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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quantum mechanics violate Bell's inequality

I have this function $$ \begin{aligned} F\big(\theta_a,\theta_b,\phi_a,\phi_b\big) = \ & – \big[\cos \theta_a \cos \theta_b \big] – \big[\sin\theta_a \sin\theta_b \sin\phi_a \sin\phi_b\big] \\ ...
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14 views

How to Construct $N$-dimensional Unitary Matrix Basis

Galitski's Exploring Quantum Mechanics says on its page 29, (There are $N^2$ linearly) independent Hermitian matrices of rank $N$. The number of independent unitary matrices is also $N^2$, since ...
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16 views

Quantum probability and quantum measure theory

Do quantum probability and free probability mean the same thing - that is, they deal with noncommutative random variables? What about quantum measure theory? Is quantum measure theory the foundation ...
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2answers
55 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 ...
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17 views

Gaussian unitary dilation of Gaussian channels

I am starting with a few definitions. All these are standard and can be accessed from some quantum information or quantum physics books, for instance the books by Holevo or Parthasarathy. The question ...
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58 views

Hermitian conjugate of an antiunitary operator

In certain fields of quantum mechanics, one often considers symmetry transformations which are defined in terms of operators which do not change the norm of states in the Hilbert space. For the ...
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1answer
54 views

Projection of a 3D spherical function to a carteasian axis

I have a 3D function defined in a spherical coordinate system $(r,\theta,\phi)$, which is written as a product of a radial function $R_{nl}(r)$ and a spherical harmonic $Y_{lm}(\theta,\phi)$ I.e $$ ...
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1answer
46 views

Make mathematical sense of the Dirac well Potential Equation

A classical problem in quantum mechanics involving the Dirac Delta function is given by $$ y''+(\delta(x)-\lambda^2)y=0 $$ Then, to find ''bound states'', you solve on the right and find the ...
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26 views

On block density matrices and CPTP maps

Consider a $n\times n$ density matrix $\rho$ and decompose it as $$ \rho=\left[\begin{array}{c|c} \rho_A & \rho_B \\ \hline \rho_B^\dagger & \rho_C\end{array}\right], $$ with $\rho_A$, ...
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37 views

Questions to the virial theorem

Let $H = H_0 + V$ be the Hamiltonian of the single electron where $H_0 = - \Delta, V = - \frac{\gamma}{|x|}$. Now one defines the dilation group $U(s) \psi(x) = e^{-ns/2} \psi(e^{-s}x), s \in \mathbb ...
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26 views

A Question about Quantization and Partition Function

I have a question about quantization and partition function, which sound a little bit inappropriate. But I still want to ask for help. I think that quantization is a unitary representation of the ...
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56 views

Pureness of Vector States

How does one show that irreducibility is equivalent to a vector state being pure? In what follows I will fill in the details of the question: Let $\mathcal{H}$ denote a Hilbert space and let ...
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52 views

Please help: My MATLAB code for solving a 2D Schrödinger equation keep giving me weird output.

I've been trying to solve the following Schrödinger equation numerically, \begin{equation} -(\frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2})\Psi + \frac{\sinh^2(y) + ...
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1answer
31 views

x-momentum operator $p_x$ expressed as multiple of Translation operator

On this page https://en.wikipedia.org/wiki/Rotation_operator_%28quantum_mechanics%29 under "The translation operator," they use Taylor expansion. As part of that proof they state $p_x = ih * ...
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16 views

Why $|j_1 - j_2 | \leq j \leq j_1 +j_2 $ holds for $J= J_1 +J_2 $, the addition of angular moment?

I wonder why the total angular momentum $$J=J_1 +J_2 $$ is given in the range of $$ |j_1 - j_2 | \leq j \leq j_1 +j_2 $$ Of course we can verify this in the course of finding Clebsh-Gordan ...
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1answer
57 views

Schrödinger operator with delta (zero range) interaction.

I am reading the book of Albeverio named Solvable models in quantum mechanics. In the first chapter it is explained how to realize the operator $"-\Delta+\delta_0"$ as a self adjoint operator on ...
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84 views

Upper bound on the Lipschitz constant of entanglement entropy

I'm looking for an upper bound for the Lipschitz constant of entanglement entropy between two subsystems with respet to the standard distance measure of pure states in the Hilbert space of the full ...
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35 views

Is there any some straightforward calculation of finite rotation operator?

