Questions tagged [quantum-mechanics]
For questions on quantum mechanics, a branch of physics dealing with physical phenomena at microscopic scales.
1,643
questions
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Prove eigenvectors are orthogonal. [closed]
Assuming the eigenvalues of a Hermitian matrix are non-degenerate,
prove that the corresponding eigenvectors are orthogonal.
0
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0
answers
24
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Proving that the limit of a given test function is a valid delta function
I am a physics student trying to gain a better mathematical understanding of the theory of distributions and namely the definition of the Dirac-Delta function. I understand that the defining properly ...
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How to represent elements from $\bar{E}=E/Z(E)$ in the form $(a|b)$
From https://arxiv.org/abs/quant-ph/9608006
Background
The group $E$ of tensor products $\pm w_{1} \otimes \dots \otimes w_{n}$ and $\pm i w_{1} \otimes \dots \otimes w_{n}$, where each $w_{j}$ is one ...
- 261
3
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0
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Transform a differential equation into Hamiltonian form
I am currently doing the following exercise in the book Modern Geometry - Methods and Applications Part I by Dubrovin, Fomenko, and Novikov.
Exercise 33.4.1: Consider the differential equation
\begin{...
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0
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29
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Having difficult with this integral [closed]
∫e^[−i(ax−bx^2)]dx
I have not idea how to do this integral
1
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0
answers
25
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Example of statistical data with the property of being contextual that is generated by quantum mechanics
Definition for the specifics of the question as well as an example of contextuality in quantum mechanics
I have a set of measurements acting on a 2 qubit state for whom the statistics of the ...
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0
answers
62
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Is there any mathematically rigorous definition of deriving a matrix valued function with respect to one of its matrix argument?
On my way of satudying Heisenberg matrix mechanics, I get blocked by formulas engaging derivations with respect to a matrix arguments. My question is the following : Is there any mathematically ...
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3
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0
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57
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Need help with an analytic solution of an integral of a particle within an infinite potential well
While working on a homework problem related to an infinite potential well, I encountered the following integral:
$$ \int_0^L \sin\left(\frac{n\pi}{L}x\right) \sqrt{x(L-x)}\mathrm{d}x $$
This integral ...
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1
vote
1
answer
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Spectrum of a sum of self-adjoint operators
This is a "sequel" to that question where I explain why I need the spectrum of an operator given as the sum of a convolution and a function multiplication. Here, I am considering the ...
0
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0
answers
39
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Tricky "Divergent" Integral: Correction to Groundstate
I am trying to rederive the results presented in the paper, in particular equation (30). That is, I am trying to compute the correction to the ground-state energy of a dipolar condensate due to beyond-...
1
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1
answer
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Computing expectation value of Pauli-$x$ operator for a single qubit, given a Hamiltonian matrix.
I am working through some introductory quantum mechanics materials, and I am stuck on the following problem...
Suppose we are working on a single qubit and we are given the Hamiltonian to be $$H=\...
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3
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1
answer
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Discrepancy in Results with Self-Adjoint Operator on a Special Hilbert Space in 2D Geometric Algebra
I am exploring the behavior of multivectors in 2D geometric algebra, specifically examining the product $\mathbf{u}^\ddagger \mathbf{u}$, where $\mathbf{u}=a+xe_1+ye_2+be_{12}$ and its Clifford ...
- 2,551
2
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0
answers
54
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Integral inequality with bounded frequencies
Suppose $f\colon\mathbb R^d\rightarrow\mathbb R$ is Schwarz class and $\int f\,dx=0$. Let $\nu>0$ and let $P_\nu$ be the projection operator such that $\widehat{P_{\nu}f}(k)=\mathbb{1}_{\{|k|\leq \...
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1
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Analytically solving PDEs on irregular domains in Physics
In many Physics courses you solve PDEs like heat or wave on square, circular, or spherical domains with separation of variables. Are there ways to solve PDEs and Boundary value problems on irregular ...
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1
answer
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How to rearrange $py'' + (2l+2-2p)y' + 2(n-l-1)y = 0$ into $2py'' + (2l+2-2p)y' + (n-l-1)y = 0$?
I have a 2nd order homogenous ODE:
$$py'' + (2l+2-2p)y' + 2(n-l-1)y = 0$$
where y is a function of p and n and l are variables
Its solution is the associated Laguerre polynomials $L_{n-l-1}^{2l+1}(2p)$...
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2
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When solving a second order linear homogenous differential equation why does $i$ appear?
I am following a walkthrough solution to the schrodinger equation and the lecturer solved this second order differential equation (the phi part of the angular equation) like so:
$d^2y/dx^2 + m^2*y = 0$...
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1
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1
answer
72
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Suppose $tr(A(P\otimes Q))\geq0$ for all semi-definite positive matrices P and Q, does it implies that A is semi-definite positive? [closed]
Suppose $\text{tr}(A(P\otimes Q))\geq0$ for all semi-definite positive matrices P and Q, does it imply that A is semi-definite positive? If it is not true, please provide some ideas on restricting $A$ ...
