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
51 views

eigenfunctions of hamiltonian in 'natural units'

Let $H=0.5(p^2+q^2)$ be the hamiltonian in natural units. Let $f_{n}$ be the eigenfunctions of H. Show that $<f_{n},f_{m}>=1$ if n=m, and equal to 0 otherwise. Do this by using the ...
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0answers
87 views

Takhtajan's “Quantum Mechanics for Mathematicians”

I want to know the math that is required to read Takhtajan's "Quantum Mechanics for Mathematicians". From the book preview on Google, I gather that algebra, topology, (differential) geometry and ...
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0answers
33 views

Expectation value in a Quantum derivation

I'm reading a physics paper (John Bell's 1964 paper on the EPR paradox if anyone is physics-curious) and I'm having an issue following his derivation. It's the probability distribution stuff -not ...
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0answers
71 views

Physical dimension of a complex wave function.

If the probability density function $\rho$ is: $$\rho=\psi^{*}\psi=|\psi|^2$$ where $\psi$ is a complex wave function that is a solution to the Schrodinger equation, what are the physical dimensions ...
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42 views

How many projectors do two commuting self-adjoints have in their common spectral decomposition?

If $A$ and $B$ are two commuting observables on a Hilbert space of dimension $n$ say. So, $$A = \sum_{j \leq a} \lambda_j P_j $$ $$B = \sum_{i \leq b} \mu_i Q_i $$ $$I_n = \sum_{i \leq b} Q_j = ...
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1answer
175 views

1D Schrodinger/Laplace equation via finite differences: incompatible eigenvalues

I need to solve a variant of the 1D Schrodinger's equation equation using finite differences, so I decided to play a little bit with the real-space representation of some operators. Using the ...
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0answers
37 views

Statement about density operators, proof feedback

I'm trying to solve the following exercise: Let $V,W$ be finite dimensional inner product spaces over $\mathbb{C}$. Show that for every $\psi \in V\otimes W$ with $\langle \psi, \psi ...
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136 views

How to Diagonalize an Extremely Large Sparse Matrix in SLEPc/PETSc

Dear Friends, Recently I have started with learning SLEPc/PETSc, but I didn't find a way to solve my problem. I have to solve a big sparse matrix which is a two dimensional quantum ...
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36 views

Demonstrate basic property of Hermitian

I want to demonstrate that $(A|v\rangle)^* = \langle v|A^*$ Given only this $(AB)^*=A^*B^*$ and $(|v\rangle, A|w\rangle) = (A^*|v\rangle, |w\rangle)$ Is it possible? The difficulty is that i don't ...
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60 views

Pure Phase Number

I am read a solution (4.9) Here say: ... both $a, d$ are pure phases, so that it is always possible to find (non unique) real numbers $\alpha, \beta, \delta$ such that $a = ...
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2answers
748 views

Bessel function recursion relation

I'm reading a paper and the following set of radial equations is derived: $ -i \lbrack \partial_r + \frac{1}{r} \left( \frac{1}{2} - \nu \right) \rbrack u(r) = \pm k v(r) $ $ -i \lbrack \partial_r ...
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34 views

Three body problem with point interactions

I've studied the HVZ theorem for the three body problem interacting with regular potentials. I'd like to extend this result to the three body problem with point interactions (delta potentials). Is ...
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27 views

References for three body problems with Fermi statistic

I'm studying the three body problem with two fermions of unitary mass and another different particle. I need references of the HVZ theorem in this case. Is there someone who knows them?
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69 views

Fourier transform of integral kernel of the free resolvent

The free resolvent in $\mathbb{R}^3$ has this rapresentation $$(R_0(z)f)(x)=\int_{\mathbb{R}^3}\frac{e^{i\sqrt{z}|x-y|}}{4\pi|x-y|}f(y)dy$$with $\Im \sqrt{z}>0$. Then its integral kernel is ...
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0answers
109 views

Can a Hermitian operator on a tensor product space be represented as a sum of tensor products of Hermitian operators?

Consider a Hilbert space (or just a vector space over $\mathbb{C}$), which is a tensor product of several smaller Hilbert spaces: $$ H = H_1 \otimes \cdots \otimes H_n, $$ and let $\mathcal{H}$ be a ...
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172 views

Fourier transform of $\frac{g_i}{e^{\frac{\epsilon_i-\mu}{kT}}-1}$? Not Gaussian like with Fermi-Dirac statistics?

