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.
1,423
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log det on density matrix plus identity
A very naive question: given a pure quantum state $|\phi\rangle$, and the associated density matrix $\rho=|\phi\rangle\langle\phi|$, does there exist an efficient quantum operator/procedure that gives ...
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Inner Product of Quantum States - Is this Calculation Correct?
Paz and Zurek (in ISBN 3-540-43367-8, p.91) give the following calculation:
For
$|E_1(t)\rangle= \bigotimes_{k=1}^N\Bigl(\alpha_{k}e^{ig_kt}|{0}\rangle+\beta_ke^{-ig_kt}|{1}\rangle\Bigr)=|E_2(-t)\...
- 237
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Inner automorphisms of Pauli strings in the unitary subgroup of matrices
Statement
What is the set $\mathcal{T}_n$ of matrices $T \in GL(2^n)$ such that for all Pauli strings $P \in \mathcal{P}_n=\{\otimes_{i=1}^n \sigma_{m_i}\mid m\in \{1,2,3\}^n, \sigma_{m_i} \text{ ...
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Spherical harmonics as simultaneous eigenfunctions
If I consider operators $L^2$ and $L_z$ in spherical coordinates where $L^2$ is the angular momentum squared operator and $L_z$ is the $z-axis$ component of the angular momentum, is a function like $\...
- 31
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Does the von Neumann equation hold for continuous functions of the density matrix?
Let $\rho$ be a density operator obeying
$$i\hbar \partial_t \rho = [H,\rho]$$
where $H$ is a time-dependent Hamiltonian and $\rho(x,0)=\rho_I(x)$. The evolution is given by
$$\rho(t) = U(t) \rho_I U(...
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Question regarding conjugate operators and the harmonic operator.
Let us consider the operator $\hat{n}=\hat{a}^\dagger\hat{a}$ as the number operator of the harmonic oscillator. Let $|n\rangle$ be the eigenstates. Then we can say : $$\hat{n}|n\rangle=n|n\rangle$$
I'...
2
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Summation of squared 3-j symbol
There is a property for the 3-j symbols as
$$
\sum_{m_1 m_2 m_3}
\begin{pmatrix}
j_1 & j_2 & j_3 \\
m_1 & m_2 & m_3
\end{pmatrix}^2 = \Delta (j_1, j_2, j_3),
$$
where $ \Delta (j_1, ...
- 123
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What does it mean for the momentum operator P to satisfy P|n>=hn|n>?
Using periodic boundary conditions to analyze momentum eigenstates on the circle $S^1$, we have the momentum operator $P=-i\hbar\frac{d}{d\phi}$ which satisfies $P|n>=\hbar n|n>$ (for clarity, |...
2
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1
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Eigenvalue problem of time dependent Hamiltonian
I would like to solve an eigenvalue problem of a Hamiltonian. I was able to find the lowest eigenvalue by converting the Hamiltonian into a matrix and applying linear algebra eigenvalue techniques. ...
- 45
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1
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Given $[A,B]=0$ then $[A,f(B)]=0$
Given $A$ and $B$ two operators that commute ($[A,B]=0$) then $A$ commutes with an arbitrary function of $B$
I recently saw this property of the commutators on a quantum mechanics course. We didn't ...
- 321
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Equivalence between two definitions of hermitian adjoint
Given the two definitions of hermitian adjoint:
$(1): <\Psi|A|\Phi>=(<\Psi|A^+|\Phi>)^*$
$(2): <\Psi|A\Phi>=<A^+\Psi|\Phi>$
I want to show that they are equivalent
However I ...
- 321
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Leibniz's integral rule for infinite limits of integration
Given $\Psi(x,t)$ a normalized function for $t=0$, $\int_{-\infty}^{\infty}\Psi^*(x,0)\Psi(x,0)dx = 1.$ Prove that $\Psi(x,t) dx$ is normalized $\forall t \gt 0$.
This is a question I've found for a ...
