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

6
votes
0answers
356 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 ...
4
votes
0answers
46 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
0answers
85 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 ...
3
votes
0answers
114 views

Solving Schrodinger for harmonic oscillator(griffiths analytic method)

I was just getting into quantum mechanics. But I'm having a bit of trouble following griffiths for the analytic method... it goes like so The Schrodinger equation $$ ...
3
votes
0answers
76 views

The intuition behind a matrix of a Hamiltonian?

We have derived an elegant partition function for a problem which resembles a quantized model taking the particles to be Bosons. The related Hamiltonian for every $i$th ensemble is there: ...
3
votes
0answers
99 views

Defining entanglement in subspaces of tensor product

Let $\mathcal{H}=\mathbb{C}^n$ be a Hilbert space. A state $\rho\in\mathcal{B(H)}$ is a positive semi-definite operator with unit trace. $\rho\in \mathcal{B(H)}$, where ...
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 ...
2
votes
0answers
46 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 ...
2
votes
0answers
46 views

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 ...
2
votes
0answers
62 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 ...
2
votes
0answers
59 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 ...
2
votes
0answers
57 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 ...
2
votes
0answers
35 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 ...
2
votes
0answers
61 views

Asymptotic behavior of $L^2$ norm for increased matrix dimensions

I am playing with matrices which are linear combinations of identity matrix, Pauli spin matrices, $\sigma_x$ and $\sigma_z$ or their tensor products. For example, let the matrix be $H$. So, $H$ could ...
2
votes
0answers
71 views

Change of basis and spectral theorem

I've been having trouble with such a rudimentary problem. Let us define a matrix $A$: $$A = \begin{pmatrix} 3 & 0 & -i \\ 0 & 3 & 0 \\ i & 0 & 3 \end{pmatrix}$$ A is a 3 by 3 ...
2
votes
0answers
48 views

Fock Subspaces and Weight Vectors

I've got an assignment due in a few hours, and I'm at a complete loss as to how to even start it, really. I haven't encountered any Dirac notation before, so I'm having a lot of trouble attempting the ...
2
votes
0answers
43 views

Experimental calculation and $\mathbb{Q}$

I have been reading this article and have a question about the first line of the second paragraph on the first page. It says: The basis for this suggestion is the simple fact that all experimental ...
2
votes
0answers
53 views

To construct a Schrödinger wave with prescribed mean position and momentum

Let $\lambda_x, \lambda_\xi>0$ be fixed parameters which we will call position lengthscale and momentum lengthscale respectively. Fix a (position-)point $x_0\in \mathbb{R}$ and a (momentum-)point ...
2
votes
0answers
109 views

At large times, $\sin(\omega t)$ tends to zero?

While doing a calculation in quantum mechanics, I got a expression $\sin(\omega t)$, and my prof said if I consider the consider at large times, then i can assume that this goes to zero because at ...
2
votes
0answers
48 views

Doubt about the spectrum of an operator

I consider the Laplacian operator $$A=-\Delta$$ in the domain $$H^2(\mathbb{R}^3)$$ where it is selfadjoint. We know that its spectrum is $[0,+\infty)$. Now I want to consider the restriction of $A$ ...
2
votes
0answers
130 views

How do I solve this integral

How do I solve this integral (expectation value) : $$\int_{-\infty}^{\infty} \psi (x)^* \hat p \psi (x)\ dx.$$ where the $\hat p =-i\hbar \frac {\partial}{\partial x}$ is an operator and $\psi (x)$ is ...
1
vote
0answers
43 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 ...
1
vote
0answers
29 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 ...
1
vote
0answers
65 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 ...
1
vote
0answers
111 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 ...
1
vote
0answers
46 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 ...
1
vote
0answers
37 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 ...
1
vote
0answers
33 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 ...
1
vote
0answers
140 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, ...
1
vote
0answers
65 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 ...
1
vote
0answers
101 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 ...
1
vote
0answers
97 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 ...
1
vote
0answers
81 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 ...
1
vote
0answers
43 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 = ...
1
vote
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 ...
1
vote
0answers
140 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 ...
1
vote
0answers
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 ...
1
vote
0answers
63 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 = ...
1
vote
0answers
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 ...
1
vote
0answers
28 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?
1
vote
0answers
71 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 ...
1
vote
0answers
113 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 ...
1
vote
0answers
182 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 ...
1
vote
0answers
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 ...
1
vote
0answers
248 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 ...
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
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
0answers
12 views

equivalence between dispersion relation and pde

I'm seeing a filmed lectures on quantum mechanics and the lecturer stated that thare is an eqivalence between dispersion relation $\omega(k)$ and wave equation. the relation in one direction is of ...
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 ...