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Computing Fourier transform of energy $E=-\nabla_xW*f$
If $W=\frac{1}{4\pi|x|}$ is the Coulomb potential and we define the energy to be $E=-\nabla_xW*f$, then I am trying to prove that $\widehat E=i\frac{k}{|k|^2}\widehat{f}$ on the Fourier side with ...
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1
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1
answer
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Integration of a composite function containing hermite polynomials.
I was trying to solve the quantum harmonic oscillator problem. It is almost done, just the normalization of the wave function is left.
$Ψ_n = β_n\mathcal{H}_n(\frac{x}{\alpha})e^{\frac{-x^2}{2\alpha^2}...
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votes
2
answers
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Compactness of subset of trace-class operators on a Hilbert space
Consider an infinite-dimensional, complex and separable Hilbert space $H$ and let $\mathcal I(H)$ denote the space of trace-class operators.
The set of density operators is defined by $$\mathcal S(H):...
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3
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2
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Operators and KdV equation
Consider
Given the linear differential operators
$$
L=-\partial_x^2+u(x, t), \quad A=4 \partial_x^3-3 u \partial_x-3 \partial_x u,
$$
considered as acting on a vector space of functions of $x$ verify ...
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$L^\infty$ decay for the Klein–Gordon equation
By Fourier transform, the flow of the Klein–Gordon wave flow is $e^{it\sqrt{1-\Delta}}$. That is, if we have initial data $\phi(0,x)=\phi_0(x)$, then the solution will be given by $\phi(t,x)=e^{it\...
- 307
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Applying an operator on both sides of an equation
I am doing a quantum mechanics question involving the positivity of the norm. So I'm using the fact that the norm will be greater than zero but i want to apply an operator onto the ket on one side of ...
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Show that $\hat{L}^2=\hat{r}^2\hat{p}^2 + i\hslash \hat{r} \cdot \hat{p} - (\hat{r} \cdot \hat{p})^2$ using Einstein Notation
Problem
Show that $\hat{L}^2=\hat{r}^2\hat{p}^2 + i\hbar \hat{r} \cdot \hat{p} - (\hat{r} \cdot \hat{p})^2 $ using Einstein notation and commutation relations in QM.
Attempt
We know that $\hat{L}_i =...
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Trouble understanding Berry's phase derivation
Following the the wikipedia article about the adiabatic theorem, and Sakurai's Modern QM, we start with the definition of the geometric phase that we get when doing a loop with a parameter R which ...
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John von Neumann theorem on self adjoint extentions
Let $H$ be a Hilbert space and $A:D(A)\subset H \rightarrow H$ be symmetric and closed.
Assume $A$ has a selfadjoint extention $B$. Then the Cayley transform of $A$ has also a unitary extention i.e. ...
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I'm confused about The Matrix Representation of Linear Operators Formula when the basis is not the standard basis
My Question/Work
I read that a Linear Operator in Quantum Mechanics can be represented as
A = $\sum_{ij} A_{ij}|u_{i}\rangle \langle u_{j}|$.
If A =
$$
\begin{pmatrix}
3 & 2\\
4 & 1
\end{...
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1
answer
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operation rules involving tensor product of perturbed wavefunctions
Let $i, j$ represent a wavefunction.
In Bra - Ket notation, I have an expression like
$\langle ij | g | i j \rangle $ which is also just $(\langle i | \otimes \langle j | ) | g | (|i \rangle \otimes |...
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Power Series Representation of matrix expression $(A - \lambda B)^{-1}$
I am a physics graduate student studying quantum mechanics, and in going over linear algebra fundamentals, we desired to come up with a power series representation of
$$
(A - \lambda B)^{-1}
$$
with A ...
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1
answer
70
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Representation of the Time Evolution Operator in QM
I was searched about the Time-evolution operator from time $t_0$ to $t$, denoted as $U(t,t_0)$ used in quantum mechanics, and which has the formula for a time-independent Hamiltonian operator $\hat H$:...
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Any reference for the mathematics of Quantum Mechanics with infinite degrees of freedom?
I am looking for a book, or lecture notes or even courses available on YouTube where there is a good and detailed discussion on the mathematical aspects of Quantum Mechanics with infinite degrees of ...
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1
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simplifying trace of product of two unitary operations
For Hermitian matrices $A$ and $B$, diagonalizing each unitary matrix $e^{iA}$ and $e^{iB}$ gives
\begin{equation}
\text{Tr}(e^{iA}e^{iB}) = \text{Tr}(U^* e^{iD_A}UV^*e^{iD_B}V),
\end{equation} where $...
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2
answers
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Quantum mechanics [closed]
When we want to calculate the probability of finding a particle in a small region of space given the time-dependent Schrödinger equation, it should be equal to $\vert\psi(x)\vert^2$ times $dx$ times ...
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0
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Collections of eigenstates and decomposition of two operators
I have a physics inspired problem which I am sure has been solved, but I am lacking the correct terminology for it - perhaps someone can point me in the right direction?
Take two finite-dimensional ...
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Limit of a particular trace norm.
I have the following problem.
Let $\mathbf{\hat{\rho}}(t)$ and $\mathbf{\hat{\sigma}}(t)$ be two trace class positive operators acting on a Hilbert space of infinite dimension for all $t > 0$. More ...