Rotation with axis $\hat{k}$ and angle $\theta$ in $\mathbb{R}^3$ is represented by $$ R = I + (\sin \theta) K + (1-\cos \theta) K^2 $$ where $K$ is the matrix for left cross product by $\hat{k}$. ...
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1answer
39 views

Why must power series count up by integers 0,1,2.. in 3D harmonic oscillator in spherical coordinates?

http://www.physicspages.com/2013/01/17/harmonic-oscillator-in-3-d-spherical-coordinates/ http://quantummechanics.ucsd.edu/ph130a/130_notes/node244.html These are two links that have roughly the same ...
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28 views

What are eigenvalues? [closed]

I am extremely interested in quantum mechanics and have been studying about it for a few years. But I don't understand the concept of eigenvalues and how is it related to matrices? I encountered this ...
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29 views

Angular momentum components in a given direction.

The exam question of interest is the following: Suppose the angular momentum of $\mathbf{v} = |1\ m \rangle$ ($j=1$, $|j \ m \rangle$ denoting the usual eigenstate of $J_3$ and $\mathbf{J}$) is ...
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1answer
74 views

Distinct eigenvalues of matrix

While studing Quantum Physics I've came to the conclusion that it would be useful to find an equivalent (or at least a sufficient) condition for a matrix (over $\mathbb{C}$) to have distinct ...
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41 views

Quantum mechanics question?

A particle of mass $m$ is confined within an infinite, one-dimensional potential well, $U(x)$, of width $a$. $$ U(x) = \begin{cases} \infty & x \leq \frac{-a}{2}, x \leq ...
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21 views

Given the matrix representation what is the expectation value

For a particle with spin $\frac{3}{2}$, construct the matrix representation for $S_z, S_x$ and $S_y$. If the particle is in an eigenstate of $S_z$, what is $\langle S_x\rangle$ and $\langle ...
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3answers
96 views

looking for help with a trace/norm inequality

I'm trying to understand a derivation that seems to claim that $\left\vert\text{Tr}\left[\rho U^\dagger\left[U,O\right]\right]\right\vert\leq\|\left[U,O\right]\|$, where $\rho$ is Hermitian and has ...
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1answer
92 views

Example of using the Hadamard's matrix to determine the superposition

I've came across those notes for Quantum computation from John Watrous. I am having troubles understanding the last example. We have those two vectors, or if I understood correctly, from now on ...
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1answer
33 views

Tensor product in dual-space

I am a bid confused regarding the notation for tensor products when going into dual-space If $\left| \Psi \rangle \right. = \left| A \rangle \right. \left| B \rangle \right.$ is $ \left. \langle \Psi ...
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0answers
20 views

Going into dual space for a vector product [duplicate]

I am a bid confused regarding the notation for tensor products when going into dual-space If $\left| \Psi \rangle \right. = \left| A \rangle \right. \left| B \rangle \right.$ is $ \left. \langle \Psi ...
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1answer
40 views

Position operator is self adjoint

Let $H=L^2(\Bbb{R})$ with the linear (unbounded) operator $P(f)(x)=x\cdot f(x)$ for each $x\in\mathbb{R}$. Have a look at the following domain: $$D(P)=\{f\in ...
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2answers
43 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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51 views

commutation relation of angular momentum operator in non cartesian coordinates

The angular momentum operator $J$ in quantum mechanics with the commutation relation \begin{equation*} [J_l,J_m]=i\hbar\epsilon_{lmn}J_n \end{equation*} has the structure of a Lie-algebra. It is ...
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1answer
18 views

Applying an unambiguous quantum state discrimination operator on an entangled qubit.

