# Tagged Questions

For questions on quantum mechanics, a branch of physics dealing with physical phenomena at microscopic scales, where the action is on the order of the Planck constant.

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1answer
16 views

### Left-ratio and right-ratio in (not necessarily commutative) field

I am reading Birkhoff–Neumann (1936) about (not necessarily commutative) field $F$. The authors use terms right-ratio and left-ratio in section 13. Right-ratio is denoted as \$[x_1, x_2, ... ...
0answers
30 views

### To construct a Schrödinger wave with prescribed mean position and momentum

Let $\lambda_x, \lambda_\xi>0$ be fixed parameters which we will call position lengthscale and momentum lengthscale respectively. Fix a (position-)point $x_0\in \mathbb{R}$ and a (momentum-)point ...
1answer
241 views

### Do these two sets of matrices form groups?

Stimulated by some Physics backgrounds, consider the following two sets of matrices. Notations and definitions:Let $A,B$ be two complex $n\times n$ matrices, then \$\left [ A,B \right ...
2answers
114 views

### Can the $0$-norm represent determinism?

In Scott Aaronson's Quantum Computing since Democritus, he presents classical probability theory as based on the $1$-norm, and QM as based on the $2$-norm. Call $\{v_1,\ldots,v_N\}$ a unit vector ...
1answer
44 views

### Dense in the special unitary group

I was going through the PhD thesis of Daniel Nagaj. In the last paragraph on page 30, there is this following sentence. A universal gate set must be dense in the group $SU(n)\ldots$ My question ...
0answers
17 views

### QFT basics for klein gordon fields [migrated]

Hi I am teaching myself QFT from Peskin for next years maths course and I have two questions: 1) What is a c-number is it a complex number and if so why does it mean, ...
1answer
69 views

### Quantum Information: Deutsch-Jozsa Algorithm

There is a step in the construction of this algorithm which I'm not understanding: \$\displaystyle \left[\sum_x \frac{| x \rangle}{\sqrt{2^n}}\right]\left[\frac{ | 0 \rangle -| 1 \rangle ...
0answers
29 views

### Demonstrate basic property of Hermitian

I want to demonstrate that $(A|v\rangle)^* = \langle v|A^*$ Given only this $(AB)^*=A^*B^*$ and $(|v\rangle, A|w\rangle) = (A^*|v\rangle, |w\rangle)$ Is it possible? The difficulty is that i don't ...
0answers
19 views

### Circuit identities HTH

Using this circuit indetities $HXH=Z, HYH=-Y, HZH = X$ prove $HTH=R_x(\pi/4)$. here $H$ is Hadamard matrix, $X,Y$ and $Z$ are Pauli matrix, $R_x$ is a rotation matrix and \$T=\left[ \begin{array}{cc} 1 ...
0answers
77 views

### Is quantum game theory reducible to classical game theory?

Quantum game theory is an extension of classical game theory to the quantum domain. It differs from classical game theory in three primary ways: Superposed initial states, Quantum ...
3answers
208 views

### Quantum Mathematics?

As per my last question, this has less to do with cold, hard, and fast calculations and more to do with the interplay between mathematics and philosophy...but as armchair philosophers aren't as hard ...
0answers
21 views

### Pure Phase Number

I am read a solution (4.9) Here say: ... both $a, d$ are pure phases, so that it is always possible to ﬁnd (non unique) real numbers $\alpha, \beta, \delta$ such that \$a = ...
1answer
27 views

### Wave-function with boundary conditions

I'm given a complex wave-function $\psi(t, x)$, in one spatial dimension, which satisfies $i \partial_t \psi = \partial_x^2 \psi$, a simplified form of Schrodinger's equation in one spatial dimension. ...
2answers
99 views

### Quantum uncertainty can explain the Riemann Hypothesis?

In the recent paper "Riemann Hypothesis as an Uncertainty Relation" (http://arxiv.org/abs/1304.2435) the author claims that the presence of zeros out of the critical line may lead to the violation of ...
2answers
89 views

### Bessel function recursion relation

I'm reading a paper and the following set of radial equations is derived: $-i \lbrack \partial_r + \frac{1}{r} \left( \frac{1}{2} - \nu \right) \rbrack u(r) = \pm k v(r)$ \$ -i \lbrack \partial_r ...
0answers
49 views

### At large times, $\sin(\omega t)$ tends to zero?

While doing a calculation in quantum mechanics, I got a expression $\sin(\omega t)$, and my prof said if I consider the consider at large times, then i can assume that this goes to zero because at ...
1answer
18 views

### Expectation value of pure state in quantum mechanics

It's well known that in quantum mechanics, the expectation value of a self-adojint operator $A$ in pure state $|\psi\rangle$ is \$\langle\psi |A|\psi\rangle = \operatorname{Tr}(A |\psi \rangle ...
1answer
61 views

