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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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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0answers
23 views

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 ...
-2
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0answers
12 views

product of two spherical harmonics. [closed]

From Sakurai eq. 6.4.21a we have that $$\langle {\bf k} \vert E,l,m \rangle=\frac{\hbar}{\sqrt{M k}}\delta\left(E-\frac{\hbar^2 k^2 }{2M}\right) Y_l^m({\bf\hat k}),$$ where $M$ is the mass of the ...
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2answers
36 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
19 views

How does the hydrogen atom know which frequencies it can emit photons at? [migrated]

At university, I was shown the Schrodinger Equation, and how to solve it, including in the 1/r potential, modelling the hydrogen atom. And it was then asserted that the differences between the ...
3
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0answers
31 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 ...
2
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1answer
13 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 ...
0
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0answers
15 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
13 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
20 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 ...
0
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0answers
15 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
29 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 ...
1
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1answer
34 views

How can we show that a map is a completely positive map?

I am doing a homework problem where I have to find whether the map $$ \rho ~\rightarrow~ {\rm tr}(\rho) I - \rho $$ is completely positive. If the map is not completely positive, a counter-example ...
0
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1answer
28 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 ...
0
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1answer
17 views

Interchange exponential of operators in quantum mechanics

What is the formula for interchanging products of exponential operators in quantum mechanics., i.e. I want to write $e^Ae^B = e^{B+...}e^A$
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0answers
15 views

Constructing a median-median inequality for a pair of distributions

Let $X_1$ and $X_2$ be continuous real random variables with medians $\mu_1$ and $\mu_2$ respectively. Can an effective upper bound on $|\mu_1-\mu_2|$ be found in terms of the variances $\sigma_{ij}$? ...
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0answers
20 views

Index notation for unitary matrices

I was wondering if someone could confirm this for me. I'm attempting to re appropriate a paper into matrix notation but i keep getting confused. I first have a unitary matrix that makes a ...
3
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0answers
76 views

Time-independent magnetic Schrodinger equation

Let $B$ be a magnetic field, let $\mathbf{a}$ be a vector and let $\Psi$ be the wave function. If $\mathcal{H}(B) = (-i\nabla + B\mathbf{a})^2$, where $\nabla$ is the gradient, then the ...
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0answers
35 views

Short examples that are/are not quantum-ergodic

Are there any considerably short examples of manifolds that are/aren't quantum ergodic, or quantum unique ergodic? Note that a (compact) Riemannian manifold is said to be quantum ergodic if ...
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0answers
11 views

Reversing Summation and Product of Sequences [closed]

I am working on a proof for some homework, so I will leave all details out. I can prove it if this step is mathematically sound: ...
2
votes
5answers
177 views

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

how to prove $e^{A+B} = $$e^A$$e^B$ where A and B are matrices? The operations '+' and '*' are defined such that AI + IB = A+B, where I is the identity matrix. I suppose these are called Matrix ...
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1answer
25 views

Hermitian transformation

I am studying Quantum Mechanics, and the book by Griffths introduces some concepts that I have never come across in my Math courses. I will try to summarize my questions, and hopefully someone will be ...
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0answers
21 views

Quantum: Toffoli gates

How do I prove that a toffoli gate is a controlled CNOT gate. i.e, $G_{Toffoli} = |0\rangle\langle 0| \otimes I + |1\rangle\langle 1|\otimes G_{CNOT}$. I am not sure how to approach this, thoughts? ...
0
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0answers
14 views

relationship between the CNOT gate and the I and X single-qubit gates

I need to prove this relationship: $G_{CNOT} = |0\rangle\langle 0| \otimes I + |1\rangle\langle 1|\otimes X$. So I think I need to show that both sides are a linear map on $H \otimes H \otimes H$ So ...
8
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1answer
174 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 ...
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0answers
25 views

Quantization and evaluating a complex function on multiple Riemann sheets

In Lecture 3 of his mathematical physics course, Carl Bender mentions that the evaluation of a complex function on multiple Riemann sheets can be used to describe the quantization of the laws of ...
2
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0answers
39 views

Axiom of choice in proof of Wigner's theorem?

In Appendix A of chapter 2 of "The Quantum Theory of Fields," vol. 1, Weinberg presents a proof of Wigner's theorem: given a symmetry transformation $T$ of rays, one can extend this to a symmetry ...
0
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1answer
19 views

Is the spectrum of a product of two operators, $AB$, invariant under $UAU^{\dagger}$ for unitary $U$?

This question is about linear operators on a Hilbert space. If necessary, the Hilbert space can be assumed to be finite dimensional. I have two Hermitian operators, $A$ and $B$. Do we have $$ ...
5
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1answer
71 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 ...
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1answer
44 views

Problem with a wavefunction in Quantum Mechanics (math) (Book solution possibly wrong?)

