Tagged Questions

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

0
votes
0answers
9 views

Rotation in configuration space.

Let $R_\psi$ be the rotation in configuration space around a vector $\bf{e}_\psi$ for an angle $\psi$. How is that the space rotation in configuration space have: ...
4
votes
1answer
57 views

How to derive the hamiltonian from a non-classical lagrangian

For the non-classical lagrangian of a hydrogen atom: $$L = -mc^2 \sqrt{1-\frac{v^2}{c^2}} + \frac{e^2}{4 \pi \epsilon r}$$ We get that two conserved quantities are: $J = \gamma mr^2 \dot{\phi}$ and ...
3
votes
1answer
26 views

identify a tensor product by virtue of pure and entangled elements

If I take a tensor product of vector spaces (for simplicity - this could be more general) $V\otimes W$ then of course it is a vector space, but it has additional structure. One way to think about ...
1
vote
0answers
21 views

Eigenvector of a linear combination of operators is an eigenvector of each operator

Assume $H$ is a Hilbert space and $a_1,\dots,a_n$ are operators with Hermitian adjoints $a_1^*,\dots,a_n^*$, satisfying the canonical commutation relations. Define $N_j=a_j^*a_j$. Assume $v$ is an ...
0
votes
0answers
11 views

Show two quantum differential operators are the same $\hat{A} \psi(x)$ = $\psi(x+b)$

Consider an operator $\hat{A}$ = $e^{b*d/dx}$, where b is a constant. Show that $\hat{A}$$\psi(x)$ = $\psi(x+b)$ I'm guessing taylor exp
0
votes
0answers
20 views

Motivating Complexification of Lie Algebras? [migrated]

What is the motivation for complexifying a Lie algebra? In quantum mechanical angular momentum the commutation relations $$[J_x,J_y]=iJ_z, [J_y,J_z] = iJ_x, [J_z,J_x] = iJ_y$$ become, on ...
1
vote
1answer
41 views

Why is $\int_{\mathbb{R}^3} |p\rangle \langle p| d\lambda(p)=id$?

As I have written in the headline, I am curious how the relation $\int_{\mathbb{R}^3} |p \rangle \langle p| d\lambda(p)=id$ that physicists use, where $|p\rangle$ is the eigenfunction to the ...
1
vote
1answer
27 views

Clarifying understanding of Poisson Brackets in Hamiltonian Dynamics

I'm just reading through my textbook and would like to clarify my understanding of 'Canonically related variables'. In my textbook, it says that if $Q_i$, $P_i$ are related to $q_i$, $p_i$ by a ...
0
votes
1answer
26 views

Spectrum of Rank 1 Operators

Given $\psi$ and $\phi$ in a Hilbert space $H$, we let $T$ be the rank-1 operator such that $$T\varphi=<\psi,\varphi>\phi.$$ It is easy to find the eigenvalues of $T$, they are $0$ and ...
0
votes
0answers
21 views

Momentum Representation vs Position Representation

I have a question involving the representation of operators in momentum representation and position representation. The question is a little long, so I'll do my best to explain it. We are given an ...
0
votes
1answer
19 views

Can a general time-dependent finite-dimensional Schrödinger equation with complex Hamiltonian be transformed to one with real Hamiltonian?

Consider a general-form time-dependent Schrödinger equation: $$i\partial_tv=\hat Hv,$$ where Hamiltonian $\hat H$ is an Hermitian matrix (finite-dimensional for simplicity), and $v(t)$ is a complex ...
1
vote
1answer
23 views

When is the product of a hermitian unitary and another unitary hermitian?

I have a Hermitian unitary Ĥ and I want to know, if Û is some other unitary, when is Ĥ Û a Hermitian unitary? Specifically, what are the conditions on Û such that ...
1
vote
3answers
43 views

Does an operator of x commute with the differential operator with respect to x?

