality into a TeX successor and use it  - or we do simply do the last step (generation of missing functions) manually currently the concept is the latter:  - there is only one single unique way of providing a variant from a given    base function, eg, having the base function \foo:nnn I'll get \foo:noo by    doing:   \def:Npn \foo:noo {\exp_args:Nnoo \foo:nnn }  - so either the variant is defined in expl3 or a package could define it    andone can be sure that two package would define it in exactly the same way    so the double definition wouldn't hurt (a simple def is extremly fast and    assuming nobody makes a mistake the definitions would always be the same)  - (one oculd also think of providing a more abstract interface for defining    variants from base functions) so far the approach was to not provide all possible permutations but wait until it becomes clear that certain forms are actually needed and if so gradually move them from packages to the core language. it is certainly questionable where to draw the line here, but for functions with 3 or more arguments the number of permutations become huge and in many cases most of them are useless. So rather than providing everyting for the sake of uniformity I would suggest to allow for gaps (as they can be very very easily filled ... for any developer who understood the construction method). cheers frank


text:
# Changeset 949 for palm/trunk/TUTORIAL/SOURCE

Ignore:
Timestamp:
Jul 18, 2012 10:25:11 AM (11 years ago)
Message:

update/new tutorial presentations

Location:
palm/trunk/TUTORIAL/SOURCE
Files:
 r945 \end{frame} % Folie 11 \begin{frame} \frametitle{Exercise} \begin{itemize} \item<2-> Set the job time ($-t$ ...) of a run (e.g. the example-run) in a way that at least one restart job is necessary to reach the end time of the simulation. \item<3-> Alternatively, create a job chain containing at least one restart run by setting the \texttt{d3par}-parameters \texttt{restart\underline{ }time} and \texttt{dt\underline{ }restart} manually. \item<4-> Compare, if the results from the job chain are the same as if the simulated is carried out in one single job. (They should be!) \item<5-> Carry out this job and watch what happens. \item<6-> Delete the results of the restart job(s) and carry out the restart from the initial run manually. \end{itemize} \end{frame} \end{document}


text:
# Tag Info

68

I am late to this party here, but I can maybe advertize something pretty close to a derivation of quantum mechanics from pairing classical mechanics with its natural mathematical context, namely with Lie theory. I haven't had a chance yet to try the following on first-year students, but I am pretty confident that with just a tad more pedagogical guidance ...

37

Why would you ever try to motivate a physical theory without appealing to experimental results??? The motivation of quantum mechanics is that it explains experimental results. It is obvious that you would choose a simpler, more intuitive picture than quantum mechanics if you weren't interested in predicting anything. If you are willing to permit some ...

20

A wave function is a complex-valued function $f$ defined on ${\mathbb R}^1$ (if your electron is confined to a line) or on ${\mathbb R}^2$ (if your electron is confined to a plane) or ${\mathbb R}^3$ (if your electron ranges over three-space), and satisfying $$\int |f|^2=1$$ (where the integral is defined over the entire line or plane or 3-space). Every ...

17

As StephenG mentioned in a comment, the paper you're asking about is the subject of a commentary in arXiv:quant-ph/0509130, by Markus Bier; Li and Li attempt a rebuttal of that comment in Appendix C of v2 of their paper. The comment by Markus Bier is phrased in dry academic language, and there are certain aspects of the phrasing that simply do not fit ...

16

The immediate problem with obtaining the Born rule in the many-worlds interpretation is quite elementary: you can't even begin to attach probabilities to "worlds" (or to events within worlds), in your theory of many worlds, if the theory isn't even clear on what a world is. Physical states according to various interpretations In classical physics, a ...

16

Part of you problem is "Probability amplitude is the square root of the probability [...]" The amplitude is a complex number whose amplitude is the probability. That is $\psi^* \psi = P$ where the asterisk superscript means the complex conjugate.1 It may seem a little pedantic to make this distinction because so far the "complex phase" of the amplitudes ...

13

The theory of quantum mechanics has been developed to explain observations, i.e. measurements. Without observations it is a floating mathematical construct. One of the postulates to connect the mathematics with reality is: To every observable there corresponds an operator which operating on the state function will give an eigenvalue. So the question ...

11

Strictly speaking, the Born rule cannot be derived from unitary evolution, furthermore, in some sense the Born rule and unitary evolution are mutually contradictory, as, in general, a definite outcome of measurement is impossible under unitary evolution - no measurement is ever final, as unitary evolution cannot produce irreversibility or turn a pure state ...

