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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23 views

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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Checking an “equivalence of products” in the notes on deformation quantization by Allen C. Hirshfeld and Peter Henselder

These notes are a great introduction to deformation quantization but I failed to check the validity of the statement p.9, right before (5.18). Context: let $(\mathcal{A},+,\mu)$ be an algebra. $\mu:\...
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Proof using taylor series

I am currently trying to solve a quantum mechanics problem in which i need to prove that $e^A e^{-A} = 1$ where $A$ is an operator and the exponent function is defined by a taylor series. However, I ...
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1answer
28 views

Integrate using given formula

I would like to ask for help with exercise I was given on QM. I am suppose to calculate: $$ \int_{0}^{L}|A|^2|x(L-x)|^2 dx = |A|^2\int_{0}^{L}(x(L-x))^2 $$ That is not a problem of course. I am ...
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Deficiencs index of symmetric operator is locally constant

Let $A: D_A \rightarrow H$ be a symmetric linear operator defined on $D_A \subseteq H$. Define the deficiency index of $A$ at $z \in \mathbb{C} \setminus \mathbb{R}$ to be $\dim( \ker (A^* - \bar{z}))$...
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57 views

Compute the Resolvent of derivative with domain $H^1(0,\pi)$ where $f(0)=0$ [closed]

So this is a question from Teschl G. - Mathematical methods in quantum mechanics- Problem 2.19 \ Compute the resolvent of $$Af=f' \enspace D(A)=\{f\in H^1[0,1] \enspace | \enspace f(0)=0\}$$ and show ...
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25 views

Output register in Shor's algorithm

In Shor's original paper he creates two registers, sets the first to uniform superposition for each possible number $a \text{(mod $q$)}$, and then computes $x^a \text{(mod $n$)}$ into the second ...
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1answer
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Show that an hermitian operator is represented by an hermitian matrix.

Let $\hat{A}$ be an hermitian operator. I have to show that $\overline{A_{\alpha\beta}}=A_{\beta\alpha}$ So, if an operator is hermitian then $\hat{A} = \hat{A}*$ I started with this: $\hat{A}*=\sum_{\...
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51 views

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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Does the time-independent Schrodinger equation has an exact and general solution?

The Schrodinger equation (time-independent) is for sure the most important equation in quantum mechanics; $$-\frac{ℏ^2}{2m}∇^2ψ(x)+V(x)ψ(x)=Eψ(x)$$ let consider the one-dimensional equation; $$\frac{d^...
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Is the measurement postulate different for infinite dimensional Hilbert spaces?

When reading about the measurement postulate the papers always seem to be specific to discrete measurements although after watching this lecture on MIT open courseware https://www.youtube.com/watch?v=...
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Help with demonstration that the system $|p\rangle$ is complete

I have the following demonstration $\langle x|x'\rangle = \langle x|(\int{|p\rangle \langle p|dp})|x'\rangle = \int{\langle x|p\rangle\overline{\langle x'|p\rangle}dx}=\int{e_{p}(x)\overline{e_{p}(x')}...
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25 views

Representation of an operator in another base

We have the following operator $\hat{H} = a|u_{1}\rangle \langle u_{2}| + a|u_{2}\rangle \langle u_{1}|$, with a = real constant and $|u_{i}\rangle $ an orthonormal base. The matrix representation, H, ...
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Computing a certain functional derivative.

I'm trying to see what happens when we extended the notion of a convex roof definition to infinite dimensions. In finite dimensions if we can define an entanglement measure $e$ on pure states of $\...
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1answer
16 views

Show that the coefficients satisfy the following condition

Hi I was asked for my QM homework to show that coefficients $c_n$ in superposition $$ \psi(x) = \sum_{n=1}^\infty c_n\psi_n(x) $$ Satisfy the following condition $$ \sum_{n=1}^\infty \lvert(c_n)\rvert^...
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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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21 views

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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32 views

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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91 views

Differential equation in $ L^2(\mathbb{R}^n) $ related to the equality condition for uncertainty principle

Define closed operators $ X_j $ and $ D_j $ on $ L^2(\mathbb{R}^n) $ as follows: \begin{align*} X_j f = x_j f &\quad \text{for $ f \in L^2(\mathbb{R}^n) $ such that $ x_j f \in L^2(\mathbb{R}^n) ...
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Spectral theory and Taylor expansion in quantum mechanics for unbounded operators

I am reading spectral theory and quantum mechanics. We know that for an unbounded self-adjoint operator $A$, operator-valued functions such as $\exp(iAt)$, $t\in\mathbb{R}$ can be defined using ...
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1answer
34 views

Can we solve a matrix equation when the vectors are given and the matrix is variable?

Usually, a matrix equation means that $$ Ax = b $$ when A and b are given and x is the variable we want to know. However, when x and b are given and we want to know the value of the matrix, is it ...
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31 views

How to understand the indexing in Tensor product?

Recently, I learn the tensor product in physics. Here are some equations I meet: $$X_{\gamma \omega \rho \sigma}=\sum_{\alpha, \beta, \delta, \nu, \mu} A_{\alpha \beta \delta \sigma} B_{\beta \gamma \...
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Is this notation to denote how an equation has been manipulated to produce a resulting equation standard?

