# Questions tagged [quantum-mechanics]

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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### Compute the PVM (Projection Valued Measure) of Parity Operator

This question is duplicate of the question Find projection-valued measure associated with parity operator.\ But in that question @Jacky Chong does not state how he found the operator \begin{align} P_\...
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### How to solve the eigendifferential integral for a continuous eigenvalue spectrum?

I don't understand how to solve the following integral in order to obtain the reported solution found in the book "Fundamentals of atomic mechanics" written by Enrico Persico. The following ...
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### Best mathematics books to study quantum mechanics and engineering

I am an electrical engineer. I want to study quantum mechanics.So, can anyone suggests me some of the best mathematics books to study quantum mechanics and electrical engineering.
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### Calculate the Euler-Lagrange for a functional with two nested integrals?

I've been reading papers about a fairly unknown topic in quantum mechanics called the quantum backflow effect. And in many of the papers they find an eigen value problem corresponding to the maximal ...
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### Find H for a particle in an infinite well

So I have this problem: 1- A particle in an infinite square well has a wave function that is $\psi(x,0)=A[\psi_1(x)+\psi_2(x)]$ with $\psi_n$ the n-th steady state. a)normalize $\psi(x,0)$ I already ...
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### Understanding a proof that “the columns of a unitary matrix are orthonormal”

Goal: Let $u_i$ and $u_j$ be the $i$th and $j$th columns of unitary matrix $U$, respectively. We wish to show that $$\langle u_i, u_j \rangle = 0, i \ne j \\ \langle u_i, u_j \rangle = 1, i = j \\$$...
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### Help with eigenvectors of an operator

We have the following operator $\hat{A} = 2|u_{1}\rangle \langle u_{1}| + 2|u_{2}\rangle \langle u_{2}| + 1|u_{3}\rangle \langle u_{3}|$ with $|u_{i}\rangle$ an orthonormal base. The matrix ...
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### Open questions in formalization of quantum theory

From time to time I come across the statement that the mathematical formalization of the mathematical instruments, which the quantum theory uses, is not complete yet (admittedly, more often in popular ...
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### Small turn in Hilbert space: Does it make vertical vectors?

Can you help me with this problem? Suppose that we have a vector $r\in\mathbb{R}^3$, that makes a small turn in the 3d space, which results in $r+dr$. Show that if $dr\to 0$, the vectors $r$ and $dr$ ...
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### Help with proving ($\langle$i[$\hat{A}$, $\hat{B}$]$\rangle$)$^{2}$ = |$\langle$[$\hat{A}$, $\hat{B}$]$\rangle$|$^{2}$ [closed]

Let $\hat{A}$ and $\hat{B}$ be hermitian operators. Show that ($\langle$i[$\hat{A}$, $\hat{B}$]$\rangle$)$^{2}$ = |$\langle$[$\hat{A}$, $\hat{B}$]$\rangle$|$^{2}$
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### Show that this operator is Hermitian? [closed]

The operator is defined as followed: $\hat A = a|u_{1}\rangle\langle u_{1}| + b|u_{2}\rangle\langle u_{2}|$ $a)\qquad a = b = 1$ $b)\qquad a = i, b = -i$
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### Inner product for elements of Fourier basis

Assuming I have a infinite dimensional continuous Hilbert $\mathcal{H}$ space and an orthonormal basis $\{|\,x\,\rangle\}_{\mathbb{R}}$, I want to transform this basis with the Fourier transform to ...
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### Reducing the expression for the Lambert $W(w e^{-w})$

I was solving an equation in Mathematica and I stepped in the following expression (2 a A m + h^2 ProductLog[(2 a A E^(-((2 a A m)/h^2)) m)/h^2])/(2 a h^2) Which ...
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### When can you have discontinuous solution to Schrodinger equation?

For a usual KE+PE Schrodinger equation (which is just a second order ODE), I know that you cannot have a physically preparable discontinuous wavefunction solution. However I am interested in ...
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### Delta function integral tricks

Often in physics we have integrals like the following containing delta functions inside derivatives. Eventhough I know the 'correct' way to compute this integral is to integrate by parts until the ...
Can someone please provide an insightful solution or verification of my solution to this question: Compute the commutator $$[p^2,q^2]$$ (in the one variable case), and compare it to quantization \...
This is part of a problem from Hall's book "Quantum Theory for Mathematicians". Determine the unitary operator $U:L^2(\mathbb{R^n})\to L^2(\mathbb{R^n})$ (unique up to a constant) such that ...