This equation $\bar n_i=\frac{g_i}{e^{\frac{\epsilon_i-\mu}{kT}}-1}$ is Fermi-Dirac statistics where variables are defined here. The classical equation i.e. the Maxwell Boltzman equation is Gaussian ...
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63 views

Convolutions of Path Integrals of Gaussian Functions

I was looking at a question on a physics forum (http://physics.stackexchange.com/questions/45955/splitting-light-into-colors-mathematical-expression-fourier-transforms) and I wanted a more ...
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2answers
65 views

Showing that $\langle p\rangle=\int\limits_{-\infty}^{+\infty}p |a(p)|^2 dp$

How do I show that $$\int \limits_{-\infty}^{+\infty} \Psi^* \left(-i\hbar\frac{\partial \Psi}{\partial x} \right)dx=\int \limits_{-\infty}^{+\infty} p \left|a(p)\right|^2dp\tag1$$ given that ...
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0answers
201 views

The general recipe for finding the conjugate of a complex function

I have the general recipe for finding the complex conjugate of a function down as follows: Suppose I have $f(z)$: Separate $f(z)$ into a sum of real and imaginary functions such that ...
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1answer
40 views

Quantum measurement - How does this commutator correspond to the following?

From the book Quantum Measurement by Vladimir B. Braginsky and Farid Ya.Khalili How do they go from 5.18 to 5.19?
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1answer
170 views

Solving $-\chi''(\epsilon)+\Big[\epsilon^2+\frac{2F}{hw}\sqrt{\frac{h}{hw}}\epsilon \Big]\chi(\epsilon)=\mu\chi(\epsilon)$

I am trying to solve this differential equation: $$-\chi''(\epsilon)+\Big[\epsilon^2+\frac{2F}{hw}\sqrt{\frac{h}{hw}}\epsilon \Big]\chi(\epsilon)=\mu\chi(\epsilon) \tag1$$ This was found ...
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4answers
58 views

Is raising a value to the second power the same as multiplying it it's complex conjugate?

I was watching a video on YouTube on Quantum Mechanics Concepts and saw that if you wanted to convert a probability amplitude to a probability, you square it. In the video he said that this was ...
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1answer
46 views

Squaring an operator

There is an excercise of squaring an operator in my book of quantun mechanics. The operator is $$\hat{A}=\frac{\mathrm{d}}{\mathrm{d}x}+x$$ And I should compute $\hat{A}^2$. He gives me a result ...
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1answer
371 views

Show that $A^{\dagger^{\dagger}} = A $

How do we show that $A^{\dagger^{\dagger}} = A $ without assuming $A$ to be a explicit matrix. That is, given a linear operator $A$, let us define $A^\dagger$ to be a unique operator such that ...
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1answer
53 views

What is the smallest non-trivial Hilbert space?

I came to know without proof or explanation that smallest non-trivial Hilbart space is generated by two basis vectors. What is its proof? One example I know. Denote $a = (0 , 1)$ and $b = (1 , 0)$. ...
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1answer
227 views

Solving time dependant Schrodinger equation in matrix form

If we have a Hilbert space of $\mathbb{C}^3$ so that a wave function is a 3-component column vector $$\psi_t=(\psi_1(t),\psi_2(t),\psi_3(t),)$$ with Hamiltonian $H$ given by $$H=\hbar\omega ...
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1answer
71 views

Inequality from Von Neumann entropy.

I am looking over some old course notes. First, Von Neumann entropy is defined. The Von Neumann entropy of a system described by a density matrix $\rho$ is defined by $S(\rho)\equiv ...
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1answer
115 views

Partial trace of a system with isolated evolution

Let $\rho_{AB}$ be the state of a composite quantum system with state space $H_A\otimes H_B$ (two finite dimensional Hilbert spaces). Now assume that $A$ and $B$ are isolated and suffer a unitary ...
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3answers
1k views

Re-writing in sign basis.

$\newcommand\ket[1]{\left\vert #1\right\rangle}$ Let $\ket\phi = 12 \ket{0} + 1 + 2\sqrt{i2}\ket{1}$. Write $\ket\phi$ in the form $\alpha_0\ket{+} + \alpha_1\ket{-}$. What is $\alpha_0$? I came ...
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1answer
432 views

Vector space generated by the tensor products of pauli matrices

Let $\sigma_0,\sigma_x,\sigma_y,\sigma_z$ stand for the $2\times 2$ identity matrix and the well known pauli matrices: \begin{equation} ...
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1answer
39 views

What math do I want to focus on for 1) quantum computing and 2) quantum physics? [closed]

What math do I want to focus on for 1) quantum computing and 2) quantum physics? I am interested in studying quantum computing and the Higgs Boson (quantum physics?) and ultimately working in some ...
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1answer
16 views

Re-expressing the Schrodinger Equation as a first order expansion.