- 321
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How to showing a function is square-integrable in Quantum mechanics
Looking the A-1 section in Cohen-Tannoudji et al Quantum mechanics Vol. 1 book I get lost in the following:
$\psi(r)=\lambda_1\psi_1(r)+\lambda_2\psi_2(r) \in F$
In order to show tha $\psi(r)$ is ...
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Is the Sy basis of Spin not orthogonal?
According to Wikipedia on Spin $1/2$ particles, the vectors of the $S_y$ basis, with respect to the $S_z$ basis, are:
\begin{bmatrix} 0.707 \\ 0.707*i \end{bmatrix}
\begin{bmatrix} 0.707 \\ -0.707*i \...
- 177
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Is there a characterisation for the convex set which is fully embedded inside a convex polytope?
As per the Krein-Milman theorem, any convex compact set (in a finite-dimensional vector space) is equal to the convex hull of its extremal points.
I am now imposing that a given convex compact set $C$ ...
3
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0
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29
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Time evolution and matrix ODE
Let $M$ be a linear operator on a finite-dimensional complex Hilbert space (thus, $M$ is just a matrix). Assume that $\text{Tr}M = 1$, $M$ is self-adjoint and $M \ge 0$ (positive semi-definite).
In ...
- 1,451
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How to find $\langle x\rangle$ and $\langle p\rangle$ of a particle in an infinite square well if wave function at $t=0$ is $Ax(a-x)$?
I have tried finding it and i got $\langle x\rangle = 4a/15$. The method I used is as follows: Integrate $x|\Psi(x,0)|^2$ from 0 to $a$.
But I don't know if it is correct because we always do it by ...
- 11
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0
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Integral over operators equal -When are operators equal?
This is a physics motivated question from quantum mechanics. When I have two operators $\hat{A}$ and $\hat{B}$ acting on the Hilbertspace $\mathbb{C}-L^2(\mathbb{R})$ with
$$\int\psi^*(x)\hat{A}\psi(x)...
- 242
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What are the elementary properties of dirac-delta function from which every other properties of it could be deduced?
I am studying dirac-delta function first time in my undergraduate course and different books have defined this function in different ways which when graphed together contradicts each other. I want to ...
- 11
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0
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Maximise a parametrised $\sin^2$ function
How to find the optimal value of $k$ that maximises $\sin^{2}((2k + 1)\theta)$ assuming $ \theta \in \left(0,\frac{\pi}{4}\right)$ and $k$ is a natural number.
As per my understanding the value of $k$ ...
0
votes
1
answer
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Computing $f(x+\frac{d}{dx})$ operator using Taylor series
According to Taylor's theorem, one can write $$f(x+h)=\sum_{k=0}^\infty\frac{f^k(x)}{k!}h^k,\tag{1}$$ where $f^k$ is the $k^{th}$ derivative of $f$ at $x$. Let us assume that the series converges for ...
- 53
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A Geometrical Derivation of Quantum Mechanics Spin Operators
I'm trying to see if there is a way to geometrically derive a general form for the quantum mechanics spin operators. I'm trying to deduce their commutation relations without using any knowledge of ...
- 1,012
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What is $ d B_t d t^{1/2} $, where $B_t$ is the Brownian motion on $\mathbb{R}$?
Consider the time-evolution operator $U(dt) = \exp(-id tJH_0 - i d B_t V)$ where $H_0$ and $V$ are some Hamiltonians, and $J$ is some coupling. $B_t$ is the Brownian motion on $\mathbb{R}$ with $B_0 = ...
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Path Integral in QM - Fourier Transform with respect to a Function?
Consider the Fourier transform of a multivariable probability density function $Pr(\{x_n\})$, i.e. its characteristic function:
$\int Pr(\{x_n\})e^{-i2\pi\sum\limits_{n}f_nx_n}\prod\limits_{n}{dx_n}$,
...
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False Negative Probability of a probabilistic black box function
Suppose we have black-box access to a function that returns the roots of some unknown 5-degree polynomial in each invocation. (So there will be 5 roots.)
