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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4
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1answer
99 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 ...
0
votes
1answer
21 views

Complex conjugate part of TISE

i would like to ask about complex conjugate part of this equation why ? I know that and so
3
votes
1answer
110 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 ...
4
votes
3answers
47 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 $\mathcal H$. ...
0
votes
2answers
28 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 ...
2
votes
3answers
42 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
1
vote
0answers
15 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
62 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 ...
0
votes
1answer
17 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 ...
0
votes
1answer
18 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 $\psi_1$...
0
votes
1answer
41 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 ...
1
vote
0answers
19 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 * \...
7
votes
0answers
122 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 ...
1
vote
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 \sum_{m=-l}^{...
0
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0answers
30 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 we ...
1
vote
1answer
50 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
43 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
44 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 \...
2
votes
1answer
48 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
0answers
25 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 ...
1
vote
3answers
76 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.
1
vote
1answer
48 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
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 $0<p<...
8
votes
3answers
267 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 + \...
3
votes
1answer
70 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, ...
2
votes
0answers
30 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 ...
-2
votes
1answer
45 views

Differential equation

Hello if I have differential equation which is a function of x = differential equation which is function of t Can I say that the differential equation which is function of x = C= the ...
0
votes
1answer
68 views

What kind of geometric object is the Pauli spin matrix vector $\vec{\sigma}$?

In quantum mechanics we learn about the Pauli spin matrices: $$ \sigma_1 = \left( \begin{array}{cc} 0 & -1 \\ 1 & 0\end{array} \right) \hspace{0.25in} \sigma_2 = \left( \begin{array}{cc} i &...
0
votes
0answers
18 views

How to compute a complex-valued function specifying its amplitude and the ampltiude of its Fourier transform

I want to set the amplitude of a complex function (independent of its phase), and also the amplitude of its Fourier transform (again independent of its phase). Given these two functions (amplitude of ...
0
votes
0answers
5 views

Classify sub $C^*-$ algebras of $\mathbb{C}^{2 \times 2}$

Apparently if $A$ is a sub $C^*-$ algebra of the complex $n \times n$ matrices then we can characterize these subalgebras as block matrices.Now, for the case $n=2$ I was wondering if there is an easy ...
1
vote
1answer
39 views

Proving Ehrenfest Theorem $m\frac{d}{dt}\langle\overrightarrow{\hat{x}}\rangle\;=\; \langle\overrightarrow{\hat{p}}\rangle$

I'm trying to prove Ehrenfest Theorem: $$m\frac{d}{dt}\langle\overrightarrow{\hat{x}}\rangle\;=\; \langle\overrightarrow{\hat{p}}\rangle$$ We can just consider one component of $\overrightarrow{x}$, ...
0
votes
1answer
13 views

Potential such that the time-independent Schroedinger equation has an explicit solution

Consider the time-independent Schroedinger equation $\phi'' (x) +V(x)\phi(x)=\lambda \phi(x)$, $\quad x\in\mathbb{R}$. For testing my numerics, I would like to now: For which choices of $V$ are ...
0
votes
0answers
41 views

Solution to the Schrödinger equation

Let $$\begin{cases} i\frac{\partial}{\partial t} \Psi(x,t) = \Delta \Psi(x,t);\\ \Psi(x,0) = \varphi(x) \end{cases}$$ Why do physicists seek a solution of this equation in the form: $$ \Psi(x,t) = ...
0
votes
1answer
53 views

How to prove the Weyl identity?

In the post, there is a formula called Weyl identity: \begin{align} e^{-\frac{k}{n}a_n z^{-n}} e^{\frac{\ell}{n} a_{-n} w^n} = e^{\frac{\ell}{n} a_{-n} w^n} e^{-\frac{k}{n}a_n z^{-n}} e^{\frac{k\ell}{...
0
votes
0answers
26 views

How to normalize the order of an expression?

Suppose that for $p, p'>0$, $i,j>0$, \begin{align} [y_{i,p}, y_{-j, p'}] = -\frac{1}{p}(1-q_1^{p})(1-q_2^p)\tilde{c}_{i,j}^{[-p]} \delta_{p,p'}, \end{align} where $\tilde{c}_{ij}^{[-p]}$ is some ...
1
vote
0answers
56 views

Why is $\sum{\frac{1}{(j/2)!}(\xi)^j}\approx\sum{\frac{1}{j!}(\xi)^{2j}}$?

