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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1answer
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If $L$ is a diagonalizable linear operator, why is $f(L)$ well-defined for $f: \mathbb{C} \rightarrow \mathbb{C}$?

In a book on quantum mechanics, I encountered a statement equivalent to the following. Suppose $V$ is a finite dimensional inner product space over $\mathbb{C}$. Let $L : V \rightarrow V$ be a normal ...
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1answer
32 views

Find the eigenvectors of a hermitian matrix as a function of angles

I have a problem which seems flawed to me. Consider the hermitian matrix M \begin{pmatrix} a/2 & b^* \\ b & -a/2 \end{pmatrix} with a>0 real and b complex. Let $\theta$,$\phi$ ...
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1answer
26 views

How do outer products differ from tensor products?

From my naive understanding, an outer product appears to be a particular case of a tensor product, but applied to the "simple" case of tensor products of vectors. However, in quantum mechanics there ...
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1answer
41 views

Quantum Mechanics - Eigenvalue and Eigenvector of a Matrix

I'm attempting to find the eigenvalues and eigenvectors from the following matrix : \begin{pmatrix} -3\cos\theta&\sqrt{3}\sin\theta e^{iφ}&0&0\\\sqrt{3}\sin\theta ...
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1answer
26 views

counter example of sum of closable operators

Let $A$, $B$ and $A+B$ are closable operators. I am not sure the relation $\overline{A+B}\supseteq \overline{A}+\overline{B}$ is true or not(with equality if one operator is bounded. ) And I have a ...
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16 views

Intuitive interpretation of negative probabilities

I have heard that in quantum physics negative probabilities show up in certain distributions. Could you give an example that aids int he intuitional interpretation of a negative probability? For ...
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18 views

Commutative diagram for hidden subgroup representation of graph automorphism

The hidden subgroup representation of the graph automorphism problem is defined in the section 10.2 of QUANTUM ALGORITHMS FOR PROBLEMS IN NUMBER THEORY, ALGEBRAIC GEOMETRY, AND GROUP THEORY. It is as ...
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18 views

General Schrodingers Equation Quetions

I have som very general questions about the Schrodingers equation, 1) In seprating the variables into radial and angular, why do we equal it to $l(l+1)$, I can not seem to understand it from the books ...
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1answer
28 views

Geometric quantization: not understanding the curvature form and Weil's theorem

I am reading a bit about geometric quantization. The texts that I am following are N.M.J. Woodhouse's "Geometric Quantization" and an article from arXiv. I am having troubles understanding the ...
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31 views

How to find the ground energy state solution in a quantum harmonic oscillator?

Recently, I came across a question which asks to solve the Schrödinger equation for a harmonic oscillator on $ [a, b] $ : $-\frac{\hbar^2}{2m}\frac{d^2\psi}{d x^2} + \frac{1}{2} m \omega^2 x^2 \psi = ...
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30 views

Existence of a canonical map of quadratic forms

For $X=\mathbb C^N\oplus \mathbb C^N$ equipped with a real structure $J^2=1$ and symplectic structure $S$ satisfying $$J(z_1,z_2)=(\bar z_2,\bar z_1),~~~~~S(z_1,z_2)=(z_1,-z_2)$$ we see that $X$ has ...
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10 views

Separation of Variables PDE on Klein Gordon Equ

I have to use separation of variables on the 3-D Klein-Gordon equation: $ c^2 \bigtriangledown^2 \Psi - \frac{\partial ^2}{\partial t^2}\Psi + (\frac{mc^2}{\hbar})\Psi = 0 $ where $ \Psi (r,t) = ...
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37 views

quantum matrices, quantum determinante

Homework, except I'm completely clueless, so if someone could potentially point me to similar worked examples or help explain this one step at a time it would be much appreciated. Could you explain ...
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2answers
55 views

N-Representation of an Operator

Calculating $\langle{n} \; {| \; \hat{X}^2 \; |} \; m\rangle$ in the N-representation, where $| \; m\rangle$ and $| \; n\rangle$ are harmonic oscillator states and $\hat{X} = \sqrt{\frac{\hbar}{2mw}}( ...
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8 views

Positive semi-definite vs. semi-positive definite?

I've heard and read the phrase positive semi-definite in many places. However, the only place I can recall seeing semi-positive definite is in my quantum mechanics text, John S. Townsend's ...
2
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1answer
70 views

Quantum Mechanics Project Ideas!!! [closed]

I am in my first year in uni and I have to write a project in Quantum Mechanics. But I have been struggling with an idea for the project since I have recently started studying quantum mechanics and my ...
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1answer
40 views

A question regarding Eigenvalues

Note: $\psi,\psi^{\dagger} :\Bbb{R} \to \Bbb{C}$ and $x, \lambda_i , \hbar, m \in \Bbb{R}$ Say we know that $\lambda_1$ is a solution to the eigenvalue equation: $$\hat{\Pi}\psi(x)= \lambda_1 \psi(x) ...
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35 views

Dirac Notation: What Do $\langle u|v \rangle$ and $\langle u|T|v \rangle$ Represent?