Given a quantum system $|\psi\rangle=\alpha_0|\psi_0\rangle\otimes |0\rangle+\alpha_1|\psi_1\rangle\otimes |1\rangle$, such that each subsystem $|\psi_i\rangle$ is entangled with a qubit is state ...
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18 views

Harmonic Oscillator & Normalised Energy Eigenfunctions

A Harmonic oscillator is described by the Hamiltonian operator H = -1/2*(d^2)/(dx^2) +1/2*x^2 The Lowering operator is given by D_ = d/dx + x Given that the integral over all space for x^2 ...
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1answer
20 views

Complex Vector spaces inner product superposition axiom

In my studies of Quantum mechanics, the following problem with complex vector spaces has come up, specifically as regards the inner product in such a space. Now in Shankars "Principles of Quantum ...
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1answer
43 views

What does does [H1,H2] = 0 mean if each H is a hamiltonian of a quantum system

What does does [H1,H2] = 0 mean if each H is a hamiltonian of a quantum system I'm trying to get through a research paper on theoretical quantum biology and I just want to make sure I'm interpreting ...
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8 views

The tensor product of two poly operators

How can I calculate the poly operator sigma x = [0 1:1 0] with itself? In other words; How can I calculate |1><1| tensor |1><1|+|0><0|tensor|0><0|?
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1answer
129 views

Linear algebra references for a deeper understanding of quantum mechanics

I'm a graduate student studying quantum mechanics/quantum information and would like to consolidate my understanding of linear algebra. What are some good math books for that purpose?
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28 views

Self-adjoint operator and vertex conditions in quantum graphs

Let $\Gamma$ be a metric graph with finitely many edges. Consider the operator $H$ acting as $\frac{-d^2}{dx_e^2}$ on each edge $e$, with the domain consisting of functions that belong to $H^2(e)$ on ...
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29 views

What does it mean to square a partial derivative with a one dimensional vector (scalar)?

Please bear with me.. In the above image, we need to substitute p2 from the partial derivative (pay attention to content in red boxes). If we're considering the one dimensional case, we can either ...
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27 views

I would like to find a generalization of the plane wave expansion to Hankel functions.

The plane wave expansion is \begin{equation} e^{i\vec{k}\cdot \vec{x}}=\sum_{\ell=0}^{\infty}i^\ell(2\ell+1)j_{\ell}(kx)P_{\ell}(\cos(\theta)) \end{equation} where $j_\ell$ is the spherical Bessel ...
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1answer
30 views

Diagonalising an infinite-dimensional Hermitian square matrix

I have a quantum state which takes the following form: $$\rho = \sum_{b, \,c \, = \,0}^\infty \frac{(-igt)^b(igt)^c}{\sqrt{b!c!}}\vert b\rangle\langle c\vert.$$ This is an infinite Hermitian matrix ...
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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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25 views

adjoint of projection operator

|$\phi_m$> are the eigenstates of a Hermitian operator H. Assume that the states |$\phi_m$> form a discrete orthonormal basis. The operator U(m,n) is defined by: U(m,n) = ...
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48 views

How can I derive Polarization Operator algebraically

Can anyone describe for me the algebraic steps needed to create what Prof. Leonard Susskind calls, in one of his early Quantum Mechanics courses, the "Polarization Operator Matrix"? It is a 2 by 2 ...
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1answer
52 views

Watrous’s definition of Quantum Cellular Automata

I need some help understanding the second-to-last and final equations of the introduction to quantum cellular automata included below. My specific questions: What does it mean when the capital Pi ...
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1answer
25 views

quantum harmonic oscillator and the mean energy U(T)

The energy of a quantum harmonic oscillator is given by: $$E(n)=\hbar\omega\left(n+\frac{1}{2}\right)$$ The canonical partition function is given by: $$Z(T)=\sum_{n=o}^\infty e^{-\beta ...
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1answer
38 views

Prove that $e^{\lambda A}Be^{-\lambda A}=B$

Prove that $$e^{\lambda A}Be^{-\lambda A}=B$$ if $[A,B]=0$. $A$ and $B$ are operators and $\lambda$ is a complex number. Can anyone explain how I should go about this question? How do I calculate ...
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29 views

Binomial-like expansion for non-commuting operators

I would like to evaluate the following and express it in a compact form: $(\hat{a}^\dagger(x)+\hat{a}(x))^n\,\vert0\rangle$, where the $n^{\text{th}}$ power of the sum of the annihilation and creation ...
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derivation of an equation concerning Fourier transform of a wave packet

My question arises from Fourier transform of a wave packet in quantum mechanics. In the following context, $\{\phi_i, E_i | i=1,2,...,n\}$ are eigenstates and eigenvalues of a $n\times n$ Hermitian ...
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2answers
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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 ...