Why is $\sum{\frac{1}{(j/2)!}(\xi)^j}\approx\sum{\frac{1}{j!}(\xi)^{2j}}$? This equation is used for solving the Schrödinger equation of the harmonic oscillator at one point in Griffiths, ...
1
vote
2answers
48 views

Prove: the density operator of a pure state has exactly one non-zero eigenvalue equal to unity

What is the proper way of proving : the density operator $\hat{\rho}$ of a pure state has exactly one non-zero eigenvalue and it is unity, i.e, the density matrix takes the form (after diagonalizing):...
1
vote
1answer
45 views

Solving $\ddot{x} + \omega^2x = 0$: Classical Path for a Simple Harmonic Oscillator

I have posted in Math rather than Physics as the problem is mainly abuse(?) of trig identities rather than physics. This comes from an exercise within some lecture notes on Feynman Path integrals and ...
0
votes
1answer
67 views

Prove: Expectation value is the weighted average

How do we mathematically prove that the expectation value of an operator is the weighted average, $$ \langle\hat{A}\rangle=\langle \psi|\hat{A}|\psi\rangle=\sum_{n}a_{n}P(a_{n}) \space \space \space? ...
1
vote
1answer
77 views

Prove $\exp{i\frac{\pi}{2}(-1+\sigma_{i})}=\sigma_{i}$

How do we prove $e^{{i\frac{\pi}{2}(-1+\sigma_{i})}}=\sigma_{i}$ ? where $\sigma_{i}:$Pauli matrix and $1=$ Identity matrix Note: I understand that $i\frac{\pi}{2}(-1+\sigma_{i})$ is anti-hermitian ...
0
votes
1answer
59 views

Help With the Proof of the Spectral Decomposition (Quantum Mechanics)

$\newcommand\dag\dagger$ I need to present a proof of the spectral decomposition and I need help in some parts. I will state the theorem and the proof indicating where help is needed. I know this is ...
0
votes
0answers
12 views

Ladder operator identity

Define $n=(x + iy)/\sqrt{2}L$ and $\overline n=(x - iy)/\sqrt{2}L$. Also, $\partial_n$ = $L(\partial_x - i \partial_y)/\sqrt{2}$ and $\partial_\overline n$ = $L(\partial_x + i \partial_y)/\sqrt{2}$....
0
votes
1answer
39 views

Partial derivative of a complex number

Given $n=(x + iy)/\sqrt{2}L$ and $\overline n=(x - iy)/\sqrt{2}L$. $\partial_n=\partial/\partial n$, $\partial_x=\partial/\partial x$, $\partial_y=\partial/\partial y$ Show that $\partial_n$ = $L(\...
1
vote
2answers
48 views

2D linear schrodinger equation

I am trying to solve a 2D Schrodinger equation of the following form. This is in the context of Partial Differential Equations. \begin{align} iu_t + \frac{1}{2} (u_{xx}+u_{yy}) & = 0 \\ \end{...
1
vote
2answers
38 views

Integral of an operator

In quantum mechanics we know that if $q$ corresponds to a complete set of parameters characterizing a quantum system, then the state vectors $|q\rangle$ satisfy the following identity: $$\int |q\...
2
votes
0answers
45 views

Functional integration and Feynman path integrals in wolfram alpha

Is it possible to do Feynman path integrals in wolfram alpha? Say for a free quantum mechanical particle. The reason I am interested in this is because I would like to see how it arrives at the ...
0
votes
0answers
22 views

Why can lorentz transformation be expressed like this

Why can the lorentz transformation: https://upload.wikimedia.org/math/e/3/e/e3ee37f49f0adb02bc81590cb697d4d0.png Also be expressed as https://upload.wikimedia.org/math/0/d/6/...
2
votes
0answers
44 views

Lorentz transformation proof

For an occurence, we can choose coordinates ct and x (calculating both time and space in length) ct and x thereby take on the form of a two dimensional vector. Show that the same coordinates in S' ...
1
vote
0answers
33 views

Kronecker delta representation of a matrix (Quantum raising / lowering operators)

The Kronecker Delta is commonly used to represent a diagonal matrix: $$ a_i \delta_{ij}=\left( \begin{array}{ccc} a_1 & 0 & 0\\ 0 & a_2 & 0\\ 0 & 0 & a_3 \end{array}\right) $$ ...
0
votes
0answers
48 views

Nonlinear Schrödinger Equation

I have to find equation and starting condition to solve Nonlinear Schrödinger Equation with periodic edge condition. This method should control the propagation of fiber optical signal. In details I ...