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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Expectation value in a Quantum derivation

I'm reading a physics paper (John Bell's 1964 paper on the EPR paradox if anyone is physics-curious) and I'm having an issue following his derivation. It's the probability distribution stuff -not ...
3
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0answers
171 views

Solving Schrodinger for harmonic oscillator(griffiths analytic method)

I was just getting into quantum mechanics. But I'm having a bit of trouble following griffiths for the analytic method... it goes like so The Schrodinger equation $$ -\frac{\hbar^2}{2m}\frac{d^2\psi}{...
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98 views

The intuition behind a matrix of a Hamiltonian?

We have derived an elegant partition function for a problem which resembles a quantized model taking the particles to be Bosons. The related Hamiltonian for every $i$th ensemble is there: $$H_i=\sum_{...
3
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1answer
163 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 ...
3
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0answers
125 views

Defining entanglement in subspaces of tensor product

Let $\mathcal{H}=\mathbb{C}^n$ be a Hilbert space. A state $\rho\in\mathcal{B(H)}$ is a positive semi-definite operator with unit trace. $\rho\in \mathcal{B(H)}$, where $\mathcal{H}=\mathcal{H}_1\...
3
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1answer
81 views

Positive Operator Value Measurement Question

I'm attempting to understand some of the characteristics of Posiitive Operator Value Measurement (POVM). For instance in Nielsen and Chuang, they obtain a set of measurement operators $\{E_m\}$ for ...
2
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3answers
107 views

How to do this integral $\int_{-\infty}^{\infty}{\rm e}^{-x^{2}}\cos\left(\,kx\,\right)\,{\rm d}x$ [duplicate]

How to do this integral $$\int_{-\infty}^{\infty}{\rm e}^{-x^{2}}\cos\left(\,kx\,\right)\,{\rm d}x$$ for any $k > 0$ ?. I tried to use gamma function, but sometimes the series doesn't converge.
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2answers
1k 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 ...
2
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2answers
806 views

A vector space with countable and uncountable basis at the same time

Let $V$ be a vector space over $\mathbb{C}$. Two self-adjoint, commutable linear operators $\xi$ and $\eta$ act on it. Both of their eigenvectors form a complete set of $V$, but $\xi$'s eigenvalues ...
2
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1answer
568 views

Arbitrarily using Sin and Cos as eigenfunctions of a Hamiltonian?

In the context of quantum optics, the rotating wave Hamiltonian can be written: $\hbar\begin{pmatrix} -\Delta & \Omega/2\\ \Omega/2 & 0 \end{pmatrix}$ The eigenvalues can then be calculated ...
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3answers
42 views

Example of a Projection Operator in $\mathbb{R^3}$

I'm looking for an operator $\hat P$ in $\mathbb{R^3}$ such that $\hat P^2=\hat P$ that is also Hermitian
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2answers
91 views

Understanding Bell's inequality vs. quantum mechanics

I have difficulty to understand how Bell's inequality rules out local hidden variable theory. It seems to me that there is some hidden variable in the Kolmogorov's axiomatization of probability theory,...
2
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2answers
275 views

Trace in non-orthogonal basis

In Dirac notation we can define the trace of an operator in Hilbert space $\rho$ as the follows, $Tr(\rho)=\sum\limits_{|s\rangle \in B} \langle s| \rho |s\rangle$ where B is some orthonormal ...
2
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2answers
226 views

Perturbation theorem of Weyl

Does anyone know where to find something about the perturbation theorem of Weyl, preferably on the internet. The theorem I'm talking about states: let $A$ be a self-adjoint operator on a Hilbert ...
2
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1answer
96 views

What does $\operatorname{supp}(A)$ mean?

I'm looking at a paper (specifically this one). In the paper, we have a positive operator $A$, and the operator $\operatorname{supp}(A)$ is supposed to be a projection operator. Does anybody know ...
2
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3answers
341 views

Why does the Method of Successive Approximations for a Differential Equation work?

