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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Show the first Chern class of a $U(1)$ bundle is integral.

I am working from John Baez's book: "Gauge Fields, Knots and Gravity". So I will stick to the notation used in that book. I am stuck at exercise 122 of part II (page 283), it reads: Show that if ...
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1answer
23 views

Showing that $HTH = e^{-i \frac{\pi}{8} \sigma_x}$ (quantum gates)

I'm trying to prove that an arbitrary single qubit unitary (read: unitary two by two matrix, and thus rotation up to a phase) can be composed from Hadamard and T gates, given by $ H = ...
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0answers
19 views

Construct bivariate symmetric (polynomial) nonnegative functions (distributions) over the unit square with certain properties

Construct bivariate symmetric polynomials $f(x,y) = f(y,x) \ge 0$ over $[0,1]^2$, with $f(1,y) = f(x,1)=0$, such that the univariate marginal distributions are both proportional to $$(1-u^2)^4$$, ...
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0answers
7 views

how to fix a state to configure expected eigenvector

using Gleason Theorem , Trace(wE) is state, w = Summation of rP where P is projection matrix, and summation of r = 1 it sounds fix state need to change E or w, how to change in real practice? does ...
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27 views

qubit measurment, i need to calculate the probabiliy

Let $$\operatorname {qubit} = \frac{1}{6^{0.5}}\times\big(\left.i\, \lvert 00\right\rangle + \left\lvert 10\right\rangle - 2\left\lvert 11\right\rangle\big)$$ If we measure only the second qubit in ...
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31 views

Can Schroedinger equation be derived from the unitary representation of Galilean group? [migrated]

I have been trying to understand quantum mechanics as a unitary representation of spacetime symmetries. My first question is: Can Schroedinger equation be derived from the unitary representation of ...
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1answer
98 views

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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1answer
31 views

Commutation of a partial trace with an operator

Let the partial trace $\mathrm{tr}_B$ be a mapping from an endomorphism End$\left( H_A\otimes H_B \right)$ onto an endomorphism End$\left( H_A \right)$. Then the partial trace is defined as $$ ...
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17 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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28 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
57 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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18 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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1answer
49 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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1answer
56 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
452 views

Studying quantum mechanics without physics background

I am a PhD math student, and I am wondering if I should study quantum mechanics while I don't have an undergrad background in physics. I posted this question in physics stackexchange, but there ...
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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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28 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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39 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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3answers
309 views

Orthogonal projections with $\sum P_i =I$, proving that $i\ne j \Rightarrow P_{j}P_{i}=0$

I am reading Introduction to Quantum Computing by Kaye, Laflamme, and Mosca. As an exercise, they write "Prove that if the operators $P_{i}$ satisfy $P_{i}^{*}=P_{i}$ and $P_{i}^{2}=P_{i}$ , then ...
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2answers
179 views

Trace in non-orthogonal basis

In Dirac notation we can define the trace of an operator in Hilbert space $\rho$ as the follows, $Tr(\rho)=\sum\limits_{|s\rangle \in B} \langle s| \rho |s\rangle$ where B is some orthonormal ...
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28 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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0answers
57 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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2answers
49 views

Expectation value of pure state in quantum mechanics

It's well known that in quantum mechanics, the expectation value of a self-adojint operator $A$ in pure state $|\psi\rangle$ is $\langle\psi |A|\psi\rangle = \operatorname{Tr}(A |\psi \rangle ...
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71 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
41 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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0answers
86 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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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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43 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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29 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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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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0answers
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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43 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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1answer
94 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
57 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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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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24 views

A detail in [ the time evolution operator]

If $$\hat C=\Delta \hat S_{22}+\lambda _1[\hat a_1 \hat S_{21}+\hat a_1^\dagger \hat S_{12}]+\lambda _2[\hat a_2 \hat S_{32}+\hat a_2^\dagger \hat S_{23}] ,$$ $$ \hat \beta= ...
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3answers
99 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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2answers
44 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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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
44 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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57 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
20 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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19 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
23 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
49 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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9 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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212 views

Tensor product of affine space and algebra

When reading about Quantum Mechanics, I always feel a bit disappointed when physicists consider that (for example) the 3-dimensional position of a particle must be decomposed into 3 coordinates $x, y, ...
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42 views

classic derivation of the proportionality between angular momentum and magnetic moment problem

The question is given in parentheses. It is a classic derivation of the proportionality between angular momentum and magnetic moment $m=-IA=-I\pi r^2$ We start with the charge on a ring (say, the ...