# Questions tagged [quantum-mechanics]

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.

996 questions
Filter by
Sorted by
Tagged with
48 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 ...
545 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 ...
60 views

### I would like to find a generalization of the plane wave expansion to Hankel functions.

The plane wave expansion is \begin{equation} e^{i\vec{k}\cdot \vec{x}}=\sum_{\ell=0}^{\infty}i^\ell(2\ell+1)j_{\ell}(kx)P_{\ell}(\cos(\theta)) \end{equation} where $j_\ell$ is the spherical Bessel ...
115 views

### Diagonalising an infinite-dimensional Hermitian square matrix

I have a quantum state which takes the following form: $$\rho = \sum_{b, \,c \, = \,0}^\infty \frac{(-igt)^b(igt)^c}{\sqrt{b!c!}}\vert b\rangle\langle c\vert.$$ This is an infinite Hermitian matrix ...
45 views

### Can I allow a value of n to be defined if that value of n gives a 1/0 , BUT that 1/0 has another 1/0 that cancels it out?

I'm working through an integral for Quantum Mechanics 2, Harmonic oscillator with time-dependent perturbation, and I have encountered this situation when evaluating the integral. The part in ...
92 views

### Watrous’s definition of Quantum Cellular Automata

I need some help understanding the second-to-last and final equations of the introduction to quantum cellular automata included below. My specific questions: What does it mean when the capital Pi ...
51 views

290 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 ...
117 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 ...
73 views

### A simple quantum mechanical system

I am studying a Quantum Mechanics course and I have come across something that I am a little stuck on, mathematically. Physically it seems to make sense but I'm not sure which equations to use to ...
29 views

166 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 ...
98 views

### Find the distance travelled by $P$ before it changes direction. (Mechanics)

A particle $P$ starts at the point $O$ and travels in a straight line. At time $t$ seconds after leaving $O$ the velocity of $P$ is v $m/s$, where $v = 0.75t^2 − 0.0625t^3$. Find (i) the positive ...
7k views

110 views

### Basis for Clifford algebra $Cl^2 (W)$ and quotient space $Cl^3(W)/Cl^2(W)$

Consider a basis $(c_1 ^ {\dagger}, c_2 ^ {\dagger}, c_1 ^ {\dagger}, c_1, c_2, c_3 )$ of creation and annihilation operators for $W=V \oplus V^*$. I need help to write the basis for Clifford ...
490 views

### About the solution to a non-linear non-constant coefficient second-order ODE

The ODE $$−y'' (x)−\frac{2a^2}{\cosh^2 (ax)} y(x)=k^2 y(x)$$ can be made into the form $$\frac{\cosh^2(ax)}{k^2\cosh(ax) - a^2} = \frac{y}{y''}.$$ Observing that $y'' = k^2\cosh(ax) - a^2$, we get ...
721 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 ...
279 views

### How to sum up this series and simplify yet another one?

Primarily, I would like to know what could be done with this series: $$\sum_{n=2}^{\infty}\frac{n^3}{(n^2-1)^3}\left(\frac{n-1}{n+1}\right)^{2n}$$ As hardmath says in his comment, the series ...
321 views

### Dirac Gamma matrix identity

In my library's (old -- 1996) copy of Peskin and Schroeder, there's an identity I'm struggling to prove. In my copy it occurs on page 42, between equations 3.28 and 3.29, but I don't know how well ...
490 views

### $C^*$-algebras, von Neumann algebras, unbounded operators and quantum mechanics in connection

I am studying the theory of $C^*$-algebras, von Neumann algebras and unbounded operators in courses on Functional Analysis and Opertor Algebras. Now I want to apply this knowledge to (algebraic) ...
203 views

### Position Operator on $l^2(\mathbb{Z})$

I'm very familiar with the position operator $(Q\varphi)(x)=x\varphi(x)$ on $L^2(\mathbb{R^d})$, but I'm trying to figure out how to interpret the same operator on $l^2(\mathbb{Z})$ (the space of ...
189 views

### Separation of variables and quantum mechanics

In the book Quantum mechanics by Eugen Merzbacher, third edition, at page 462 he claims that this differential equation (for the unknown operator $F_0=F_0(x,y,z)$) can be solved by separation of ...
667 views

### Probability and Quantum mechanics

I don't quite understand how the probability language of sample spaces, $\sigma-$algebra, random variables, etc, fit into the quantum mechanics' formalism. To wit, we usually say that an observable ...
I've recently begun learning about Projection Valued Measure and I'm a little confused. I understand that a Projection Valued Measure is a family of orthogonal projections $P(\Lambda)$ indexed by the ...