### A strange quantum potential: $V(x) = \frac{x^2}{5}+\mu \left(\left\lfloor x+\frac{1}{2}\right\rfloor \right).$

So I have a strange quantum potential I have been playing with: $$V(x) = \frac{x^2}{5}+\mu \left(\left\lfloor x+\frac{1}{2}\right\rfloor \right).$$ where $\mu$ is the Möbius function. This is what ...
0answers
14 views

### Opposite vector state in Bloch Sphere

I will like understand geometrically Why the opposite vector coordinate $\theta$ of $|\psi\rangle = \cos (\theta/2)|0\rangle + \exp(i\phi)\sin(\theta/2)|1\rangle$ is $\pi - \theta$. Anybody will be ...
0answers
39 views

### Doubt about the spectrum of an operator

I consider the Laplacian operator $$A=-\Delta$$ in the domain $$H^2(\mathbb{R}^3)$$ where it is selfadjoint. We know that its spectrum is $[0,+\infty)$. Now I want to consider the restriction of $A$ ...
0answers
28 views

### Three body problem with point interactions

I've studied the HVZ theorem for the three body problem interacting with regular potentials. I'd like to extend this result to the three body problem with point interactions (delta potentials). Is ...
1answer
56 views

### Tensor and Kronecker product

I have a follow exercise. Let $\left|\Psi\right\rangle = \tfrac{1}{\sqrt{2}}\left(\left|0\right\rangle + \left|1\right\rangle\right)$. Write out $\left|\Psi\right\rangle^{\otimes 2}$ xplicitly, both ...
0answers
60 views

### Translation of an article

I need to read this article "On the spectrum of an energy operator for atoms with fixed nuclei in subspaces corresponding to irriducible representations of permutation groups" authors:G.Zhislin, A. ...
1answer
77 views

### Is the partial trace congruent under a change of basis?

My intuition tells me that the partial trace should be congruent under a change of basis. That is, if I have some matrix $A$ in the space of linear operators acting on a joint hilbert space: \$A \in ...
0answers
25 views

### References for three body problems with Fermi statistic

I'm studying the three body problem with two fermions of unitary mass and another different particle. I need references of the HVZ theorem in this case. Is there someone who knows them?
1answer
44 views

### Dirac's notation? (QM)

I have a question regarding Dirac's notation in quantum physics. As far as I understand: $\langle a|b\rangle=(a1^*,a2^*)*(b1,b2)^T$ But what does $\langle1/2,1/2|J|1/2,-1/2\rangle$ mean?
0answers
46 views

### Fourier transform of integral kernel of the free resolvent

The free resolvent in $\mathbb{R}^3$ has this rapresentation $$(R_0(z)f)(x)=\int_{\mathbb{R}^3}\frac{e^{i\sqrt{z}|x-y|}}{4\pi|x-y|}f(y)dy$$with $\Im \sqrt{z}>0$. Then its integral kernel is ...
1answer
35 views

### General theory behind ladder operators

To derive the representation of SO(3) one uses the ladder operator method. What is the theoretical basis for this method? Often the ladder operators are simply stated in the textbooks of quantum ...
0answers
32 views

### Solvable Models In Quantum Mechanics

Is there anyone who studied on the book "Solvable Models In Quantum Mechanics" by Albeverio? I don't succed in understanding the proof of page 116 about the eigenvalues of the Hamiltonian with point ...
0answers
35 views

### Qubit state finding

Suppose we have two qubits in the state $x|00\rangle+y|11\rangle$. What is the resulting state of the second qubit in that case? Use and to denote and respectively.
1answer
34 views

### Is multiplying by a measurable function $V$ always self-adjoint?

There are a handful of results establishing conditions on the measurable real-valued function $V(x)$ under which the operator: $$-\Delta + V(x)$$ Is (essentially) self-adjoint on ...
1answer
1k views

### Qubits and vector projections

In $\Bbb C^2$, how many real unit vectors are there whose projection onto $|1\rangle$ has length $\sqrt{3}/2$? I would think zero as $\bigl(\frac{\sqrt{3}}{2}\bigr)^2 + x^2 = 1$, therefore there are ...
1answer
101 views

### Question on complex number calculation for transmission coefficient of finite potential well

This is actually in my quantum mechanics textbook (pure math question though), and I just cannot see why this equality is true. Any help would be greatly appreciated! Let $F$ and $A$ be nonzero ...
1answer
90 views

### Kernel of adjoint operator

This problem is puzzling me, even though it should be really simple. Let $L=-\partial_x^2 + \frac 1 2 x^{-2}$ be an operator defined on $D(L)=C^\infty_c(0,+\infty)\subset L^2(0,+\infty)$. Its adjoint ...
0answers
24 views