But we also know that the function is ...
0
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0
answers
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How to perform a functional integral?
I'm completely beginner to the quantum field theory and try to learn the basics of functional integrals. However, I could not understand clearly. Could someone please explain the idea with the help of ...
- 5
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0
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Gaussian Integral like $\exp{[-r^2 - r_\alpha^2 +2 r r_\alpha \cos{(\theta-\theta_\alpha)}]}[r -r_\alpha \cos{(\theta-\theta_\alpha)}]f(r,\theta)$
I have a Gaussian kernel that I wish to evaluate $$\int_{0}^{\infty} \int_{0}^{2\pi} \exp{[-r^2 - r_\alpha^2 +2 r r_\alpha \cos{(\theta-\theta_\alpha)}]} [r -r_\alpha \cos{(\theta-\theta_\alpha)}]f(r,\...
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Algebraic Bethe Ansatz - Lax operator as $2\times 2$ matrix
The following part is taken from the Algebraic Bethe Ansatz paper https://arxiv.org/abs/hep-th/9605187 (Page 7).
The definition of the Lax operator involves the local quantum space $h_n$ and the ...
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Singular value decomposition for a convolution kernel
I am trying to get some insight into how to solve the following linear equations:
$$ \frac{dA_n}{dt} = i\sum\limits_{m} B^*_{n-m} C_m$$
$$ \frac{dC_n}{dt} = i\sum\limits_{m} B_{n-m} A_m$$
where $B_n$ ...
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1
answer
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Why is $\Phi(X) = \mathrm{Tr}_\mathcal X(J(\Phi)(1_\mathcal Y \otimes X^T))$?
I'm reading about the Choi representation from John Watrous' textbook on quantum information. On page 78, he says that for any choice of complex Euclidean spaces $\mathcal X$ and $\mathcal Y$, one may ...
- 317
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1
answer
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Is $\nabla^2 f/f\in\mathbb{R}$ for a complex-valued $f\in L^2$?
Given a complex-valued function $f: \mathbb{R}^N\rightarrow\mathbb{C}$ with $f\in L^2$, which vanishes at the domain boundaries, is twice differentiable, and antisymmetric upon two-coordinate exchange—...
- 115
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1
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Estimate $\Vert \Delta u \Vert_{2}$ for wave equation [closed]
We consider the wave equation
\begin{equation}\label{1}
\left\{
\begin{array}{ll}
u_{tt}(x,t)-\Delta u(x,t)=0, x \in \Omega, t>0\\
u=0, \quad u \in \partial \Omega, t>0 \\
u(0,x)=u_{0}(x), \quad ...
- 11
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0
answers
13
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Given the Symplectic Matrix acting on phase space, find the Gaussian Unitary acting on the Hilbert space
In Gaussian Quantum Mechanics, a unitary preserving the Gaussian nature of the state is a called a Gaussian Unitary. In the phase space picture, a Gaussian state is fully characterized by its first ...
0
votes
1
answer
72
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Questions about the eigenfunctions and eigenvalues of the momentum operator $\hat{p}$
I'm studying Quantum Mechanics right now and working through an example in the book of an eigenfunction with a continuous spectrum - the momentum operator, $\hat{p} = -i\hbar\frac{d}{dx}$.
In the ...
- 1,305
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Sampling harmonic oscillator eigenfunctions
What is the most computationally efficient way of sampling $x$ from the distribution
$|\psi_n(x)|^2$ for a given $n$, where $\psi_n(x)=\frac{1}{\sqrt{2^n
n!}}\pi^{-1/4}\exp(-x^2/2)H_n(x)$ is a quantum ...
- 11
1
vote
0
answers
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Correlations in implementations of random unitary channels
A random unitary channel (RUC) acting on a quantum system $S$ is any completely positive and trace preserving (CPTP) linear map $\mathcal{E}$ that can be expressed as
\begin{align}
\mathcal{E}(\...
