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
57 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
25 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
40 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
31 views

Binomial-like expansion for non-commuting 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
14 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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2answers
32 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
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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0answers
23 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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45 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 ...
2
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1answer
30 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
41 views

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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33 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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1answer
76 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
38 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
42 views

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

The question is given in parentheses. 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 (say, the ...
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0answers
60 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
28 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
24 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
64 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
19 views

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
24 views

A detail in [ the time evolution operator]

If $$\hat C=\Delta \hat S_{22}+\lambda _1[\hat a_1 \hat S_{21}+\hat a_1^\dagger \hat S_{12}]+\lambda _2[\hat a_2 \hat S_{32}+\hat a_2^\dagger \hat S_{23}] ,$$ $$ \hat \beta= ...
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4answers
39 views

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
67 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 ...
2
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1answer
32 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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0answers
50 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 ...
2
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1answer
51 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
16 views

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
43 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
29 views

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
59 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
23 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
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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1answer
36 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
15 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
50 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
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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0answers
41 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? $$\sum_{n=2}^{\infty}\frac1{n(n^2-1)} \sum_{k=1}^\infty \frac{(k-1/n)^{2n-2}}{(k+1/n)^{2n+2}}$$ ...
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1answer
75 views

Bloch vector time evolution in magnetic field

I'm wondering if there is a smart way of solving the system of equations $$\frac{d\vec{n}}{dt} = \gamma (\vec{n} \times \vec{B}(t)),$$ where $\vec{n}(t) = \big(x(t),y(t),z(t) \big)$ is the Bloch ...
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0answers
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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0answers
22 views

How can projection operator be expressed in form $\frac{1}{d} (I + \sum_i r_i \lambda_i)$?

How can projection operator be expressed in form $\frac{1}{d} (I + \sum_i r_i \lambda_i)$? I was reading a papar and found out that the desity matrix in $d$-dimensional Hilbert Space can be expressed ...
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1answer
33 views

Can the Kernel of the commutator of two matrices with empty Kernel sets be non-empty?

The motivation for this question arises from the following: Is it possible, given two Quantum Mechanical observables $A$ and $B$ with associated operators $\hat{\mathbf{A}}$ and $\hat{\mathbf{B}}$ ...
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1answer
48 views

Distinct ways three integers can sum to a constant

So I am doing some quantum mechanics and it has led to some combinatorics. I need to know how many distinct ways I can have $N_1+N_2+N_3=N$ where $N$ is fixed so we can change $N_1$, $N_2$ and $N_3$. ...
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0answers
52 views

Proof of Heisenberg Uncertainty Principle Exercise

I'm not very knowledgeable in QM, and I know many physics books derive the uncertainty principle using commutators, but as an exercise in my PDE book (by Asmar), I should be able to derive it from one ...
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1answer
60 views

Diagonalization versus s.d. product for non-commuting Hermitian matrices

Although the application of the following is in quantum physics, the question per se is mathematical: I have seen two characterizations of the problem in measuring a discrete variable of a state ...
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1answer
63 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 ...
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2answers
104 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 ...
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3answers
159 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 ...
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0answers
163 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 ...
6
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1answer
116 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 ...