I am reading an online text on quantum computing and the author expands and re-expresses the Schrodinger equation. I am not really sure as to the intermediate steps he used or what happened to the ...
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2answers
46 views

Constructing 5 by 5 Unitary matrices

I am trying to construct an arbitrary 5 x 5 Unitary matrix. Any example will be appreciated.
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1answer
28 views

Find $\psi\in L^2$ so that the overlap $\langle \psi(x+\delta),\psi(x-\delta)\rangle$ is as small as possible

Is there a way to find / characterise functions $\psi\in L^2$ which make the value of this integral small? $$ \text{Re}\int_{\mathbb R} \overline{ \psi (x+\delta)}\,\psi(x-\delta)\;\text d x $$ ...
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1answer
39 views

Complex Projective Line

How can I go about showing that a collection of all states is the complex projective line $CP^1$? All I understand at the moment is that an element in $CP^1$ is of the form ...
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1answer
47 views

Angular Momentum Operators: A quick question [closed]

Given $L_+|l,m\rangle \propto |l,m+1\rangle$ and $L_-|l,m\rangle \propto |l,m-1\rangle$ Why isn't it the case that $\langle l,m|L_+L_-|l,m\rangle = 1$? Perhaps naively, but I assumed $\langle ...
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1answer
31 views

Pauli Spin Operator General Rotation

I would like to calculate the Pauli spin operator rotation $$ U^{\dagger } \overset{\rightharpoonup }{\sigma } U$$ where $$\overset{\rightharpoonup }{\sigma }=\sigma _x \overset{\rightharpoonup ...
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1answer
53 views

pre-quantization on cotangent bundle $T^*M$

Let M be a manifold. Is there any pre-quantization on cotangent bundle $T^*M$?. In which case this pre-quantization is unique?
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1answer
70 views

Wave-function with boundary conditions

I'm given a complex wave-function $\psi(t, x)$, in one spatial dimension, which satisfies $i \partial_t \psi = \partial_x^2 \psi$, a simplified form of Schrodinger's equation in one spatial dimension. ...
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1answer
370 views

triple integral (quantum mechanics)

I recently started a quantum mechanics course after a long time with no serious maths and I'm having some problems with the most basic maths operations. Please, help me solve this triple integral ...
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1answer
132 views

Solution of the complex Ginzburg - Landau equation

Can someone show that it's possible to find a solution of the kind: $$\Phi(x,t)=R(x,t)\exp[i\Psi(x,t)]$$ of the complex Ginzburg - Landau equation: ...
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1answer
7 views

Sign of energy and solving the Schrodinger equation.

The particular problem that triggered my question is as follows: A particle of mass m is confined within the box $0 < x < a$, $0 < y < a$ and $0 < z < c$. The potential vanishes ...
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0answers
12 views

Numerical methods for computing exponential, if I have computed an exponential of a perturbated matrix

I need to compute the product $e^{H_1}\,e^{H_2}\,\ldots\,e^{H_n}$ for antihermitian matrices $H_j$ that do not commute and $H_i-H_{i+1}$ is small. Is there a numerically convenient way to compute ...
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0answers
14 views

Is this a compact group?

Consider $$x(t)=e^{-iHt}x(0)$$ and define $$G=\{ e^{-iHt}\mid t\in \mathbb{R}_{\geq 0}\}$$ Also write $\bar{G}$ to the closure of $G$ wrt the Euclidean topology. Q: is $\bar{G}$ a compact group? ...
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Probability of Measurement in a QM System - Angular Momentum and Spin

below is my question. Please read Question 2; I have done Q1, but Q2 references Q1, hence it is included. I think I have done the first part ($j=1/2$): $m = \pm {1 \over 2}$ as $-j \le m \le ...
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14 views

Perturbation Theory for Interacting Quantum Mechanical System

Hello all! I am rather stuck at the start of this question; once I can get going, I should be ok. The issue that I'm having is that I don't know (/ can't work out) what Hamiltonian I am supposed to ...
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21 views

Wavefunction of electron above grounded conductor

Consider a non-relativistic electron moving above a large, flat grounded conductor while it is attracted by its image charge, but cannot penetrate the conductor's surface. What is the Hamiltonian of ...
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1answer
25 views

How to show Thomas-Reiche-Kuhn sum rule

Given: $$ f_{ni} = \frac{2m\omega_{ni}}{\hbar}\big|\langle n | x |i\rangle\big|^2, $$ and a Hamiltonian in the form $$ H_0=\frac{p^2}{2m}+V(x), $$ I would like to show the following sum rule (known as ...
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31 views

Quantum Mechanics: time-dependent perturbations

I must solve this problem, but I'm not good at non constant differential equation system. Can somebody help me?
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30 views

A puzzling derivation about the expectation of [$\hat{X}$, $\hat{H}$]

a free particle of mass $m$, with Hamiltonian $\hat{H} = \frac {\hat{P}^2} {2m}$, where $\hat{P} = -i \hbar \frac{\partial} {\partial x}$. The commutative relation is given by $[\hat{X}, \hat{H}] ...