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

4
votes
1answer
717 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
0answers
62 views

Baker Campbell Hausdorff formula for unbounded operators

Baker Campbell Hausdorff formula says that for elements $X,Y$ of a Lie algebra we have $$e^Xe^Y=\exp(X+Y+\frac12[X,Y]+...),$$ which for $[X,Y]$ being central reduces to ...
4
votes
1answer
79 views

proving a quadratic form is closed

I'm trying to show that, given a spectral measure $d\mu_\psi(\lambda)$ for a self-adjont operator $A$, for the following quadratic form $$q_\lambda(\psi)=\int_{\mathbb ...
4
votes
1answer
40 views

Tensor product in dual-space

I am a bid confused regarding the notation for tensor products when going into dual-space If $\left| \Psi \rangle \right. = \left| A \rangle \right. \left| B \rangle \right.$ is $ \left. \langle \Psi ...
4
votes
1answer
85 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
4answers
246 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
1k 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
401 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
2answers
492 views

Expected Values of Operators in Quantum Mechanics

I've recently started an introductory course in Quantum Mechanics and I'm having some trouble understanding what the expectation of an operator is. I understand how we get the formula for the ...
3
votes
1answer
37 views

Do infinite dimensional Hermitian operators admit a complete basis of eigenvectors?

I'm currently taking a quantum mechanics course. We have proven that hermitian operators always have real eigenvalues, that we can choose the eigenvectors to be orthonormal, and that finite ...
3
votes
2answers
79 views

Understanding Bell's inequality vs. quantum mechanics

I have difficulty to understand how Bell's inequality rules out local hidden variable theory. It seems to me that there is some hidden variable in the Kolmogorov's axiomatization of probability ...
3
votes
1answer
45 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 ...
3
votes
2answers
117 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
1answer
88 views

Bra-ket multiplication

I'm studying a little bit of bra-ket notation and I found this property: $$\langle n| H_1 H_2|m\rangle=\sum_{k} \langle n|H_1|k\rangle \langle k|H_2|m\rangle$$ Is this property true? Why? Thank you! ...
3
votes
2answers
197 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
110 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
511 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 ...
3
votes
1answer
91 views

Function space in QM

I need to understand how one can think of a function as a vector (in Hilbert space, more specifically) so I can follow along QM texts. I've read this question's answers, but they were uninspiring to ...
3
votes
1answer
54 views

How to rigorously understand continuous bases?

In Quantum Mechanics it is quite common to see the idea of a continuous basis of a Hilbert space. In truth if $\mathcal{H}$ is the state space of a quantum system and if $X : U\subset \mathcal{H}\to ...
3
votes
1answer
34 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 ...
3
votes
1answer
83 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 ...
3
votes
1answer
197 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: ...
3
votes
1answer
98 views

Solving a PDE arising from physics

Is there a way to find an analytic solution to the following PDE? $i \partial _t \psi = - \gamma \partial _x ^2 \psi - c x $cos$(\omega t) \psi $, where $\psi (x,t)$ is defined (in $x$) on the ...
3
votes
1answer
393 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 ...
3
votes
1answer
231 views

A question about Moyal product

In Geometric quantization theory the Moyal product is one of main tools. We know Moyal product for the smooth functions $f$ and $g$ on $ℝ^{2n}$ takes the form $f\star g = fg + \sum_{n=1}^{\infty} ...
3
votes
1answer
145 views

Is the partial trace congruent under a change of basis?

My intuition tells me that the partial trace should be congruent under a change of basis. That is, if I have some matrix $A$ in the space of linear operators acting on a joint hilbert space: $A \in ...
3
votes
1answer
121 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 ...
3
votes
1answer
232 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, ...
3
votes
1answer
120 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 ...
3
votes
1answer
256 views

To show a set of projectors summing to the identity implies mutually orthogonal projectors

The general setting is the study of positive operator measures in quantum mechanics http://en.wikipedia.org/wiki/Positive_operator-valued_measure instead of the projector operator measures. Going ...
3
votes
1answer
59 views

Momentum is quantised in compact spaces?

