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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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 ...
2
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0answers
38 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. ...
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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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0answers
42 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 ...
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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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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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13 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 ...
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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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0answers
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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2answers
126 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 ...
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25 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 ...
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98 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 ...
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1answer
50 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
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1answer
19 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
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1answer
39 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: ...
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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
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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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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0answers
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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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.
2
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1answer
36 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 ...
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1answer
19 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 ...
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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}] ...
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2answers
123 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 ...
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1answer
42 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: $$ \sigma_1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} $$ $$ \sigma_2 = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} $$ ...
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2answers
64 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 ...
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1answer
32 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 ...
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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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0answers
68 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 ...
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2answers
52 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 ...
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1answer
37 views

Transpose of a differential operator

Let $H$ be a diagonalizable matrix (not necessarily Hermitian). Then, it induces a biorthogonal left and right vectors, such that $$ ...
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0answers
18 views

When can we get discrete spectrum?

Suppose that $T$ is a densely defined closed operator on a separable Hilbert space $H$. Form $N = T^*T$. Assume further that $T$ has a finite dimensional kernel and satisfies the commutation relation ...
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17 views

Interesting question about a measurement using $J^2$

I really dont understand how to do part d)iv) on this question. This seems strange as it is only worth 2 marks? What step am I missing, I feel this may be rather obvious to others. for part d)i) I ...
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1answer
39 views

Finding eigenvalues of a two-state system

Let $$A=\left[\begin{matrix} 2 & -i \\ i & 2 \end{matrix}\right],$$ Show that $U_1 = \dfrac{1}{\sqrt{2}}(\Psi_1+i\Psi_2)$ and $U_2 = \dfrac{1}{\sqrt{2}}(\Psi_1-i\Psi_2)$ are ...
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2answers
30 views

Operators in Quantum Theory

Let $U$ be a unitary operator on a Hilbert Space, and let $\phi$ be an eigenvector of $U$ with eigenvalue $\mu$. Show that $|\mu|=1$ ? I know that if $U$ is unitary then $UU^{+}=UU^{-1}=I$ but I'm ...
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36 views

Problem with commutator relations

part a) is fine. part b) is not. A commutator is defined as, for operators $A$ and $B$, $[A,B]=AB-BA$. [SOLVED]I get that $H(\lambda)=e^{-\lambda D}Ce^{\lambda D}$, $H'(\lambda)=-De^{-\lambda ...
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0answers
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group cohomology for SO(3) and SO(3,1)

I am studying relativistic quantum mechanics and I have encountered the concept of projective representation for a group. I have read in http://groupprops.subwiki.org/wiki/Projective_representation ...
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1answer
24 views

Easy exercise operators on Hilbert space

Let $(H,\langle\,|\,\rangle)$ be a separable Hilbert space on $\mathbb{C}$. $\rho_{\psi}:=\langle\psi,\,\rangle\psi$, where $\psi\in H$ is such that $\|\psi\|=1$. ...
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2answers
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Orthogonal Projector

Let $(H,\langle\,|\,\rangle)$ be a separable Hilbert space on $\mathbb{C}$. $P_{\psi}:=\langle\psi,\,\rangle\psi$, where $\psi\in H$ is such that $\|\psi\|=1$. I have to prove that $P_{\psi}$ is an ...
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0answers
54 views

Infinite double potential well

So I'm having problems with the double infinite potential well given by I have to use the fact that the potential well is symmetric about $x=0$. I have solved the Schrödinger equation in all the ...
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6answers
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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 ...
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Cyclic Properties of the trace in Quantum Field Theory

I'm trying to figure out what's going on in this paper here between lines 10 and 11. But I'll give a brief rundown of what's going on. We are trying to compute the trace of the exponential operator ...
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2answers
70 views

Trace in non-orthogonal basis

In Dirac notation we can define the trace of an operator in Hilbert space $\rho$ as the follows, $Tr(\rho)=\sum\limits_{|s\rangle \in B} \langle s| \rho |s\rangle$ where B is some orthonormal ...
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0answers
43 views

Standard deviation of a quantum walk?

The standard deviation of a classical random walk with $n$ steps is $\sqrt n$ - Standard deviation of a random walk. I have read in many places that the standard deviation of a quantum walk $n$ with a ...
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1answer
24 views

Unitary transformation between complete and orthonormal bases

I'm using the Dirac notation for vectors here, since this is a quantum mechanics question. Suppose the complete orthonormal bases $\{|\psi_n\rangle\}$ and $\{|\psi{'}_n\rangle\}$ are related by the ...
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22 views

In the Stinespring dilation theorem, what is the minimum dimension for which a dilation Hilbert space of this form is guaranteed to exist?

This may look like a problem that could easily be looked up, but it's not quite as easy as it first appears, hence my asking. I'm going to phrase my question in terms of the "Schroedinger picture" ...
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1answer
19 views

Positivity and Complete positivity of Simon Map

Simon map in a specific basis is defined as $$ \left[ {\begin{array}{ccc} A & B & C \\ D & E & F \\ G & H & I \\ \end{array} } \right] \rightarrow \left[ ...
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0answers
21 views

Does a better formula exist for converting quantum state into binary?

First, would like to say that I am not a mathematician so by no stretch of the imagination could I dream of resolving this problem. I have been working with a concept for copying real matter into ...
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1answer
48 views

Domain of the quantum free Hamiltonian in 1D

Consider the quantum free Hamiltonian $H_0 =-\frac{d^2}{dx^2}$ (the Laplacian on the real line). I want to show that it is (essentially) self-adjoint in its domain of definition. The usual approach ...
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1answer
42 views

Commutator proof

I have a proof in my book I dont fully understand. The author is proving that if $[A,B]=1$ then $[A,B^n]=nB^{n-1}$. The proof is really short, it is only one line of equations, but I dont understand ...