Time dependent perturbation theory in quantum mechanics is often derived using the Method of Successive Approximations for a Differential Equation. I have not seen an explanation or a more rigorous ...
2
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1answer
224 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 ...
2
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2answers
115 views

Is there a reason for the similarity between $\exp(-x^2-x^4-x^6)$ and $\cos(0.5\pi x)$

I was wondering whether the similarity between the functions $\exp(-x^2-x^4-x^6)$ and $\cos(0.5\pi x)$ was due to some more fundamental limiting relation between the two functions (or similar ...
2
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1answer
81 views

Quantum Mechanics Project Ideas!!! [closed]

I am in my first year in uni and I have to write a project in Quantum Mechanics. But I have been struggling with an idea for the project since I have recently started studying quantum mechanics and my ...
2
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1answer
48 views

Eigenvalue equation and separable solutions

Recently, studying Quantum Mechanics I found a doubt regarding separable solutions and eigenvalue equations for differential operators. Suppose we are considering some space $\mathcal{H}$ of functions ...
2
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1answer
45 views

How to calculate this Delta Dirac Integral?

I'm trying to understand the result of this integral. $$\int_{-\infty}^a\delta(x-b)\,\mathrm dx$$ The book gives me the following answer. If $a>b$, $1$; If $a<b$, $0$; I'm not sure but ...
2
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2answers
78 views

A question on use of square integrable functions

I'm approaching this from a physicist's perspective, so apologies for any inaccuracies (and lack of rigour). As far as I understand it, a square-integrable function $f(x)$ satisfies the condition $$\...
2
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1answer
36 views

Zero Tensor Product

Suppose we have a space $|\psi_1\rangle \otimes |\psi_2\rangle \otimes |\psi_3\rangle$, and operators (matrices) A ⊗ B ⊗ C acting on this Hilbert space (like in quantum mechanics). I'm trying to ...
2
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2answers
54 views

Can someone explain the notion of “unbounded” operator as simple as possible?

I've read about these operators in quantum mechanics, but I have never seen them in action. I think that is because I absolutely do not intuitively understand this concept. I've read some stuff online ...
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2answers
106 views

Question about Dirac notation

So from what I understand $\langle w | v \rangle=\vec w^* \cdot \vec v$. Ok. I'm fine with that notation. But then I've seen $\langle x | y \rangle=\delta(x-y)$ and $\langle x | p \rangle=e^{-ixp/\...
2
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1answer
169 views

How can we show that a map is a completely positive map?

I am doing a homework problem where I have to find whether the map $$ \rho ~\rightarrow~ {\rm tr}(\rho) I - \rho $$ is completely positive. If the map is not completely positive, a counter-example ...
2
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1answer
84 views

Quantum Mechanics- Quantum Zero Paradox

Im struggling with this question after doing other question on my example sheet. I'm struggling with showing both results in the question and help will be very much appreciated. Thanks
2
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1answer
85 views

Delta function proof in QM

I'm actually working with some QM problems at the moment but I've hit a wall with a delta potential involved. The problem asks me to verify that $$ \frac{d \phi_{x=0^{+}}}{dx} -\frac{d \phi_{x=0^{-}...
2
votes
1answer
151 views

Simon's algorithm for n = 3

The Simon's problem is as follows: Suppose we are given a function $f : \{0, 1\}^n \to \{0, 1\}^m$, with $m \ge n$, and we are promised that either $f$ is 1-to-1, or there exists a non-trivial s such ...
2
votes
2answers
60 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 \...
2
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1answer
316 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 ...
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1answer
137 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 $L^2(\mathbb{R}^n)$....
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2answers
392 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 ...
2
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2answers
220 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} \rangle}{...
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1answer
131 views

How are Tr(AB) results restricted?

In quantum mechanics one postulates that for each state $i$ there is a matrix $A_i$ and for each measureable dynamical variable $j$ (velocity etc.) there is a matrix $B_j$. Both are Hermitian matrices ...
2
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1answer
22 views

Computate the commutator $[p^n,x]=-ihnp^{n-1}$

Computate the commutator of $[p^n,x]=-ihnp^{n-1}$. With $p=-ih \frac{\delta}{\delta x}$ the impulse operator. $h$ stands for $\frac{h}{2\pi}$. Answer: I do it with induction over $n$. For $n=1$ it ...
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1answer
40 views

A question regarding Eigenvalues

Note: $\psi,\psi^{\dagger} :\Bbb{R} \to \Bbb{C}$ and $x, \lambda_i , \hbar, m \in \Bbb{R}$ Say we know that $\lambda_1$ is a solution to the eigenvalue equation: $$\hat{\Pi}\psi(x)= \lambda_1 \psi(x) ...
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2answers
39 views

Different definitions for the complex inner product.