While solving a problem in Quantum Mechanics I got an expression $ \frac{d}{dx}V(x)-V(x)\frac{d}{dx} $. The first term is just the derivative of the potential but the second one seems a bit weird. Is ...
3
votes
1answer
110 views

Gradient and Laplacian in $S^1$

I'm trying to solve the particle in a ring problem without embedding the circle in $\Bbb R^3$, by instead taking the entire space to be $S^1$. Unfortunately, I haven't taken differential geometry yet ...
1
vote
0answers
48 views

Diagonalization of total angular momentum over creation operators for an isotropic harmonic oscillator?

You have an isotropic three dimensional quantum harmonic oscillator so the Hamiltonian is $$ H=\frac{p^2}{2}+\frac{r^2}2 $$ If you do the creation-annihilation operator-algebra trick and define ...
0
votes
0answers
19 views

Checking some work on an expectation value problem

I am working on a pretty simple problem (or so it seems it should be) from Griffith's QM text. The problem states: for the probability density function $\rho (x) = Ae^{-\lambda(x-a)^2}$ a) find A ...
2
votes
2answers
65 views

Question about Dirac notation

So from what I understand $\langle w | v \rangle=\vec w^* \cdot \vec v$. Ok. I'm fine with that notation. But then I've seen $\langle x | y \rangle=\delta(x-y)$ and $\langle x | p ...
2
votes
1answer
54 views

Quantum translation operator

Let $T_\epsilon=e^{i \mathbf{\epsilon} P/ \hbar}$ an operator. Show that $T_\epsilon\Psi(\mathbf r)=\Psi(\mathbf r + \mathbf \epsilon)$. Where $P=-i\hbar \nabla$. Here's what I've gotten: ...
0
votes
2answers
19 views

Quantum mechanics, conmutative operators.

If two operators $A$ and $B$ commute then any eigenvector of $A$ is an eigenvector of $B$? I know that if that happens there is a basis in which the eigenvectors of $A$ and $B$ are equal, but I don't ...
2
votes
3answers
65 views

How to do this integral $\int_{-\infty}^{\infty}{\rm e}^{-x^{2}}\cos\left(\,kx\,\right)\,{\rm d}x$ [duplicate]

How to do this integral $$\int_{-\infty}^{\infty}{\rm e}^{-x^{2}}\cos\left(\,kx\,\right)\,{\rm d}x$$ for any $k > 0$ ?. I tried to use gamma function, but sometimes the series doesn't converge.
2
votes
0answers
33 views

Invariant set under the flow defined by Schroedinger equations

I have to show that the set of functions of the form $$\psi(x,t)=c(t)^{-1}e^{\frac{-(x-q(t))^2}{2c(t)^2}}e^{ip(t)x}\hspace{1cm}c(t),p(t),q(t)\in\mathbb{R}$$ is invariant (as set) under the flow ...
0
votes
1answer
19 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 ...
3
votes
0answers
55 views

Understanding the Quantum Fourier Transform

I have a question about the Quantum Fourier Transform. I would like to understand it because I have a re-take for an exam. I have studied the provided / recommended literature extensively. ...
2
votes
0answers
44 views

Adding a delta function to a differential equation

So say I have a differential equation of the form: $$ \left(\alpha \frac{d^2}{dx^2}+fx^2 \right)y(x)=\lambda y(x) $$ Whose solutions are known (a Gaussian multiplying a Hermite polynomial.) I am now ...
0
votes
0answers
13 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 ...
0
votes
0answers
17 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? ...
0
votes
0answers
27 views

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 ...
0
votes
0answers
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 ...
1
vote
1answer
28 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 ...
2
votes
2answers
209 views

A vector space with countable and uncountable basis at the same time

Let $V$ be a vector space over $\mathbb{C}$. Two self-adjoint, commutable linear operators $\xi$ and $\eta$ act on it. Both of their eigenvectors form a complete set of $V$, but $\xi$'s eigenvalues ...
1
vote
0answers
30 views

Square Root of matrices and multiplication

In David Tong's Quantum field theory lecture notes, page 101, line 5, he shows that: $$(p.\sigma)(p.\overline{\sigma})=m^2.I_2$$ (I have placed the identity matrix $I_2$ for clarity) Then I don't ...
7
votes
1answer
113 views