### If $H=\frac{P^2}{2m}+V(X)$, prove $\mathbb{E}_\psi(\frac{P^2}{m})=\mathbb{E}_\psi(XV'(X))$

Suppose we have a quantum mechanical Hamiltonian $$H=\frac{P^2}{2m}+V(X)$$ How can we show that $$\mathbb{E}_\psi(\frac{P^2}{m})=\mathbb{E}_\psi(XV'(X))$$ Any help would be greatly apprecaited
1answer
28 views

### Essential selfadjointness preserved under unitarily transfomration?

I am wondering if essential selfadjointness of an operator in a Hilbert space is preserved under unitarily transformations. In other terms: let $H,H'$ be two isomorphic Hilbert spaces, with an ...
5answers
90 views

### General solution of $\frac{d^{2}u}{d\rho^{2}}=\frac{l\left(l+1\right)}{\rho^{2}}u$

In his discussion of the radial wave function of hydrogen Griffiths (Introduction to Quantum Mechanics, 2nd ed, p.146) gives the general solution ...
2answers
79 views

### Show $e^{-a\sigma_3}\sigma_1e^{a\sigma_3} = \sigma_1e^{2a\sigma_3}$

How do you show that $$e^{-a\sigma_3}\sigma_1e^{a\sigma_3} = \sigma_1e^{2a\sigma_3}$$ where $\sigma_i$ are the Pauli matrices.
0answers
38 views

### Can a Hermitian operator on a tensor product space be represented as a sum of tensor products of Hermitian operators?

Consider a Hilbert space (or just a vector space over $\mathbb{C}$), which is a tensor product of several smaller Hilbert spaces: $$H = H_1 \otimes \cdots \otimes H_n,$$ and let $\mathcal{H}$ be a ...
0answers
31 views

### A non-integrable representation of the Heisenberg Algebra

Let $\mathfrak h$ be the Heisenberg algebra in dimension 1, generated by vectors $P$, $Q$ and $I$ satisfying $[P,Q] = I$, $[P,I] = [Q,I] = 0$. A representation of $\mathfrak h$ on a Hilbert space $X$ ...
2answers
188 views

### “proof” A is a Hermitian Matrix

For an arbitrary complex matrix A show that $$A*A^\dagger$$ is Hermitian. Where the dagger "$\dagger$" stands for the "complex conjugate and transpose" operators. From what I understand this must ...
4answers
186 views

### Applications of Operator Algebras to modern physics

I think that recently I've started to lean in my interest more towards operator algebras and away from differential geometry, the latter having many applications to physics. But while taking physics ...
2answers
174 views

### Instance of Ehrenfest's Theorem

Please Help me to fill in the gaps Show $$\frac{\text d \langle {p} \rangle}{ \text{d} t} =\left\langle - \frac{ \partial V }{\partial x} \right\rangle .$$ \frac{\text d \langle {p} ...
1answer
261 views

### triple integral (quantum mechanics)

I recently started a quantum mechanics course after a long time with no serious maths and I'm having some problems with the most basic maths operations. Please, help me solve this triple integral ...
0answers
74 views

### Fourier transform of $\frac{g_i}{e^{\frac{\epsilon_i-\mu}{kT}}-1}$? Not Gaussian like with Fermi-Dirac statistics?

This equation $\bar n_i=\frac{g_i}{e^{\frac{\epsilon_i-\mu}{kT}}-1}$ is Fermi-Dirac statistics where variables are defined here. The classical equation i.e. the Maxwell Boltzman equation is Gaussian ...
0answers
48 views

### Convolutions of Path Integrals of Gaussian Functions

I was looking at a question on a physics forum (http://physics.stackexchange.com/questions/45955/splitting-light-into-colors-mathematical-expression-fourier-transforms) and I wanted a more ...
2answers
152 views

### Solving the time-independent Schrodinger equation for particle in a potential well

I'm solving a quantum mechanics problem for the particle in a potential well, and the equation I have to solve is $$\frac{d^2\psi}{dx^2}+k\psi=0$$where $$k=\frac{2mE}{\hbar^2}$$ This seems easy enough ...
0answers
50 views

### Applying boundary conditions on the general solution of the time-indpendent Schrodinger equation

I'm using the boundary conditions for the general solution of the time-independent Schrodinger equation for a particle in a potential well, and I get the following 2 equations: ...
2answers
167 views

### “Fun” question: anyone know why $e$ (Euler's Number) was chosen for wave functions?

First, let me say that this is merely something I have always wondered about, and can never seem to find a good reference for. I simply want to know... the geek in me. Why was $e$ (Euler's Number) ...
1answer
32 views

### Manipulating derivatives after substitution: $\xi=\gamma x$

I am following a quantum mechanics text book which uses a simple looking substitution in a derivative. The substitution is $$\xi=\gamma x\tag1$$ It then says that ...
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