Articles of quantum mechanics

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 large times, the graph of $\sin$ oscillates very rapidly and so you can take it to be zero. When […]

Quantum translation operator

Let $T_\epsilon=e^{i \mathbf{\epsilon} P/ \hbar}$ an operator. Show that $T_\epsilon\Psi(\mathbf r)=\Psi(\mathbf r + \mathbf \epsilon)$. Where $P=-i\hbar \nabla$. Here’s what I’ve gotten: $$T_\epsilon\Psi(\mathbf r)= e^{i \mathbf{\epsilon} P/ \hbar}\Psi(\mathbf r)=\sum^\infty_{n=0} \frac{(i\epsilon \cdot (-i\hbar \nabla)/\hbar)^n}{n!} \Psi(\mathbf r)=\sum^\infty_{n=0} \frac{(\mathbf \epsilon \cdot \nabla)^n}{n!}\Psi(\mathbf r)= \Psi(\mathbf r) + (\epsilon \cdot \nabla) \Psi(\mathbf r) + \frac{(\epsilon \cdot \nabla)^2 \Psi(\mathbf r)}{2} + […]

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 inner product. Let $S$ be the shift operator: $$ (S a)_n = a_{n-1}. $$ If there a linear operator $A:\ell^2\to\ell^2$ such that $S=e^A$? I really doubt there is, but I’m not […]

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

I’ve been trying to solve the following Schr√∂dinger equation numerically, \begin{equation} -(\frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2})\Psi + \frac{\sinh^2(y) + \sinh^2(z)}{(\cosh(y) + \cosh(z))^2}\Psi = E\Psi \end{equation} So far I have written a simple MATLAB code that should solve this, but I keep running into a very weird error. Here’s the code: clc; clear all; N = […]

Eigenvalue of an Euler product type operator?

Background Let us have the following orthonormal basis such that: $$ \langle m | n \rangle = \delta_{mn}$$ Consider the following operators defined as: $$ \hat 1 = | 1 \rangle \langle 1 | + | 2 \rangle \langle 2 | + | 3 \rangle \langle 3 | + \dots $$ $$ \hat 2 = […]

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 expectation of position, if we assume that the complex function $\psi$ describes a particle, and $f=|\psi|^2$ is a probability density of finding the particle […]

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 = A\otimes I_m + I_n\otimes B, $$ where $I_m$ and $I_n$ are the identity matrices of size […]

Why do physicists get away with thinking of the Dirac Delta functional as a function?

For instance they use it for finding solutions to things like Poisson’s Equation, i.e. the method of Green’s functions. Moreover in Quantum Mechanics, it’s common practise to think of the delta functions $\delta_x$ as being a sort of standard basis for the vector space of square integrable functions but $\delta$ is obviously not a square […]

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 analysis are needed. What level of real and complex analysis do I need, and some good books for learning them? PS – Hopefully suitable group […]

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, but cannot be represented as a rotation in odd dimensions. Even dimensional spheres do not admit continuous nonvanishing vector fields, but odd dimensional spheres […]