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
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Linear algebra references for a deeper understanding of quantum mechanics

I'm a graduate student studying quantum mechanics/quantum information and would like to consolidate my understanding of linear algebra. What are some good math books for that purpose?
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Eigenfunctions of hyperbolic Laplace–Beltrami operator [on hold]

For the Poincaré disc model, the Poincaré half-plane model, and the Lorentz model (hyperboloid model) find the eigenfunctions $u$ of Laplace operator $\Delta: -\Delta u=\lambda u$
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0answers
21 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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0answers
17 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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0answers
21 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 ...
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1answer
20 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 ...
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2answers
40 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 ...
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0answers
19 views

adjoint of projection operator

|$\phi_m$> are the eigenstates of a Hermitian operator H. Assume that the states |$\phi_m$> form a discrete orthonormal basis. The operator U(m,n) is defined by: U(m,n) = ...
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0answers
39 views

How can I derive Polarization Operator algebraically

Can anyone describe for me the algebraic steps needed to create what Prof. Leonard Susskind calls, in one of his early Quantum Mechanics courses, the "Polarization Operator Matrix"? It is a 2 by 2 ...
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1answer
24 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 ...
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1answer
24 views

quantum harmonic oscillator and the mean energy U(T)

The energy of a quantum harmonic oscillator is given by: $$E(n)=\hbar\omega\left(n+\frac{1}{2}\right)$$ The canonical partition function is given by: $$Z(T)=\sum_{n=o}^\infty e^{-\beta ...
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1answer
37 views

Prove that $e^{\lambda A}Be^{-\lambda A}=B$

Prove that $$e^{\lambda A}Be^{-\lambda A}=B$$ if $[A,B]=0$. $A$ and $B$ are operators and $\lambda$ is a complex number. Can anyone explain how I should go about this question? How do I calculate ...
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0answers
6 views

Determining constants for a wave function

A particle is bound in a potential well such that U is infinite for $x<0$, $U=0$ for $0<x<L$ and $U=U_0$ for $x>L$ ($U_0>E$ where E is the particle's energy) . A wave function is ...
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0answers
19 views

Binomial-like expansion for non-commutative operators

I would like to evaluate the following and express it in a compact form: $(\hat{a}^\dagger(x)+\hat{a}(x))^n\,\vert0\rangle$, where the $n^{\text{th}}$ power of the sum of the annihilation and creation ...
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0answers
13 views

derivation of an equation concerning Fourier transform of a wave packet

My question arises from Fourier transform of a wave packet in quantum mechanics. In the following context, $\{\phi_i, E_i | i=1,2,...,n\}$ are eigenstates and eigenvalues of a $n\times n$ Hermitian ...
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0answers
45 views

Which books would you buy (Stochastic analysis Physical mathematics PhD)? [closed]

I have a soft question for you, any help will be appreciated. I have a large availability of money to buy mathematical text books (graduate level). I'm looking for suggestions with these topics: ...
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1answer
8 views

Delta functions as eigenstates

In Quantum mechanics, it's standard to say that the eigenstate of the position operator in 1D is the Dirac delta function. More formally, define a linear map $\hat x$ on some enhanced version of ...
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0answers
34 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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0answers
14 views

Fourier Transform of $n$ functions

I would like to evaluate the Fourier Transform of $n$ functions. I am aware from the derivation of the convolution how this is done for the case of $n=2$. How could this be generalised for $n=3$? ...
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0answers
41 views

Sum over all compositions of $n$ into exactly $k$ terms (overlaps of $SU(k)$ coherent states)

Given two $k$-dimensional vectors, $\vec{x}$ and $\vec{y}$, I would like to find, $$ d(\vec{x}, \vec{y}) = \left|\sum_{t_1 + t_2 + \cdots + t_k = n} \prod_{1\leq q \leq k} (x_q\times ...
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1answer
20 views

Zero Tensor Product

Suppose we have a space $|\psi_1\rangle \otimes |\psi_2\rangle \otimes |\psi_3\rangle$, and operators (matrices) A ⊗ B ⊗ C acting on this Hilbert space (like in quantum mechanics). I'm trying to ...
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0answers
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A mixed state integrated over Haar measure in QFT

I think that we can write a state in Quantum Mechanics like: $$ \int dx\lambda _x \vert x\rangle\langle x\vert $$ and when we talk about QFT, we would replace integral with functional integrals: $$ ...
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0answers
21 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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2answers
39 views

Operators and eigenvectors [closed]

Hello, can anyone help me with this problem?
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1answer
70 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 ...
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1answer
34 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 ...
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0answers
38 views

classic derivation of the proportionality between angular momentum and magnetic moment problem

The question is given in paranthesis. It is a classic derivation of the proportionality between angular momentum and magnetic moment $m=-IA=-I\pi r^2$ We start with the charge on a ring of Say the ...
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0answers
45 views

complex potentials in plane polar coordinates - stream function

Determine the stream function and the potential in plane polar coordinates and sketching streamlines We need to take the value of m=1. I have an idea on how to do the parts and i know what a ...
2
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1answer
25 views

