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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1answer
98 views

Invertible Fourier integral

I am reading a book on Schroedinger's equation and it says that "The relation between $\psi(x, 0)$ and $\phi(p)$ [where the latter is the amplitude in the $\psi(x,t)$ integral] is obtained by ...
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21 views

Show the first Chern class of a $U(1)$ bundle is integral.

I am working from John Baez's book: "Gauge Fields, Knots and Gravity". So I will stick to the notation used in that book. I am stuck at exercise 122 of part II (page 283), it reads: Show that if ...
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1answer
98 views

quantum mechanics violate Bell's inequality

I have this function $$ \begin{aligned} F\big(\theta_a,\theta_b,\phi_a,\phi_b\big) = \ & – \big[\cos \theta_a \cos \theta_b \big] – \big[\sin\theta_a \sin\theta_b \sin\phi_a \sin\phi_b\big] \\ ...
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17 views

How to Construct $N$-dimensional Unitary Matrix Basis

Galitski's Exploring Quantum Mechanics says on its page 29, (There are $N^2$ linearly) independent Hermitian matrices of rank $N$. The number of independent unitary matrices is also $N^2$, since ...
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28 views

Quantum probability and quantum measure theory

Do quantum probability and free probability mean the same thing - that is, they deal with noncommutative random variables? What about quantum measure theory? Is quantum measure theory the foundation ...
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18 views

Gaussian unitary dilation of Gaussian channels

I am starting with a few definitions. All these are standard and can be accessed from some quantum information or quantum physics books, for instance the books by Holevo or Parthasarathy. The question ...
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39 views

Questions to the virial theorem

Let $H = H_0 + V$ be the Hamiltonian of the single electron where $H_0 = - \Delta, V = - \frac{\gamma}{|x|}$. Now one defines the dilation group $U(s) \psi(x) = e^{-ns/2} \psi(e^{-s}x), s \in \mathbb ...
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70 views

Please help: My MATLAB code for solving a 2D Schrödinger equation keep giving me weird output.

I've been trying to solve the following Schrödinger equation numerically, \begin{equation} -(\frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2})\Psi + \frac{\sinh^2(y) + ...
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16 views

Why $|j_1 - j_2 | \leq j \leq j_1 +j_2 $ holds for $J= J_1 +J_2 $, the addition of angular moment?

I wonder why the total angular momentum $$J=J_1 +J_2 $$ is given in the range of $$ |j_1 - j_2 | \leq j \leq j_1 +j_2 $$ Of course we can verify this in the course of finding Clebsh-Gordan ...
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43 views

Is there any some straightforward calculation of finite rotation operator?

Rotation with axis $\hat{k}$ and angle $\theta$ in $\mathbb{R}^3$ is represented by $$ R = I + (\sin \theta) K + (1-\cos \theta) K^2 $$ where $K$ is the matrix for left cross product by $\hat{k}$. ...
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20 views

Going into dual space for a vector product [duplicate]

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 ...
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29 views

Self-adjoint operator and vertex conditions in quantum graphs

Let $\Gamma$ be a metric graph with finitely many edges. Consider the operator $H$ acting as $\frac{-d^2}{dx_e^2}$ on each edge $e$, with the domain consisting of functions that belong to $H^2(e)$ on ...
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34 views

What does it mean to square a partial derivative with a one dimensional vector (scalar)?

Please bear with me.. In the above image, we need to substitute p2 from the partial derivative (pay attention to content in red boxes). If we're considering the one dimensional case, we can either ...
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65 views

Partial Trace of Density Operator

Before stating my question I present my motivation: to learn more about the tensor product. Now, quantum mechanics assigns a Hilbert space to each physical system as a postulate of the theory. ...
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31 views

Antilinear correspodence of bra and ket proof

I do not understand how the antilinear correspondence arises between ket vectors in V and bra vectors in V' (dual space): So I want to know why $$ \alpha\lt f_1 | + \beta\lt f_2 | $$ corresponds to ...
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60 views

Neveu-Schwarz and Ramond sector in the free fermion CFT

My question is about the Neveu-Schwarz and the Ramond sector in the free fermion CFT. The setup is as follows. We consider two dimensional Minkowski space with a point removed $M = \mathbb{R}^{1, ...
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2answers
112 views

Can we describe quaternions using bra-ket in quantum mechanics?

For example, the rotation plus translation of a point using the language of quaternions is written as $Q(0,x,y,z)Q^* + T$ where $Q$ is the unit quaternion, $(x,y,z)$ is the point, and $T$ is some ...
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45 views

Relation between Poisson bracket and commutator.

In quantum case, we have commutators. In classical case, we have Poisson bracket. Let $T$ be a Poisson group, $a, b \in \mathbb{C}_q[T].$ It seems that we have $$ [a, b]=(q-1)\{a,b\}+o((q-1)^2). $$ ...
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53 views

Schrödinger Semigroup is compact if potential goes to infinity

Several papers (e.g. this one: arXiv:0810.3275v1 [math.SP] 17 Oct 2008 ) claim that if $H=-\Delta+V$ and $V(x)\to\infty$ if $|x|\to\infty$, then the semigroup $e^{-tH}$ is eventually compact. Does ...
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222 views

What is the meaning of “Hermitian”?

Google search-bar gives the definition of Hermitian as: Hermitian: denoting or relating to a matrix in which those pairs of elements that are symmetrically placed with respect to the principal ...
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36 views

Eigenvector of a linear combination of operators is an eigenvector of each operator

Assume $H$ is a Hilbert space and $a_1,\dots,a_n$ are operators with Hermitian adjoints $a_1^*,\dots,a_n^*$, satisfying the canonical commutation relations. Define $N_j=a_j^*a_j$. Assume $v$ is an ...
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86 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 ...
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30 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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1answer
32 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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1answer
56 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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96 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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199 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 ...
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1answer
23 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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155 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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88 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 ...
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38 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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41 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 ...
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2answers
38 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 ...
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68 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 ...
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127 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 ...
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1answer
61 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 ...
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177 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 ...
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57 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 = ...
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40 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 ...
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201 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 ...
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39 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 ...
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74 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 = ...
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2answers
1k views

Bessel function recursion relation

I'm reading a paper and the following set of radial equations is derived: $ -i \lbrack \partial_r + \frac{1}{r} \left( \frac{1}{2} - \nu \right) \rbrack u(r) = \pm k v(r) $ $ -i \lbrack \partial_r ...
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36 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 ...
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29 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?
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82 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 ...
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145 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 ...
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268 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 ...
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77 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 ...
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2answers
76 views

Showing that $\langle p\rangle=\int\limits_{-\infty}^{+\infty}p |a(p)|^2 dp$

How do I show that $$\int \limits_{-\infty}^{+\infty} \Psi^* \left(-i\hbar\frac{\partial \Psi}{\partial x} \right)dx=\int \limits_{-\infty}^{+\infty} p \left|a(p)\right|^2dp\tag1$$ given that ...