- 11
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votes
0
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Calculating Trace of Integral Operators
I could not figure out how the following formula can be derived:
$$ \operatorname{Tr} [ W(t) Q(t) ] = \frac{1}{(2\pi)^{d/2}} \int_{\mathbb{R}^d \times \mathbb{R}^d} \widehat{W}(t, q-p) \widehat{Q}(t,p,...
- 102
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Hermitian and Observables
Let $H$ be an finite dim Hilbert space with $|\phi \rangle$ in H, $\langle\phi \mid \phi\rangle=1$, and $\rho \triangleq|\phi\rangle\langle\phi| .$ Let $A$ be an observable which is represented by the ...
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Definition of the pullback of $L=Im(dS)$ and related questions on defintions
Def: We call a phase function $S: \mathbb{R}^n \rightarrow \mathbb{R}$ admissible if it satisfies the Hamilton-Jacobi equation. The image $L=Im(dS)$ of the differential of an admissible phase function ...
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Is there a connection between graph theory and many worlds QM
Obligatory disclaimer that this is my first post and first contact with the community so please redirect me to better channels if these types of questions are not for this forum.
So I was just naively ...
3
votes
1
answer
38
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Identity regarding Pseudodifferential operator
In the paper I am reading, the following identity appears:
$$ e^{-it \Delta} f(x) e^{it \Delta} = f(x- 2it \nabla) $$
where $f \in \mathcal{D}(\mathbb{R}^{d})$ and $f(x)$ on the left hand side denotes ...
- 102
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votes
1
answer
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Why does the order of eigenvectors matter when changing a matrix to a different basis?
I'm solving a basic problem in quantum mechanics to change the matrix A into a representation in the basis consisting of eigenvectors of matrix B. And I've noticed that the new mateix A looks ...
- 133
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vote
1
answer
37
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What can I infer from this scalar product identity?
Suppose I have shown for a single vector $\lvert 0 \rangle$ that
$$\langle 0 \rvert U^\dagger \phi U \lvert 0 \rangle=\langle 0 \rvert \phi \lvert 0 \rangle$$
where $\phi$ is a certain operator and $U$...
- 121
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votes
1
answer
52
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Matrices that are simultaneously Hermitian and unitary [duplicate]
My quantum mechanics professor was discussing the properties of Pauli matrices, their being both Hermitian and unitary. Then he made a remark that it is not possible to find three $n \times n$ ...
- 33
0
votes
1
answer
67
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Can the solution of a real differential equation always chosen to be real?
I read somewhere, that the solution to a real differential equation can always be chosen to be a real-valued function. Is this true? In particular, I am interested in the stationary Schrödinger ...
- 115
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votes
1
answer
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Derivative of the action functional
I am self studying the book An Brief Introduction to Physics for Mathematicians. At the second page, the equation (1.6) says that the derivative of the action functional can be derived as
$$
S'(x)(h) =...
- 925
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vote
1
answer
82
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Spherical harmonics as orthonormal basis in quantum mechanics
In this article https://mrtrix.readthedocs.io/en/dev/concepts/spherical_harmonics.html the following statement is given:
Spherical harmonics are special functions defined on the surface of a sphere. ...
- 279
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vote
0
answers
16
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Calculating Error for using an approximation for the Schrodinger Equation Evolution Operator
I've devising a Crank-Nicolson scheme for the Schrodinger equation using a time-independent potential, $V(x)$. In devising this scheme the following equation is approximated and THEN discretized, ...
- 191
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votes
1
answer
38
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What is the relation between the classical and the quantum mechanical approach in calculating the Value at Risk?
What is the relation between the quantities in classical Value at Risk (VaR) calculation and the calculation via quantum computers?
To specify this question, I would like to briefly explain my ...
- 11
0
votes
0
answers
28
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Integrating an ODE with matrices
I've been watching David Deutsch's Lectures on Quantum Computation here
https://www.youtube.com/playlist?list=PLqdVnC7OWuEcfKRZXsrooK_EPzwmWSi-N
In Lecture 2, he gives these equations to show how X,Y ...