Find the constant in the following matrix

An atom is prepared in the angular momentum state $$C\left(\begin{array}{c}1 \\ 2\end{array}\right)$$Here $C$ is a constant. This has benn written in the $S_z$ basis. a)Find C b)Work out ...
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1answer
21 views

Show that a function takes the following form using the definition for the function of an operator

If $f(z)$ is a function with a Taylor series expansion $f(z)=\sum _{ n=0 }^{ \infty }{c_n z^n }$, then we define $f(M)=\sum _{ n=0 }^{ \infty }{c_n M^n }$ First consider ...
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1answer
34 views

Calculating eigenstates of Pauli matrices

I need to find out the eigenvalues and the eigenstates of the Pauli matrices. I know that the eigenvalues of $\sigma_x$, $\sigma_y$, and $\sigma_z$ are all $\pm 1$. How do I find the eigenstates?
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0answers
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Find the Expectation Value of Basis States

Recently, I have picked up a Quantum Computing class. We have been asked to find the expectation value of $X$ tensor $Z$ and $H$ tensor $H$ $$ (|00>+|11>)/\sqrt(2); \qquad ...
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0answers
19 views

A detail in [ the time evolution operator]

If $$C=\Delta S_{22}+\lambda _1[a_1S_{21}+a_1^\dagger S_{12}]+\lambda _2[a_2S_{32}+a_2^\dagger S_{23}] ,$$ $$ \beta= ...
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4answers
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Particles that are distinguishable and indistiguishable at the same time

Thinking about a question and my answer to it and another question I asked earlier. I've come up with the following problem: Consider two otherwise very similar marbles, a red one and a blue one. Let ...
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0answers
56 views

Marbles that are distinguishable and indistinguishable at the same time

Thinking about a question and my answer to it and another question I asked earlier. I've come up with the following problem: Consider two otherwise very similar marbles, a red one and a blue one. Let ...
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1answer
28 views

Calulate the eigenvalues and the eigenstates

An observable is given by $$\sum\limits_{n= 1}^N a_n|a_n\rangle\langle a_n | $$ Here $\langle a_n |a_m\rangle = \delta_{nm}$. What are the possible measurement results corresponding to the operator ...
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3answers
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Find the eigenvalues of the operator

A projection operator $P$ is defined as $P^2$=$P$. Use this definition to find the eigenvalues of this operator. In this question is it necessary to define what the projection operator ...
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0answers
34 views

Prove the expectation value of a function of random variables

Consider a random variable $A$ and suppose we look at it the expectation value of $A^m$. Then we have the expectation value of $A^m$: $$= \sum\limits_{n= 1}^N {a^m_n}p $$ Using ...
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1answer
44 views

Application of Fubini Theorem in Quantum Mechanics

I'm afraid I'm very confused by how to correctly apply the Fubini theorem to simplify integrals? I have some integral $$ \sum_{k = 0}^{2}\int_{0}^{T} dt_2 \int_{0}^{t_1} dt_1 \bigg[ ...
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0answers
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Finding the expectation of momentum using density matrices.

I've been given the task of showing that $$ \bar{<p>}=\int\nabla_r\rho(r,r')|_{r=r'} dr$$ using the defition that the expectation of an operator $O$ is given by: $$ \bar{<O>}=\int\int ...
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2answers
39 views

Can someone explain the notion of “unbounded” operator as simple as possible?

I've read about these operators in quantum mechanics, but I have never seen them in action. I think that is because I absolutely do not intuitively understand this concept. I've read some stuff online ...
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0answers
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Just what is the importance of operators that produces an eigenvalue?

For some operators, there is a well known eigenvalue associated with it, for example the energy operator in quantum mechanics $i\hbar \partial_t$, this is very important indeed and gives us physical ...
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2answers
45 views

Creating a Hermitian function

Say I have an operator $A$ such that $A^\dagger = B$. I want to construct a Hermitian function, $f$, of these operators, $f(A,B)^\dagger = f(A,B)$. Is it possible to construct a function $f$ such that ...
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0answers
20 views

Any way to rewrite/simplify $\int dp ~f(p)~\delta[c-g(p)-E(p)]$?

Im thinking of using $$\delta(g(x)) = \sum_i \frac{\delta(x-x_i)}{|g'(x_i)|}$$ but in my case $$\int dp ~f(p)~\delta[c-g(p)-E(p)]$$ the functions $f$ and $E$ are basically unknown, while $g$ is a ...
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0answers
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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
40 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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1answer
23 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 ...
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0answers
11 views

Transformation of the infinitessimal integration variable under a coordinate transformation

I always get confused when I'm facing the 3D integral over space and have to do a coordinate transformation on the given function to solve the integral. Do some of you have tips/trick how to ...
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2answers
42 views

Can a differential equation with real coefficients have solution with complex coefficients?

Can a differential equation (with constant coefficients, linear or nonlinear) with real coefficients have solution(s) with complex coefficients? If so, are there any examples related to actual ...
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0answers
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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 $$ [a, b]=(q-1)\{a,b\}+o((q-1)^2). $$ ...