11

Before trying to understand quantum mechanics proper, I think it's helpful to try to understand the general idea of its statistics and probability. There are basically two kinds of mathematical systems that can yield a nontrivial formalism for probability. One is the kind we're familiar with from everyday life: each outcome has a probability, and those ...

11

What I don't get is how he ended up with an equation that is a probability distribution. He ended up with Schroedinger's equation, but he did not think it describes probability. He thought more along the lines of de Broglie idea, that the electron is some kind of wave, and $\psi$ - solution to the equation - expresses mathematically shape and other ...

11

There is a Hilbert space H (probably of dimension 3 or more, as in Gleason's theorem?) equipped with some logical apparatus (the lattice L?). Correct, and the lattice $L(H)$ is that of orthogonal projectors/closed subspaces of a separable complex Hilbert space $H$. As a partially ordered set, the partial ordering $P\leq Q$ relation is the inclusion of ...

10

You should use history of physics to ask them questions where classical physics fail. For example, you can tell them result of Rutherford's experiment and ask: If an electron is orbiting around nucleus, it means a charge is in acceleration. So, electrons should release electromagnetic energy. If that's the case, electrons would loose its energy to collapse ...

10

Well there are multiple reasons, but a very important one is that it can be proven (from the Schrödinger equation) that $$\frac{\mathrm d}{\mathrm dt}\int \mathrm d\boldsymbol x\ |\psi(\boldsymbol x,t)|^2=0$$ so that, if at any moment in time we have $\int \mathrm d\boldsymbol x\ |\psi(\boldsymbol x,t)|^2=1$, this will remain true at any other time. On ...

9

If I would be designing an introduction to quantum physics course for physics undergrads, I would seriously consider starting from the observed Bell-GHZ violations. Something along the lines of David Mermin's approach. If there is one thing that makes clear that no form of classical physics can provide the deepest law of nature, this is it. (This does make ...

9

Those factors usually come out when separating the wave-function for a 3-dimensional problem (like the Hydrogen atom) in its radial and angular parts: $$\psi(r, \theta, \phi) = R(r) Y(\theta, \phi).$$ If you are then interested in the probability of finding the particle (or whatever you are studying) at a distance from the origin between $r$ and $r+dr$ ...

9

To be precise, it is not the complex-valued wave function $\psi(\vec{x})$ that is interpreted as a probability density, but its absolute square $|\psi(\vec{x})|^2$ is, sometimes $\psi(\vec{x})$ is called probability wave to stress the difference. This is significant, if $\psi(\vec{x})$ itself was a density, or if $|\psi(\vec{x})|^2$ itself obeyed an ...

8

Though there are many good answers here, I believe I can still contribute something which answers a small part of your question. There is one reason to look for a theory beyond classical physics which is purely theoretical and this is the UV catastrophe. According to the classical theory of light, an ideal black body at thermal equilibrium will emit ...

8

The classical limit for light is a wave theory. The (quantum) amplitude of the wavefunction wave becomes the (classical) amplitude of the wave (e.g. the magnitude of the electric field), and the (quantum) expected number of photons in a volume becomes the (classical) intensity of the wave (e.g. the squared electric field). Of course it is correct to map the ...

8

The wavefunction being zero at a single point has no physical significance because the wavefunction is a probability density (more precisely, its squared modulus is that), not a probability. All that counts are the integrals of the density over regions of non-zero measure, the values at single points are completely irrelevant. Formally, wavefunctions lie in $... 7 All the key parts of quantum mechanics may be found in classical physics. 1) In statistical mechanics the system is also described by a distribution function. No definite coordinates, no definite momenta. 2) Hamilton made his formalism for classical mechanics. His ideas were pretty much in line with ideas which were put into modern quantum mechanics long ... 7 In quantum mechanics, the amplitude$\psi$, and not the probability$\mid\psi\mid^2$, is the quantity which admits the superposition principle. Notice that the dynamics of the physical system (Schrödinger equation) is formulated in terms of and is linear in the evolution of this object. Observe that working with superposition of$\psi$also permits complex ... 7 You need quantum field theory to describe light as it is clearly a relativistic system. Photons are excitations of the electromagnetic (EM) field which is a spin 1 boson field. In the limit of large numbers of photons the usual classical limit is obtained, i.e. one can describe the field evolution classically to a good approximation. So it is not correct to ... 7 The answer is negative for two distinct reasons. (1) In QM operator means linear operator and the map$\psi \mapsto |\psi(x)|^2$is not linear, evidently. (2) Wavefunctions are elements of$L^2(\mathbb R)$and these elements are defined up to zero-measure sets. I mean that, if$\psi(x) \neq \psi'(x)$for$x\in E$where$E$has zero measure, then$\psi=\psi'...