I have asked this question on P.S.E. and have gotten some nice answers, but I felt I might get even more satisfactory answers if I post it here. "I have just now noticed that Griffiths (in his book ...
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2answers
52 views

Is there a way to factor out the middle tensor product?

Given a state $(|1\rangle \otimes |0\rangle \otimes |1\rangle) + (|0\rangle \otimes |0\rangle \otimes |1\rangle)$, is it possible to factorise out the $|0\rangle$ in the middle of both of them? ...
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2answers
43 views

Can I allow a value of n to be defined if that value of n gives a 1/0 , BUT that 1/0 has another 1/0 that cancels it out?

I'm working through an integral for Quantum Mechanics 2, Harmonic oscillator with time-dependent perturbation, and I have encountered this situation when evaluating the integral. The part in ...
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2answers
53 views

Delta functions as eigenstates

In Quantum mechanics, it's standard to say that the eigenstate of the position operator in 1D is the Dirac delta function. More formally, define a linear map $\hat x$ on some enhanced version of $L^2(...
2
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1answer
39 views

Calulate the eigenvalues and the eigenstates

An observable is given by $$\sum\limits_{n= 1}^N a_n|a_n\rangle\langle a_n | $$ Here $\langle a_n |a_m\rangle = \delta_{nm}$. What are the possible measurement results corresponding to the operator A ...
2
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1answer
68 views

Distinct ways three integers can sum to a constant

So I am doing some quantum mechanics and it has led to some combinatorics. I need to know how many distinct ways I can have $N_1+N_2+N_3=N$ where $N$ is fixed so we can change $N_1$, $N_2$ and $N_3$. ...
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1answer
36 views

Evaluation of Operator-Valued Function

Hello all; above is my question! :) I've gone through all the way up to the final "and hence deduce that". Up to this point, the question has been fairly straightforward, but I have no idea how to ...
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1answer
43 views

Unitary transformation between complete and orthonormal bases

I'm using the Dirac notation for vectors here, since this is a quantum mechanics question. Suppose the complete orthonormal bases $\{|\psi_n\rangle\}$ and $\{|\psi{'}_n\rangle\}$ are related by the ...
2
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1answer
115 views

Quantum Hamiltonian commuting with the Pauli-Runge vector.

I have to prove that $[A_j, H] = 0$, with; $$\vec{A} = \frac{1}{2Ze^{2}m}(\vec{L} \times \vec{P} - \vec{p} \times \vec{L}) + \frac{\vec{r}}{r}$$ $$H = \frac{p^2}{2m} - \frac{Ze^2}{r}$$ And, $Z, e, m$...
2
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1answer
217 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, ... x_n]_r\,...
2
votes
1answer
88 views

boundary conditions for Schr$\ddot{\textrm{o}}$dinger equation in 2D polars?

What are the boundary conditions at $r=0$ for Schr$\ddot{\textrm{o}}$dinger's equation for a quantum particle in 2D polars $(r,\theta)$, with potential $U=0$ for $r<a$ and $U=\infty$ for $r>a$? ...
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16 views

integral of product of three basis functions and Clebsh-Gordan coefficients

Suppose I have an orthonormal basis $\{b_i\}_{i=1}^\infty$ for an $L_2$ space (for example, the $b_i$ could be spherical harmonics on the round sphere with the Euclidean $L_2$ inner product). I want ...
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19 views

Commutative diagram for hidden subgroup representation of graph automorphism

The hidden subgroup representation of the graph automorphism problem is defined in the section 10.2 of QUANTUM ALGORITHMS FOR PROBLEMS IN NUMBER THEORY, ALGEBRAIC GEOMETRY, AND GROUP THEORY. It is as ...