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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21
votes
2answers
508 views

Why do odd dimensions and even dimensions behave differently?

It is well known that odd and even dimensions work differently. Waves propagation in odd dimensions is unlike propagation in even dimensions. A parity operator is a rotation in even dimensions, ...
12
votes
6answers
1k views

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 ...
12
votes
4answers
366 views

Correct spaces for quantum mechanics

The general formulation of quantum mechanics is done by describing quantum mechanical states by vectors $|\psi_t(x)\rangle$ in some Hilbert space $\mathcal{H}$ and describes their time evolution by ...
11
votes
1answer
204 views

Does this notion of morphism of noncommutative rings appear in the ring theory literature?

Definition: Let $R, S$ be two rings. A classical morphism $\phi : R \to S$ is a function from elements of $R$ to elements of $S$ which restricts to a homomorphism (of rings, in the usual sense) on ...
11
votes
1answer
400 views

Quantization of angular momentum: is Dirac's proof wrong?

I'm trying to understand the physicist's proof of the theorem on the spectral structure of angular momentum operators (I'm being told that this proof is due to Dirac). I will refer to Ballentine's ...
10
votes
1answer
350 views

Topological Quantum Field theories

I was wondering about the following on TQFTs. It is said that TQFTs have vanishing Hamiltonians $\hat{\mathcal{H}}$. Firstly, I would like to ask: Why is this so? Secondly, consider the ...
8
votes
1answer
411 views

Physical (Quantum Mechanical) Significance of completeness of Hilbert Spaces.

I'm not sure if the question is very 'mathematical',but I'm asking any way. I have a basic knowledge of quantum mechanics and I'm studying Hilbert spaces. I was wondering what is the physical ...
8
votes
2answers
130 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 ...
7
votes
2answers
223 views

“Fun” question: anyone know why $e$ (Euler's Number) was chosen for wave functions?

First, let me say that this is merely something I have always wondered about, and can never seem to find a good reference for. I simply want to know... the geek in me. Why was $e$ (Euler's Number) ...
7
votes
1answer
190 views

'Quantum' approach to classical probability

Quantum mechanics defines a state of a system as a superposition of 'classical' states with complex coefficients, thus reducing many problems to linear algebra. Can classical statistics be approached ...
7
votes
1answer
355 views

Do these two sets of matrices form groups?

Stimulated by some Physics backgrounds, consider the following two sets of matrices. Notations and definitions:Let $A,B$ be two complex $n\times n$ matrices, then $\left [ A,B \right ...
7
votes
1answer
82 views

A non-integrable representation of the Heisenberg Algebra

Let $\mathfrak h$ be the Heisenberg algebra in dimension 1, generated by vectors $P$, $Q$ and $I$ satisfying $[P,Q] = I$, $[P,I] = [Q,I] = 0$. A representation of $\mathfrak h$ on a Hilbert space $X$ ...
7
votes
1answer
117 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 ...
6
votes
2answers
247 views

Orthogonal projections with $\sum P_i =I$, proving that $i\ne j \Rightarrow P_{j}P_{i}=0$

I am reading Introduction to Quantum Computing by Kaye, Laflamme, and Mosca. As an exercise, they write "Prove that if the operators $P_{i}$ satisfy $P_{i}^{*}=P_{i}$ and $P_{i}^{2}=P_{i}$ , then ...
6
votes
1answer
133 views

How to find interesting operators for a quantum system?

How can we find "interesting" operators for a quantum mechanical system? I can think of the following method: Given some system with an associated Hilbert space $V$ and Hamiltonian $H:V\rightarrow ...
6
votes
3answers
112 views

A Weird Contradiction about angular momentum operator in quantum mechanics

I am starting with the standard definition of an angular momentum operator in quantum mechanics given as $$\mathbf{L} = k(\mathbf{r}\times\mathbf{p}) = k(\mathbf{r}\times\nabla),$$ where ...
6
votes
1answer
274 views

Kernel of adjoint operator

This problem is puzzling me, even though it should be really simple. Let $L=-\partial_x^2 + \frac 1 2 x^{-2}$ be an operator defined on $D(L)=C^\infty_c(0,+\infty)\subset L^2(0,+\infty)$. Its adjoint ...
6
votes
1answer
83 views

Dirac Gamma matrix identity

In my library's (old -- 1996) copy of Peskin and Schroeder, there's an identity I'm struggling to prove. In my copy it occurs on page 42, between equations 3.28 and 3.29, but I don't know how well ...
6
votes
0answers
365 views

Studying quantum mechanics without physics background

I am a first year PhD math student, and I am wondering if I should study quantum mechanics even though I don't have an undergrad background in physics. I posted this question in physics ...
5
votes
3answers
301 views

Quantum Mathematics?