I post this hoping for clarification, and particularly in a context of linear algebra without too much mention of matrices. (1) Let $V$ be a (perhaps infinite dimensional) Hilbert space and $V'$ its ...
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1answer
41 views

Singular value decomposition of sum of single particle operators

here is my question. Suppose to have an operator $L$ in a composite Hilbert space $A\otimes B$ which can be written as sum of single particle operator as: \begin{equation} L = (L_0\otimes \mathbb{I} + ...
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33 views

How to prove the given property of spherical harmonics?

How to prove: $$ \int^\pi_0\int^{2\pi}_0 Y_{l'',m''_l}(\theta,\phi) Y_{l',m'_l}(\theta,\phi) Y_{l,m_l}(\theta,\phi) \sin\theta \,d\theta \,d\phi = 0 $$ unless $l, l',$ and $l''$ are integers that can ...
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65 views

Pauli Matrices as Representations of Reflection Operators?

How does one derive, using, say, the Householder transformation $$ R(r) = (I - nn^*)(r),$$ the reflection representation of a vector $$ R(r) = R(x\hat{i} + y\hat{j} + z\hat{k}) = xR(\hat{i}) + ...
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19 views

Probability distribution obtained by repeatedly sampling $S_x,S_y$ on a spin-$S$ system

While trying to rework an upcoming quiz problem for a quantum physics course, I came up with the following question which turned out to be harder than I expected. (Note: I take $\hbar =1$ in all ...
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21 views

Unitary transformation with eigenvectors of fourier basis

I'm trying to understand this statement made in https://users.cs.duke.edu/~reif/courses/randlectures/UVnotes/lec18.pdf in the last paragraph: "Since U is multiplication by a group element, the ...
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2answers
30 views

Expectation of Complex Operators

Given an operator $\hat{\alpha}$, how do we obtain, $$ \sqrt{ \left\langle \left( \hat{\alpha} - \left\langle\hat{\alpha}\right\rangle \right)^2 \right\rangle } = \sqrt{ ...
4
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1answer
84 views

Intuition behind definition of spinor

Some time ago I searched for the definition of spinors and found the wikipedia page on the subject. Although highly detailed the page tries to talk about many different constructions and IMHO doesn't ...
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1answer
18 views

Complex conjugate part of TISE

i would like to ask about complex conjugate part of this equation why ? I know that and so
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1answer
93 views

Introductory book on probability for physicists

I'm a physics student looking to start learning more about probability. Is there some introductory book on measure theoretical probability theory that includes sections on quantum probability? To ...
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3answers
43 views

Two operators $X$ and $Z$ in an infinite dimensional Hilbert space satisfying $X^2=Z^2=I$ and $\{X,Z\}= 0$

I am seeking to extend the following theorem to the case of infinite dimensional Hilbert space: Suppose we have two Hermitian operators $X$ and $Z$ in a finite dimensional Hilbert space $\mathcal H$. ...
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2answers
27 views

Dirac Notation Confusion [closed]

I am trying to express the vectors $0\choose 1$ and $1\choose 0$ in dirac notation wrt the basis {|$0\rangle,$|$1\rangle$} How do I distinguish between the above two vectors given that all vectors ...
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3answers
40 views

Example of a Projection Operator in $\mathbb{R^3}$

I'm looking for an operator $\hat P$ in $\mathbb{R^3}$ such that $\hat P^2=\hat P$ that is also Hermitian
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Book Recommendation for Poisson Manifold and Deformation Quantisation

Can someone please recommend a basic introduction to the concept of Poisson Manifolds and Deformation Quantization. I'm new to Theoretical Physics and had to go through a lot of books before I even ...
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1answer
55 views

Riesz representation theorem finite dimnesional case

I am taking a Quantum Mechanics course not a Functional Analysis course so I have only had a very basic introduction to Hilbert Spaces. I don't understand where the if and only if statement arises ...
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1answer
16 views

$\frac{d^2}{dx^2} = \lim_{\delta x\to 0} [f(x-\delta x) - 2f(x) + f(x+\delta x))/\delta x^2]$ find a matrix

Given that $$\frac{d^2}{dx^2} = \lim_{\delta x\to 0} [f(x-\delta x) - 2f(x) + f(x+\delta x))]\cdot\frac1{\delta x^2}$$ find an appropriate matrix that could represent such a derivative operator, in a ...
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1answer
17 views