Well there is a problem in my book which lists this problem: Calculate the probability that a particle will be found at $0.49L$ and $0.51L$ in a box of length $L$ when it has (a) $n = 1$. Take the ...
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0answers
62 views

Tensor product of affine space and algebra

When reading about Quantum Mechanics, I always feel a bit disappointed when physicists consider that (for example) the 3-dimensional position of a particle must be decomposed into 3 coordinates $x, y, ...
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1answer
22 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 ...
2
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0answers
56 views

Does $Z_A$ exist such that $\exp(X+A) = \exp(X) Z_A$?

I am considering an exponential on the following form: $$\exp(X + A \otimes I_B),$$ where $X$ is a Hermitian operator on a tensor Hilbert space $\mathcal{H}_A \otimes \mathcal{H}_B$, $A$ is a ...
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0answers
33 views

Proving commutation relation in Algebraic Bethe Ansatz

I have a problem with proving a certain commutation relation. For my Bachelor's thesis I give a more mathematically rigurous 'treatment' of a select set of chapters of a paper by L.D. Faddeev. Noting ...
3
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1answer
41 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_+} ...
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2answers
22 views

Hermitian - Using the Spectral Theorem

Ok so I am trying to show that if A is normal and has real eigenvalues, then A is hermitian. It was suggested that I try using the spectral theorem. So if we assume A is normal and has real ...
3
votes
1answer
53 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 ...
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0answers
24 views

Algebraic formulation of quantum mechanics and unbounded operators

Posted in the physic site: In AQFT one specifies the structure of the observables as a $C^*$-algebra. This seems to excludes algebras that don't have a norm, such as the Heisenberg algebra. ...
0
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0answers
25 views

Homeomorphism between the space of all Ashtekar connections and spacetime?

This is a question I've asked in physics.stackexchange: Excerpt from an essay of mine: Let $\Psi(\varsigma)$ be the wavefunction in the loop representation, where $\varsigma:[0,1]\to\mathcal{M}$, ...
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0answers
59 views

Continuity in a physical context

I'm currently trying to solve an exercise for my quantum mechanics class and have run into a bit of a jam: Suppose we have the following potential : $V(x) = 0$ if $x > |a/2|$ but $V(x) = V_0$ if ...
3
votes
1answer
37 views

Can a unitary matrix be constructed from any doubly stochastic matrix?

Here is a question that came up while I was thinking about the foundations of quantum mechanics: Consider a unitary $n\times n$ complex matrix $U$, with elements $u_{ij}$. We know that the rows and ...
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0answers
72 views

How to solve analytically or simplify this coupled system of ODEs?

I have a coupled system of ODEs: $$\cases{ i\frac{\text{d}y_1}{\text{d}t}=A f(t)y_2(t)+E_1 y_1(t)\\ i\frac{\text{d}y_2}{\text{d}t}=A f(t)y_1(t)+E_2 y_2(t) }\tag1$$ Here $f(t)$ is a periodic function ...
0
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1answer
43 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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0answers
47 views

Question arising from quantum mechanics concerning groups and symmetries

I'm trying to understand a calculation my professor did in my quantum mechanics script. Here it is: Each rotation $R \in O(3)$ induces a unitary transformation in $L^2(R^3)$, i.e. the space of square ...
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1answer
75 views

Reference request for differential geometry/quantum chaos text

I'm looking for a differential-geometry based exposition of chaos theory and quantum chaos. Ideally, it would start with the Hamiltonian formalism (on symplectic manifolds) and discuss as many of the ...
2
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1answer
74 views

Finite element method for the 'Particle-In-a-Box' problem in quantum mechanics

(Apologies in advance for the lengthy question, but it really is needed for a precise description of what I've done!) In suitable units, the 'Particle-in-a-box' problem is described by the following ...
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0answers
19 views

Could anyone help me differentiate an ansatz for the Calogero-Marchioro model?

Could anyone help me substitute the following ansatz, $$\psi=(\prod_{i<j}^{N}\bf{r_{ij}^{2}})^{\frac{\Lambda_D}{2}}\phi(p)$$ into $$H=-\frac{{\hbar}^{2}}{2m}\sum_{i=1}^{N}{\nabla}_i^{2}$$ So ...
1
vote
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 ...
20
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2answers
383 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, ...
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1answer
164 views

Integrals involving Hermite Polynomials

Could you please tell me, How to evaluate this integral which involve hermite polynomials, $\int_{-\infty}^\infty e^{-ax^2}x^{2q}H_m(x)H_n(x)\,dx=?$ where $H_n$ is the $n$-th Hermite polynomial ...