As per my last question, this has less to do with cold, hard, and fast calculations and more to do with the interplay between mathematics and philosophy...but as armchair philosophers aren't as hard ...
5
votes
2answers
441 views

Regarding Ladder Operators and Quantum Harmonic Oscillators

When dealing with the Quantum Harmonic Oscillator Operator $H=-\frac{d^{2}}{dx^{2}}+x^{2}$, there is the approach of using the Ladder Operator: Suppose that are two operators $L^{+}$ and $L^{-}$ and ...
5
votes
1answer
56 views

Defining conditional quantum probability

My knowledge of quantum mechanics is very limited, but I will try to ask a purely mathematical question here. If there is a text or resource that explains this, I would definitely appreciate any ...
5
votes
1answer
199 views

Physical interpretation of $q$-deformation

I am currently reading the paper Quantum Group Particles and Non-Archimedean Geometry by Volovich and Aref'eva. Here they discuss the difference between $q$-deformation and $\hslash$-deformation. In a ...
5
votes
1answer
85 views

Spectral theorem in Quantum Mechanics

In Quantum Mechanics one often looks at self-adjoint(unbounded and closed) linear operators $A,B$ that are defined dense on $L^2$. My question is: Is it true that if we have $[A,B]=0$, where $[,]$ is ...
5
votes
1answer
108 views

Quantum Information: Deutsch-Jozsa Algorithm

There is a step in the construction of this algorithm which I'm not understanding: $\displaystyle \left[\sum_x \frac{| x \rangle}{\sqrt{2^n}}\right]\left[\frac{ | 0 \rangle -| 1 \rangle ...
4
votes
4answers
362 views

Applications of Operator Algebras to modern physics

I think that recently I've started to lean in my interest more towards operator algebras and away from differential geometry, the latter having many applications to physics. But while taking physics ...
4
votes
1answer
118 views

If the expectation $\langle v,Mv \rangle$ of an operator is $0$ for all $v$ is the operator $0$?

I ran into this a while back and convinced my self that it was true for all finite dimensional vector spaces with complex coefficients. My question is to what extent could I trust this result in the ...
4
votes
2answers
73 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 ...
4
votes
1answer
1k views

Relationship between dual space and adjoint of a linear operator

I am having a hard time understanding the concept of adjoint of a linear operator. Given a finite dimensional Hilbert space $H$ over a field $F$, I know the dual space is the vector space $H^*$ of all ...
4
votes
1answer
67 views

Probability and Quantum mechanics

I don't quite understand how the probability language of sample spaces, $\sigma-$algebra, random variables, etc, fit into the quantum mechanics' formalism. To wit, we usually say that an observable ...
4
votes
1answer
80 views

Solving $y'' + (ax+b)y = 0$

This is a problem in quantum mechanics when one considers a linear potential; in physics-speak the equation would be written as $$\frac{d^2\psi}{dx^2} + \frac{2m}{\hbar^2}(E-ax)\psi = 0,$$ with ...
4
votes
1answer
113 views

General theory behind ladder operators

To derive the representation of SO(3) one uses the ladder operator method. What is the theoretical basis for this method? Often the ladder operators are simply stated in the textbooks of quantum ...
4
votes
5answers
120 views

General solution of $\frac{d^{2}u}{d\rho^{2}}=\frac{l\left(l+1\right)}{\rho^{2}}u$

In his discussion of the radial wave function of hydrogen Griffiths (Introduction to Quantum Mechanics, 2nd ed, p.146) gives the general solution ...
4
votes
2answers
127 views

Geometric meaning of block-diagonalization of a matrix

some times we need to do block-diagonalization in favor of easy computation. For instance, for a matrix like this $$ \begin{bmatrix} A_{11} & A_{12} & A_{13} & 0 & 0 & 0\\ A_{21} ...
4
votes
1answer
220 views

Uniqueness of solutions to Schrödinger's equation

Consider \begin{cases} u_t(x,t)=\sqrt{-1} u_{xx}(x,t), \quad (x,t)\in[0,2\pi]\times[0,\infty)\\ u(x,0)=f(x),\quad x\in[0,2\pi] \end{cases} where $f(\cdot) \in C^\infty$ is periodic with period ...
4
votes
1answer
431 views