7

The wave function is the solution to the Schrödinger equation, given your experimental situation. With a classical system and Newton's equation, you would obtain a trajectory, showing the path something would follow: the equations of motion. For a quantum mechanical system you get a wave function, and the rules it obeys over time. With this you can determine ...

7

This is because the values of the wavefunction are complex numbers. Squaring a complex number doesn't necessarily give you a positive number, or even a real one. For example, $i^2=-1$, and $(1+i)^2 = 2i$.

7

You see no formal proof of the Born rule because it is not a theorem of quantum mechanics, it is a postulate. See, for instance, this question and its answers. Normalization has nothing to do with the Born rule, since we can formulate the Born rule for arbitrary non-normalized states $\lvert \psi\rangle,\lvert \phi\rangle$ as  P(\lvert \psi\rangle,\lvert \...

6

There is a paper called Ruling Out Multi-Order Interference in Quantum Mechanics that, I think, answers your question in the negative (within a certain bound anyway). The authors show that the Born rule implies quantum interference comes only in pairs of possibilities (second order interference), and that by relaxing the Born rule one would expect higher ...

6

The use of the word "postulate" in the question may indicate an unexamined assumption that we must or should discuss this sort of thing using an imitation of the axiomatic approach to mathematics -- a style of physics that can be done well or badly and that dates back to the faux-Euclidean presentation of the Principia. If we make that choice, then in my ...

6

There is no integration of the radial part because, as you said yourself, we want the probability of finding the electron somewhere in the spherical shell between $r$ and $r+dr$ from the nucleus. (in a differential shell between $r$ and $r+dr$, and no need to integrate over $r$.)

Only top voted, non community-wiki answers of a minimum length are eligible


text:
# CYBERCHASE - interactive math activities for kids

By Murray Bourne, 23 Oct 2006

CYBERCHASE is produced by PBS Kids. It is an animated TV show and also has an interactive website. From their "Parents and Teachers" page:

CYBERCHASE teaches kids that math is everywhere, everyone can be good at it, and it's fun! Here at CYBERCHASE Online for Parents & Teachers, we've provided everything you need to bring the math adventure and learning fun of CYBERCHASE to your favorite kids.

CYBERCHASE is aligned with the NCTM Standards. I liked the "quest" nature of the activities, where the kids nead to solve problems before they can move on. A screenshot:

What I wonder, though, is:

1. Is this more entertainment than it is learning?
2. What do usability tests reveal about what kids really do?
3. Is this kind of animated, cartoon-like approach the best way?

These questions may indicate I am cynical about CYBERCHASE. No, I'm not, but I really want to know more about the learning effectiveness of such approaches.

Be the first to comment below.

### Comment Preview

HTML: You can use simple tags like <b>, <a href="...">, etc.

To enter math, you can can either:

1. Use simple calculator-like input in the following format (surround your math in backticks, or qq on tablet or phone):
a^2 = sqrt(b^2 + c^2)
(See more on ASCIIMath syntax); or
2. Use simple LaTeX in the following format. Surround your math with $$ and $$.
$$\int g dx = \sqrt{\frac{a}{b}}$$
(This is standard simple LaTeX.)

NOTE: You can't mix both types of math entry in your comment.


text:
# Creation of momentum on vertex (quantum field theory)

For a an interaction term like $g(\overline{\psi} \gamma^\mu \psi) \partial_\mu \phi$ in which $\psi$ is a Dirac spinor and $\phi$ a scalar field (d=4), should we expect this vertex to have a momentum associated with it (from the $\partial_\mu \phi$)? If so, what is the meaning of a vertex "generating" momentum?

• The first question is closely related to Interpretation of derivative interaction term in QFT. As for the second one, who says that the vertex generates momentum? where have you read/heard this? – AccidentalFourierTransform Jan 24 '16 at 15:29
• Thanks @AccidentalFourierTransform! The interpretation interaction was really useful. I didn't read on the generating of momentum anywhere, I was just confused for the momentum term on the vertex – Sofia Jan 24 '16 at 16:12


text:
# Haar measure coming from Pontryagin duality v/s Fourier inversion

Not research but advertising this question from mse in case someone wants to answer.

I'm struggling with some bookkeeping associated with the Pontryagin duality theorem. I'm thinking about the first equation of the second last line in the proof of $$4.32$$ in Folland's harmonic analysis, which I feel requires some explanation along the lines of my question below.

