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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27 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
37 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
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1answer
33 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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0answers
32 views

Integral of $\vec \nabla f(x)$

I was trying to prove a theorem and I came across this integral as a part of the theorem: $$ \int d^3x \left(\, \psi \vec r \,\nabla^2 \psi^* - \psi^* \vec r \,\nabla^2 \psi \right)$$ I was thinking ...
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0answers
15 views

Relationship between Lippmann-Schwinger integrals of different dimensions

Define $G_n (\mathbf{x},\mathbf{x}')$ as $$ G_n (\mathbf{x},\mathbf{x}') = \lim_{\epsilon \to 0^{+}} \left[\dfrac{1}{(2 \pi \hbar)^{n}} \int_{\mathbb{R}^{n}} \mathrm{d}^{n}\mathbf{p} \dfrac{e^{i ...
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0answers
16 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
63 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
34 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
21 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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3answers
229 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
59 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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26 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 ...
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1answer
33 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 ...
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1answer
54 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 ...
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0answers
17 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 ...
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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 ...
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1answer
37 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 $\overrightarrow{x}$, ...
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1answer
12 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 ...
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0answers
36 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) = ...
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1answer
35 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}} ...
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0answers
24 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 ...
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0answers
55 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, ...
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2answers
31 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 ...
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1answer
29 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 ...
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1answer
62 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? ...
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1answer
73 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 ...
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1answer
31 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 ...
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8 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 ...
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1answer
38 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 $\partial_n$ = ...
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2answers
39 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 \\ ...
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2answers
37 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 ...
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1answer
26 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 ...
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0answers
20 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 ...
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0answers
30 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' ...
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0answers
27 views

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

The Kronecker Delta is commonly used to represent a diagonal matrix: $$ a_i \delta_{ij}=\left( \begin{array}{ccc} a_1 & 0 & 0\\ 0 & a_2 & 0\\ 0 & 0 & a_3 \end{array}\right) $$ ...
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0answers
39 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 ...
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1answer
37 views

Do infinite dimensional Hermitian operators admit a complete basis of eigenvectors?

I'm currently taking a quantum mechanics course. We have proven that hermitian operators always have real eigenvalues, that we can choose the eigenvectors to be orthonormal, and that finite ...
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0answers
20 views

Compactness of operator related to harmonic oscillator hamiltonian

Let define the operator $$A = -\partial_x^2-\partial_y^2+2iy\partial_x+y^2$$ with domain in the Schwartz space of the rapidly decreasing functions. Using partial Fourier transform in $x$, the ...
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2answers
81 views

Understanding Bell's inequality vs. quantum mechanics

I have difficulty to understand how Bell's inequality rules out local hidden variable theory. It seems to me that there is some hidden variable in the Kolmogorov's axiomatization of probability ...
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0answers
25 views

General Solution to linear Schrodingers Equation

I am trying to find a solution to $$\displaystyle \left[-\frac{1}{2}\nabla^2 - \frac{2}{r} + C(r)\right]\phi(r) = E\phi(r)$$ where C(r) is a known function of r. I am just looking for some help on ...
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1answer
46 views

Finding eigenvectors and eigenvalues of an operator in quantum mechanics

I'm reading this book about quantum mechanics. The author wants the reader to do the following exercise: Find the eigenvectors and eigenvalues of the operator $$A = \pmatrix{\cos \theta & ...
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15 views

How do we take the limit of this quantum operation?

I am wondering how to take the following limit: \begin{align} L= \lim_{\tau \to \infty} \frac{1}{\tau} \int_{-\tau/2}^{\tau/2} dy \, \left(1 - \frac{1}{\sqrt{ \pi} \sigma } ...
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1answer
31 views

Finding hilbert space basis

I already have two vectors expressed in basis $\{|u_1 \rangle, |u_2\rangle, |u_3\rangle\}$: $$|\phi_1\rangle = \frac{1}{\sqrt{3}}(|u_1\rangle + i |u_2\rangle - |u_3\rangle) $$ $$|\phi_2\rangle = ...
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1answer
32 views

Why an unbounded operator defined everywhere fails to be closed?

The Toeplitz theorem says : If a closed operator is defined everywhere, then it is continuous. So if a non continuous operator is defined everywhere, it is not closed. But why is it not closed? What ...
2
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2answers
38 views

Different definitions for the complex inner product.

I have asked this question on P.S.E. and have gotten some nice answers, but I felt I might get even more satisfactory answers if I post it here. "I have just now noticed that Griffiths (in his book ...
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1answer
40 views

Why are projective representations of a group classified by the second cohomology group?

I'm reading about the classification of bosonic SPT's, and I came across this statement: projective representations, where $v(g_1)v(g_2)=\alpha(g_1,g_2)v(g_1g_2)$, $v(g_1)$ being the transformation ...
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10 views

Separating an Odd Partial Derivative

I'm applying a change of variables to the Schrodinger Eq, which means that my partial derivatives have gotten rather wonky. In short: I've created new variables $\alpha \equiv x_1+x_2$ and $\gamma ...
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1answer
66 views

Lagrangian invariant under left and right multiplication by unitary matrices, slick way to see?

Is there a slick way to see that the Lagrangian$$\mathcal{L} = \text{Tr}(\partial^\mu G\partial_\mu G^{-1}),$$where $G$ is an $N \times N$ unitary matrix, is invariant under left and right ...
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2answers
56 views

Finding the adjoint of a linear operator using Dirac notation.

Trying to answer this question and I am fairly new to Dirac Notation: Let $|\psi\rangle$ and $|\phi\rangle$ be two states in a Hilbert Space and consider the linear operator ...
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1answer
24 views

Why the principal quantum number (n) is maximum 7? [closed]

Can you explain why actually the principal quantum number is maximum 7?