Showing a orthogonal basis is complete

$\psi_1 = \frac{1}{\sqrt{2}}$ $\psi_2 = \sqrt{\frac{3}{2}}x$ By shwoing that any arbitrary function $f(x)=ax+b$ can be represented as linear combination of $\psi_1$ and $\psi_2$, show that ...
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1answer
37 views

general two-state system

Consider a two-state system: The Hamiltonian takes the general form $$ H= \begin{pmatrix} a_1 & c-id \\ c+id & a_2 \\ \end{pmatrix} $$ where $a_1, a_2, c, d \in\mathcal{R}$ $H$ can be ...
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15 views

Berry's curvature equation

The following textbook Heisenberg's Quantum Mechanics shows an example of calculating Berry's curvature (top page on pg 518). It led to a following equation [1] $$ V_{m} = {- 1 \over B^2 } * i * ...
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Tridiagonal matrix w/trigonometric eigenvalues

Let $N=2^n$ for a natural number $n$ and $B$ be the $N\times N$ square matrix of $0$'s and $1$'s $$ B=\begin{pmatrix} 0 & 1 & 0 & \ldots & 0 \\ 1 & 0 & 1 & \ldots ...
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1answer
40 views

How to construct a complete set in $L^2(\mathbb{R}^3)$ starting with the Spherical Harmonics?

The Spherical Harmonics form a complete set of functions on the sphere $S^2$, so that any function of $f: S^2\to \mathbb{R}$ can be written uniquely as $$f(\theta,\phi)=\sum_{l=0}^\infty ...
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0answers
29 views

Expressing a solution to a differential equation in a more compact form

We know that $Ae^{ikx}$ is a solution for $k \geq 0$ and $Be^{ikx}$ is a solution for $k \leq 0$. So does the $N$ here depend on whether or not $k \geq 0$ or $\leq 0$? Also I do not understand how ...
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1answer
34 views

Tensor product distributive property?

Is it true that for vectors $a$, $b$, $c$, $d$ we have $$|a\rangle \otimes |b\rangle \langle c| \otimes \langle d|= |a \rangle \langle c| \otimes |b \rangle \langle d|?$$ So does this kind of ...
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1answer
41 views

How to define properly the radial and angular dependence of a function?

Recently I've came across the following situation when studying Quantum Mechanics: suppose we have two operators $A,B$ on the space of functions $L^2(\mathbb{R}^3)$ and suppose they have the following ...
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1answer
43 views

How to write this down in a rigorous manner?

I'm studying Quantum Mechanics and there's something I'm in doubt in how to write it down in a rigorous way. The idea is: consider a Hilbert space $\mathcal{H}$ and one hermitian operator $A\in ...
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1answer
44 views

Eigenvalue equation and separable solutions

Recently, studying Quantum Mechanics I found a doubt regarding separable solutions and eigenvalue equations for differential operators. Suppose we are considering some space $\mathcal{H}$ of functions ...
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21 views

How to integrate with a matrix in the measure?

I've been given the following integral (actually a path integral from quantum field theory). $$ Z(a;N) = \int d^{2n}M.exp(-\frac{1}{2}tr(M^2)-\frac{a}{M}tr(M^4)) $$ where M is a square matrix of size ...
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3answers
72 views

Explanation of this integral

$$\int_{-\infty}^{\infty} e^{-\frac{i}{\hbar}(p-p')x} dx = 2\pi\hbar\delta(p-p')$$ I don't quite understand how this integration leads to the right hand side. Any explanation is appreciated.
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1answer
47 views

reduced density matrix for the given composite system

Given the composite system of two qubits $$ |\psi^{AB}\rangle=\frac{1}{\sqrt{2}}(|0^{A}\rangle \otimes|0^{B}\rangle+|1^{A}\rangle\otimes|1^{B}\rangle) $$ with the density matrix of the composite ...
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0answers
27 views

trace inequality trivial on kernel

Let $x_1, x_2>0$ such that $x_1+x_2=0$. Then the concativity theorem claims that for any $n \times n$ matrix $ K$ and any positive matrices $A_1, A_2$ following inequality holds for all ...
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248 views

Can epsilon be a matrix?

Question In the following expression can $\epsilon$ be a matrix? $$ (H + \epsilon H_1) ( |m\rangle +\epsilon|m_1\rangle + \epsilon^2 |m_2\rangle + \dots) = (E |m\rangle + \epsilon E|m_1\rangle + ...
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1answer
65 views

Momentum is quantised in compact spaces?

Background One of the first examples given when studying quantum mechanics is the particle on a cylinder, or particle on a ring. One finds that because of the periodic boundary conditions, ...
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28 views

How does Ornstein-Uhlenbeck arise in quantum mechanics?

The Ornstein-Uhlenbeck operator, defined here is a self-adjoint operator on $L^2(\mathbb R^d)$ (with respect to Gaussian measure). I've heard it comes up in QM, and I'm wondering how? Does it ...