Background One of the first examples given when studying quantum mechanics is the particle on a cylinder, or particle on a ring. One finds that because of the periodic boundary conditions, ...
3
votes
0answers
49 views

Any closed form for this expression?$ \sum_{k=0,\,l=0}^{k=n,\,l=m}\frac{\lambda^{l+k}}{k!\,l!}\sqrt{\frac{n!\,m!}{(n-k)!(m-l)!}}\delta_{n-k,\,m-l}$

I am looking for a closed form of this expression. If you have seen something like this or remember something similar, please let me know. My sincere thank! $$ ...
3
votes
0answers
66 views

Pureness of Vector States

How does one show that irreducibility is equivalent to a vector state being pure? In what follows I will fill in the details of the question: Let $\mathcal{H}$ denote a Hilbert space and let ...
3
votes
0answers
104 views

Upper bound on the Lipschitz constant of entanglement entropy

I'm looking for an upper bound for the Lipschitz constant of entanglement entropy between two subsystems with respet to the standard distance measure of pure states in the Hilbert space of the full ...
3
votes
0answers
68 views

commutation relation of angular momentum operator in non cartesian coordinates

The angular momentum operator $J$ in quantum mechanics with the commutation relation \begin{equation*} [J_l,J_m]=i\hbar\epsilon_{lmn}J_n \end{equation*} has the structure of a Lie-algebra. It is ...
3
votes
1answer
78 views

Books on random permutations

I'm looking for books or introductory/expository articles about random permutations, in particular with regards to their cycle structure. EDIT: I forgot to mention that I'm especially interested in ...
3
votes
0answers
45 views

Is it possible to eliminate the inner sum to evaluate numerically?

Any hints on how to simplify the following double sum to be able to find the sum at least numerically? $$\sum_{n=2}^{\infty}\frac1{n(n^2-1)} \sum_{k=1}^\infty \frac{(k-1/n)^{2n-2}}{(k+1/n)^{2n+2}}$$ ...
3
votes
0answers
123 views

Taking a stationary phase approximation of a multidimensional integral

I'm looking for a way to take a stationary phase approximation of an integral of the following form: $$ \int_{-\infty}^\infty d\vec{q} \exp\left(2 \pi i N \left(S(q_{n+1}, \vec{q}, q_1) - ...
3
votes
0answers
82 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. ...
3
votes
0answers
106 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
43 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 ...
3
votes
0answers
159 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
96 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
1answer
157 views

A strange quantum potential: $V(x) = \frac{x^2}{5}+\mu \left(\left\lfloor x+\frac{1}{2}\right\rfloor \right).$

So I have a strange quantum potential I have been playing with: $$V(x) = \frac{x^2}{5}+\mu \left(\left\lfloor x+\frac{1}{2}\right\rfloor \right).$$ where $\mu$ is the Möbius function. This is what ...
3
votes
0answers
124 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 ...
3
votes
1answer
79 views

Positive Operator Value Measurement Question

I'm attempting to understand some of the characteristics of Posiitive Operator Value Measurement (POVM). For instance in Nielsen and Chuang, they obtain a set of measurement operators $\{E_m\}$ for ...
2
votes
3answers
100 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
3answers
216 views

Geometric Algebra/ Calculus for Physics

I don't know if this would be a better question for physics.SE, but I'll try here first: There is at least one good book on classical mechanics using the geometric algebra/ calculus (GA): New ...
2
votes
2answers
1k views

“proof” A is a Hermitian Matrix

For an arbitrary complex matrix A show that $$A*A^\dagger$$ is Hermitian. Where the dagger "$\dagger$" stands for the "complex conjugate and transpose" operators. From what I understand this must ...
2
votes
2answers
695 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 ...