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periodic boundary system for infinite quantum periodic system
It has been a long time since I've worked on periodic boundary conditions involving the Bloch theorem (quantum mechanics) so I am hoping to get some help on my next step.
Suppose I have a lattice with ...
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3
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Whether $[A,B]=0$ if $\langle[A,B]\rangle=0$ for all states in the Hilbert space?
Let $\hat{A}$ and $\hat{B}$ are two self-adjoint operators in quantum mechanics corresponding to two dynamical variables $A$ and $B$. If $$\langle[\hat{A},\hat{B}]\rangle\equiv \langle\psi|[\hat{A},\...
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Coefficient conditions for square root of the Laplacian
I am confused about what's going on in the attached picture (from the introduction of Friedrich's Dirac Operators in Geometry). The author claims that for an operator $P$ to satisfy $P = \sqrt{\...
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Prove that a linear map from $\mathbb{C}^n$ to $\mathbb{C}^n$ that sends unit vectors to unit vectors is Unitary. [duplicate]
I believe the title contains all useful information for the problem. I tried to show that this map would have to be an isometry, but I can't seem to ensure that orthogonal vectors in the domain remain ...
1
vote
1
answer
29
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Dimension of the Hilbert space in the tensor product between Von-Neumann algebras
I'm having trouble understanding the final step in the proof of the Stone-von Neumann theorem.
So we have a Hilbert space $H$, $k \in \mathbb{N}$ and weakly continuous unitary representations $U(\cdot)...
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"Probability flux": Imaginary part
I was confronted with this problem and wasn't able to find the supposed answer:
$$ {\partial_t |ψ|^2}={\partial_t (\bar{ψ}ψ)}={\partial_t\bar{ψ} *ψ+\bar{ψ}*\partial_t ψ}={-\frac{i\hbar}{2m}Δ\bar{ψ}ψ+\...
- 1
1
vote
1
answer
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Existence of unitary operator affecting partial trace? [closed]
Let $\rho$ and $\phi$ be any two different density matrices on $H_A \otimes H_B$ such that $Tr_B (\rho) = Tr_B (\phi)$.
Does there always exist a unitary $U$ on $H_A \otimes H_B$, $U\rho U^{\dagger} = ...
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1
answer
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Reading off probabilities for measurement outcome rather than using projection operator?
Let $\alpha_{0}$ = $\alpha_{1}$ = $\frac{1}{\sqrt{2}}$.
Suppose the state vector $| \psi \rangle = \alpha_{0}| \psi_{0} \rangle + \alpha_{1} |\psi_{1}\rangle $ describes a quantum mechanical system ...
- 6,193
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0
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40
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inner product (Bra - Ket) involving projection operator
In quantum mechanics, the action of a projection operator $\hat{P}$ acting on a quantum mechanical system, prepared in a state $| \psi \rangle$, is described by the eigenvector equation
$\hat{P} | \...
- 6,193
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votes
1
answer
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Are two linear mappings $M$ and $N$ on $\mathcal{S}(H_A \otimes H_B)$ equal if they have same behaviour on product states? [closed]
Let $M$ and $N$ be mappings from $\mathcal{S}(H_A \otimes H_B)$ to itself, where $\mathcal{S}(H)$ denotes the set of density operators over the Hilbert space $H$.
If the following two conditions hold:
...
- 55
0
votes
1
answer
52
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Showing $L_j=-i\hbar~\varepsilon_{jk\ell} x_k\partial_\ell,$ is hermitian by integration by parts
Using the definition of hermitian operator $A$: $$\int \psi^*(A\phi)dV=\int(A\psi)^*\phi dV,$$ together with the orbital angular momentum $$L_j=~\varepsilon_{jk\ell}x_k p_\ell=-i\hbar~\varepsilon_{jk\...
- 641
1
vote
1
answer
84
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Fourier Transform Duals and Multi-Variable Chain Rule
EDIT: I believe that I have come to the conclusion that my original idea was fundamentally misguided, and there is no reason to expect that such a process is possible in general. There is much more ...
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0
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41
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Proving limit exists and is positive for smooth function involving integral
Let $\mu$ be a function from $\mathbb R_+\rightarrow\mathbb R_+$ in $C^\infty$ with $\Upsilon:=\sup\{|p|:\mu(|p|^2)>0\}$. Suppose that $$\lim_{|p|\rightarrow \Upsilon}\frac{\mu(|p|^2)}{(\Upsilon-|p|...
- 307
0
votes
0
answers
60
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Parity in the Schrödinger equation of the hydrogen atom
Hi I'm studying quantum mechanics with Stephen Gasiorowicz's book.
It says for the $n=2$ states of the hydrogen atom, the $l=0$ state has even parity and the $l=1$ state has odd parity.
As far as I ...
- 39
1
vote
1
answer
92
views
Fourier integral representation of dirac delta function
Is the following proposition true?
Proposition. For any $a,k\in \mathbb{R}$,
\begin{equation}
\int_a^{\infty} dx e^{ikx} = 2\pi \delta(k).
\end{equation}
(End)
I think it is true based on the ...
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