Derivation of Schrödinger's equation

I recall a famous quote of the late physicist Richard Feynman: Where did we get that from? It's not possible to derive it from anything you know. It came out of the mind of Schrödinger. This ...
3
votes
1answer
54 views

Determining Degeneracies of Operator (Quantum Mechanics / Linear Algebra)

Hello all! Above is my question. I am fine all the way up to the final part about the degeneracy. I find counting degeneracies quite difficult, and this is no exception! I have really no idea why ...
2
votes
1answer
26 views

Evaluation of Operator-Valued Function

Hello all; above is my question! :) I've gone through all the way up to the final "and hence deduce that". Up to this point, the question has been fairly straightforward, but I have no idea how to ...
3
votes
1answer
49 views

How to use the Mehler kernel to get the solution of the Quantum harmonic oscillator with a given initial condition

In this wiki-article http://en.wikipedia.org/wiki/Mehler_kernel the fundamental solution of the differential equation for the Quantum harmonic oscillator is provided by the Mehler Kernel: ...
0
votes
1answer
56 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 $b = (1 , 0)$. ...
1
vote
1answer
169 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 ...
0
votes
1answer
20 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 ...
0
votes
0answers
32 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?
0
votes
2answers
48 views

Constructing 5 by 5 Unitary matrices

I am trying to construct an arbitrary 5 x 5 Unitary matrix. Any example will be appreciated.
2
votes
1answer
37 views

Questions about Commutators

My motivation is understanding some derivations in Quantum Mechanics, but since my question is purely Algebraic, I think it is suited for these forums. I have a general question and then a specific ...
1
vote
1answer
21 views

Vectors, columns and representations

When I learned quantum mechanics, my professor frequently emphasized that a matrix is a representation of an operator, not the operator itself, and a column ($1\times M$ matrix) is not a vector, it's ...
0
votes
0answers
31 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}] ...
8
votes
2answers
127 views

Does the shift operator on $\ell^2(\mathbb{Z})$ have a logarithm?

Consider the Hilbert space $\ell^2(\mathbb{Z})$, i.e., the space of all sequences $\ldots,a_{-2},a_{-1},a_0,a_1,a_2,\ldots$ of complex numbers such that $\sum_n |a_n|^2 < \infty$ with the usual ...
0
votes
1answer
44 views

Show: $\phi: \mathbb{R}^3 \rightarrow \mathcal{su}(2)$, $h \mapsto h \cdot \sigma$ is an isometric isomorphism

I found this problem and need some help. It is given: $$ \sigma_1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} $$ $$ \sigma_2 = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} $$ ...
4
votes
2answers
68 views

Basic Quantum Mechanics Concepts with Continuous Spectra

The following are a couple excerpts of the first chapter of Sakurai and Napolitano, Modern Quantum Mechanics, 2nd edition: Prior to these formulas, the text discusses the fundamental mathematics ...
2
votes
1answer
34 views

Promise of the hidden subgroup problem for $\mathbb{Z} mod 2$

I am going through the talk, Graph isomorphism, the hidden subgroup problem and identifying quantum states, by Pranab Sen. On slide 3, the promise of the problem is defined as follows. Here is the ...
0
votes
1answer
30 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? $$ \text{Re}\int_{\mathbb R} \overline{ \psi (x+\delta)}\,\psi(x-\delta)\;\text d x $$ ...
0
votes
0answers
86 views

Solution to second order differential equation Quantum harmonic oscillator

Hi I am reviewing some quantum mechanics and and have come across a solution to a differential equation that I do not understand in the derivation of the quantum harmonic oscillator. the Schrodinger ...
1
vote
2answers
61 views

Book on periodic Schrödinger operators

I am looking for good books about the spectral theory of periodic (1-dimensional) Schrödinger operators on a compact interval. A good reference I found was Reed/Simon Analysis of Operators (and a ...