Dirac Orthonormality Proof - Can't Make Sense of Complex Integral

I'm having trouble rationalizing a particular statement that is, surely, present in many quantum mechanics textbooks. The following statement comes from the orthnormalization condition for ...
4
votes
1answer
113 views

Quantum Hermiticity Bra-Ket notation please

If $A$ and $B$ are Hermitian operators, show that $$C~:=~i[A,B]$$ is Hermitian too. My work: $$\begin{gather} C=i(AB-BA) \\ \langle\psi\rvert C\lvert\phi\rangle = i\langle\psi\rvert ...
4
votes
1answer
534 views

Derivative of a bra?

I understand that $$ \frac{\mathrm d}{\mathrm dt} \langle\psi|\psi\rangle =\left[\frac{\mathrm d}{\mathrm dt} \langle\psi|\right]|\psi\rangle + \langle\psi|\left[\frac{\mathrm d}{\mathrm ...
4
votes
0answers
40 views

Two Body Schrodinger Equations

I have a question involving the eigenvalues of a two-body Schrodinger equation. Let $$H=-\frac{1}{2m}\Delta_{x_1}-\frac{1}{2m}\Delta_{x_2}+\frac{e^2}{|{{x_1}-{x_2}}|}$$ over the Hilbert space ...
4
votes
1answer
62 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 ...
4
votes
0answers
51 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 ...
3
votes
3answers
146 views

Unitary invariance

Why is it that for any non-negative matrix $M$ and unitary matrix $U$, we have $$\sqrt{UMU^\dagger}=U\sqrt{M}U^\dagger$$? This question has to do with Problem 2c from this sheet. I think I am ...
3
votes
4answers
224 views

Which book to read on quantum-related mathematics

Recently I watched the "Big Bang Theory" and decided to google about quantum mechanics. It really intrigued me. But I also understood that I am too stupid to understand even the basic mathematics in ...
3
votes
5answers
895 views

Hydrogen atom in partial differential equations

For the hydrogen atom, if $$\int |u|^2 ~dx = 1,$$ at $t = 0$, I am trying to show that this is true at all later times. What I need help is with differentiating the integral with respect to $t$, and ...
3
votes
5answers
236 views

How to prove $e^{A \oplus B} = e^A \otimes e^B$ where $A$ and $B$ are matrices? (Kronecker operations)

How to prove that $e^{A \oplus B} = $$e^A \otimes e^B$? Here $A$ and $B$ are $n\times n$ and $m \times m$ matrices, $\otimes$ is the Kronecker product and $\oplus$ is the Kronecker sum: $$ A \oplus B ...
3
votes
1answer
58 views

Hermite Differential Equation - Non-integer values of $\lambda$

The Hermite differential equation, given by : $$ \frac{d^2y}{dx^2} - 2x \frac{dy}{dx} + \lambda y = 0 $$ has solutions of the $$ y(x) = \mathcal{H_n(x)} $$ when $ \lambda \: \epsilon \:\mathcal{Z_+} ...
3
votes
2answers
91 views

Selfadjointness of Coulomb Hamiltonian in d>=3 dimensions

I know that the Coulomb-Hamiltonian $H=-\Delta - |\cdot|^{-1}$ is self-adjoint with $dom(H)=H^2(\mathbb R^3)$. This follows by the Kato-Rellich-Theorem. Has the corresponding quadratic form a form ...
3
votes
2answers
183 views

Can the $0$-norm represent determinism?

In Scott Aaronson's Quantum Computing since Democritus, he presents classical probability theory as based on the $1$-norm, and QM as based on the $2$-norm. Call $\{v_1,\ldots,v_N\}$ a unit vector ...
3
votes
2answers
104 views

Mathematical explanation of problems behind time and space derivatives being second order

$\left(\nabla^2 - \frac{1}{c^2}\frac{\partial^2}{\partial t^2}\right)\phi = \frac{m^2c^2}{\hbar^2}\phi$ with the wave function $\phi$ being a relativistic scalar: a complex number which has ...
3
votes
1answer
276 views

What are the requirements for a “test” function to show operators commute?

To show that two operators $\hat{A}$ and $\hat{B}$ commute, we can check whether $\hat{A}\hat{B}f(x)$ = $\hat{B}\hat{A}f(x)$. My question is regarding the function $f(x)$. To check that $\hat{A}$ and ...