In the following image I am specifically referring to the step that produces the result Eq. (4.3). Is the notation off to the side $| \;\,\cdot \frac{1}{\psi(x)f(t)}$ in this step standard? (To anyone ...
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1answer
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Trouble Deriving the Canonical Commutation Relation from the Product Rule

From pg. 74 of No-Nonsense Quantum Mechanics, the author derives the canonical commutation relation from the momentum operator $\hat{p}_i$ as follows: Question: How does the product rule (for ...
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31 views

Proving $H$ is self-adjoint on bosonic Fock space

Let $x_{1},...,x_{n} \in \mathbb{R}^{3}$ and $\mathcal{H}_{n}$ be defined by: \begin{eqnarray} H_{n} := \sum_{i=0}^{n}\bigg{(}-\frac{\hbar^{2}}{2m}\nabla_{x_{i}}+W(x_{i})\bigg{)} + \frac{1}{2}\sum_{i\...
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On the boundary conditions of black body problem

The black body problem tries to find the spectral energy density of electromagnetic field per unit volume in a isolated cavity with electromagnetic free radiation (i.e., with no sources) The free ...
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1answer
41 views

Evaluating the energy of a wavefuntion (time independant GPE)

Im looking at a condensate in a spherically symeric trap so the potential is $$V(r)=\frac{m}{2}\omega_r^2r^2$$ When the interactions are weak we assume this wavefunction: $$\psi(\boldsymbol{r})=\left(\...
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31 views

Trotter product formula with complex exponent

If $A$ and $B$ are bounded operators, one version of Trotter product formula is: $$e^{A+B} = \lim_{n\to \infty}\bigg{(}e^{i\frac{A}{n}}e^{i\frac{B}{n}}\bigg{)}^{n}$$ where the limit is with respect to ...
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3answers
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Stuck and confused with $\int \frac{u^{1/2}}{(u+1)^2} du$

I'm trying to find the expectation value $\langle x^2 \rangle_{\psi}$ for $$\psi(x,0) = \sqrt{\frac{2}{\pi}}a \cdot \frac{1}{x^2+a^2}$$ and I know the result is $a^2$. I'm terribly stuck integrating ...
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11 views

Transform a constrained multidimensional integral into an unconstrained one

A quantum-mechanical "density" matrix is Hermitian (self-adjoint), positive definite, having trace 1. In terms of its four ordered eigenvalues ($\lambda_1 \geq \lambda_2 \geq \lambda_3 \geq \...
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1answer
41 views

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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Proving how to apply the adjoint to a dyad

I have to prove that $(|\Psi\rangle \langle \Phi|)^\dagger= |\Phi\rangle \langle \Psi|$ Hint: Prove that $\langle f|(|\Psi\rangle \langle \Phi|)^\dagger|g\rangle = ... = \langle f||\Phi\rangle \...
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Help with geometrical interpretation of dyad operators

$a)$ Let $\hat{A} = |e_{1}\rangle \langle e_{3}| + |e_{3}\rangle \langle e_{1}|$. Consider the space $\mathbb{R}^3$, what does $\hat{A}$ mean? $b)$ Let $\hat{B} = |e_{1}\rangle \langle e_{1}| + |e_{3}\...
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1answer
19 views

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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41 views

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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1answer
30 views

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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52 views

Questions about evaluating Dyson series

I have some questions about Dyson series in quantum theory. For perturbed hamiltonian $H=H_0+\lambda V(t)$, we have the following differential equations in interaction picture. $$ i \hbar \frac{...
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1answer
60 views

c-numbers, a-numbers, and q-numbers

I had heard of c-numbers, a-numbers, q-numbers: c-numbers: real or complex numbers as commuting numbers https://en.wikipedia.org/wiki/C-number a-numbers: anti-commuting numbers q-numbers: operators ...
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1answer
16 views

Are the multiplications in $\langle v | \bar U^{\operatorname{T}} U | v \rangle$ commutative?

I'm learning about operations on qubits, and I came across this statement: Suppose $|w\rangle = U |v\rangle$, and we want $U$ to preserve state norms. Then $\langle w|=\langle v|\bar U^{\operatorname{...
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2answers
57 views

How Different is an Eigenvalue Problem from an Ordinary Differential Equation

I have been thinking about how different an eigenvalue problem such as that of the Sturm-Liouville Equation (SL) with that of a second-order linear differential equation. It doesn't seem to be the ...
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33 views

Prove convergence of series under trace-norm topology

Let $H$ be a separable Hilbert space, $T(H)$ the set of trace-class operators, and $D(H)\subset T(H)$ the set of density operators (i.e., positive and having trace 1). For any unit vector $\vert \...
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1answer
49 views

Finding an Operator from its Commutator

I'm trying to figure out if it's possible to derive a general form for the linear operator $L_z$(rotational momentum in Quantum mechanics) which is a combination of the four linear operators $$X, Y, ...
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1answer
55 views

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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1answer
57 views

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 ...
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1answer
35 views

Compute the commutator, and compare it to quantization of the Poisson bracket.

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 $$\...
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1answer
30 views

Translation operator unitarily equivalent to multiplication by exponential

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 ...

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