Notation: Let $$(G, d_G)$$ be a locally compact abelian group with fixed choice of Haar measure. Let $$\hat{G}$$ be the set of all characters on $$G$$ equipped with the compact-open topology (or, equivalently, the weak$$^*$$-topology inherited from $$L^{\infty}(G, d_G)$$). Ponrtyagin duality says that the map $$\Phi: G\to \hat{\hat{G}};\ \ x\mapsto (\cdot)(x)$$ is a homeomorphism as well as an isomorphism of groups. Further, write $$M(G), M(\hat{G}), M(\hat{\hat{G}})$$ to be the set of regular complex Borel measures on $$G, \hat{G}, \hat{\hat{G}}$$ respectively. If $$\mu \in M(\hat{G})$$ let $$T_\mu(x) := \int \xi(x)d\mu(\xi).$$ $$T$$ gives an injective norm decreasing map from $$M(\hat{G})$$ into the bounded continuous functions on $$G$$ (In fact, onto the complex span of the set of functions of positive type on $$G$$). Thus the image will contain all convolutions of functions having compact support. We have our test function spaces, $$\mathcal{B}(G):= \{T_\mu : \mu \in M(\hat{G})\}$$ and its intersection with $$L^1(G, d_G)$$, $$\mathcal{B}^1(G)$$.

The theorem Fourier Inversion I ($$4.21$$ in the book) says that, there is a unique choice of Haar measure $$d_{\hat{G}}$$ on $$\hat{G}$$ such that: $$$$\label{1} \tag{1} \text{if \ f\in \mathcal{B}^1(G)\ with f = T_{\mu_f}, then \ \hat{f} \in L^1(\hat{G})\ and \ \mu_{f}= \hat{f}(\xi)d_{\hat{G}}(\xi)\ on the dual.}$$$$ This, in particular, implies $$\ f(x) =\int \xi(x)\hat{f}(\xi)d_{\hat{G}}(\xi)$$. Note, the fourier transform on $$L^1(G, d_G)$$ is defined by $$\hat{f}(\xi) = \int \overline{\xi(x)}f(x)d_G(x).$$ The same Fourier inversion I shows that we have a uniquely defined Haar measure $$d_{\hat{\hat{G}}}$$ on $$\hat{\hat{G}}$$ satisfying the corresponding equations in \eqref{1}.

Setup: In the proof of Fourier inversion II ($$4.32$$ in the book), there is a point where the author takes an $$f\in L^1(G, d_G)$$ with $$\hat{f} \in L^1(\hat{G}, d_{\hat{G}})$$ and arrives at the conclusion that $$\hat{f}$$ must belong to $$\mathcal{B}^1(\hat{G})$$ with the corresponding measure $$d\mu_{\hat{f}}(\eta) = f\circ \Phi^{-1}(\eta^{-1})d(\Phi_*d_G)(\eta)\ \left(= \Phi_*(f(x^{-1})d_G(x))\right)$$ on $$\hat{\hat{G}}$$. This and Fourier inversion I applied to $$\hat{G}$$ then leads the the equality of measures $$$$\label{2} \tag{2} \hat{\hat{f}}(\eta)d_{\hat{\hat{G}}}(\eta) = c f\circ \Phi^{-1}(\eta^{-1})d_{\hat{\hat{G}}}(\eta)$$$$ where the constant $$c$$ is defined by the relation $$\Phi_*(d_G) = cd_{\hat{\hat{G}}}$$. The above equation \eqref{2} leads to the equality $$\hat{\hat{f}}(\Phi(x)) = cf(x^{-1})$$ rather than what is stated in the text. This leads to:

Question: On pushing forward, do we have the equality $$\Phi_*(d_G) = d_{\hat{\hat{G}}}$$?

If I understand author correctly - For $$f \in L^1(G)$$ with $$\hat{f}\in L^1(\hat{G})$$, you get $$f(x) = c^{-1}\hat{\hat{f}}(\Phi(x^{-1}))$$ where $$\phi_*(d_G) = cd_{\hat{\hat{G}}}$$.
On the other hand, if $$h \in \mathcal{B}^1(G)$$, from 'inversion I' we know $$\hat{h}\in L^1(\hat{G})$$ and that, $$h(x) = \int \xi(x)\hat{h}(\xi)d_{\hat{G}}(\xi) =: \hat{\hat{h}}(\Phi(x^{-1})).$$ This shows $$c$$ must be equal to $$1$$.