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.

learn more… | top users | synonyms

1
vote
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 $\theta$,$\phi$ ...
1
vote
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 ...
-1
votes
2answers
55 views

Quantum mechanics. [closed]

If $P(x) =Axe^{-x^2/a^2}$ for $x > 0$ and $P(x) = 0$ for $x < 0$, find $A$ such that $$\int_{-\infty}^{\infty}P(x)dx=1$$ And hence calculated the expected value of $\left\langle x\right\rangle$ ...
3
votes
1answer
39 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 ...
1
vote
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 ...
2
votes
0answers
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 ...
0
votes
0answers
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 ...
1
vote
0answers
17 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 ...
3
votes
2answers
282 views

To show a set of projectors summing to the identity implies mutually orthogonal projectors

The general setting is the study of positive operator measures in quantum mechanics http://en.wikipedia.org/wiki/Positive_operator-valued_measure instead of the projector operator measures. Going ...
1
vote
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 ...
1
vote
0answers
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 = ...
0
votes
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}}( ...
0
votes
0answers
29 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 ...
0
votes
0answers
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) = ...
1
vote
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 ...
0
votes
0answers
36 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 ...
3
votes
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} + ...
0
votes
0answers
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
votes
1answer
69 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 ...
15
votes
4answers
512 views

Correct spaces for quantum mechanics

The general formulation of quantum mechanics is done by describing quantum mechanical states by vectors $|\psi_t(x)\rangle$ in some Hilbert space $\mathcal{H}$ and describes their time evolution by ...
2
votes
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) ...
2
votes
0answers
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 ...
5
votes
0answers
64 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}) + ...
1
vote
0answers
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 ...
1
vote
0answers
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 ...
4
votes
1answer
81 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 ...
1
vote
0answers
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 ...
4
votes
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{ ...
0
votes
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
4
votes
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 $\mathcal H$. ...
3
votes
1answer
91 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 ...
2
votes
3answers
39 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
0
votes
2answers
26 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 ...
1
vote
0answers
13 views

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 ...
0
votes
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 ...
2
votes
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 ...
0
votes
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 ...
0
votes
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 ...
7
votes
0answers
111 views

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 ...
0
votes
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 ...
0
votes
0answers
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 * ...
1
vote
1answer
33 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 ...
1
vote
1answer
38 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 ...
0
votes
0answers
28 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 ...
1
vote
1answer
39 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 ...
0
votes
1answer
41 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 ...
7
votes
2answers
507 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 ...
18
votes
7answers
4k views

Mathematics needed in the study of Quantum Physics

As a 12th grade student , I'm currently acquainted with single variable calculus, algebra, and geometry, obviously on a high school level. I tried taking a Quantum Physics course on coursera.com, but ...
8
votes
3answers
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 + ...
0
votes
0answers
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 ...