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
175 views

Two-particle operator in the second quantization

In "Quantum mechanics" by Schwabl I found a chapter (1.3.3) about one- and two-particle operators in the second quantization. The derivation was only sketched and contained this equation: ...
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76 views

Inequality from Von Neumann entropy.

I am looking over some old course notes. First, Von Neumann entropy is defined. The Von Neumann entropy of a system described by a density matrix $\rho$ is defined by $S(\rho)\equiv ...
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1answer
226 views

Why are the coefficients of the base states of a qubit complex numbers?

Why are qubits represented as $$\left|{q}\right\rangle = \alpha\left|{0}\right\rangle+\beta\left|{1}\right\rangle\equiv\alpha\left[{1 \ 0}\right]^T+\beta\left[{0 \ 1}\right]^T; ...
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2answers
40 views

Operators and eigenvectors [closed]

Hello, can anyone help me with this problem?
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1answer
71 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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1answer
107 views

Computing a Projection Valued Measure

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 ...
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1answer
47 views

Why is $\int_{\mathbb{R}^3} |p\rangle \langle p| d\lambda(p)=id$?

As I have written in the headline, I am curious how the relation $\int_{\mathbb{R}^3} |p \rangle \langle p| d\lambda(p)=id$ that physicists use, where $|p\rangle$ is the eigenfunction to the ...
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1answer
37 views

Clarifying understanding of Poisson Brackets in Hamiltonian Dynamics

I'm just reading through my textbook and would like to clarify my understanding of 'Canonically related variables'. In my textbook, it says that if $Q_i$, $P_i$ are related to $q_i$, $p_i$ by a ...
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1answer
95 views

Reference request for differential geometry/quantum chaos text

I'm looking for a differential-geometry based exposition of chaos theory and quantum chaos. Ideally, it would start with the Hamiltonian formalism (on symplectic manifolds) and discuss as many of the ...
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1answer
66 views

Commutation of abstract o(3) generators and vectors.

I've been given the following problem, and I'm quite lost with it - any help would be fantastic!! Let $L_1$, $L_2$, and $L_3$ denote the abstract o(3) algebras. You are given that $\vec{A} = (A_1, ...
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1answer
80 views

Grover Algorithm Orthogonal vectors

I'm study the Grover algorithm. About this picture my lecture say that the expression $|s'\rangle$ in the computational basis is $$|s'\rangle = \dfrac{1}{\sqrt{N-1}}\sum_{x\neq w}|x\rangle ...
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1answer
79 views

Dense in the special unitary group

I was going through the PhD thesis of Daniel Nagaj. In the last paragraph on page 30, there is this following sentence. A universal gate set must be dense in the group $SU(n)\ldots$ My question ...
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1answer
41 views

Expectation value of pure state in quantum mechanics

It's well known that in quantum mechanics, the expectation value of a self-adojint operator $A$ in pure state $|\psi\rangle$ is $\langle\psi |A|\psi\rangle = \operatorname{Tr}(A |\psi \rangle ...
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1answer
52 views

Example of using the Hadamard's matrix to determine the superposition

I've came across those notes for Quantum computation from John Watrous. I am having troubles understanding the last example. We have those two vectors, or if I understood correctly, from now on ...
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1answer
38 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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2answers
19 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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43 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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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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1answer
28 views

When is the product of a hermitian unitary and another unitary hermitian?

I have a Hermitian unitary Ĥ and I want to know, if Û is some other unitary, when is Ĥ Û a Hermitian unitary? Specifically, what are the conditions on Û such that ...
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1answer
26 views

Vectors, columns and representations

When I learned quantum mechanics, my professor frequently emphasized that a matrix is a representation of an operator, not the operator itself, and a column ($1\times M$ matrix) is not a vector, it's ...
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2answers
29 views

Orthogonal Projector

Let $(H,\langle\,|\,\rangle)$ be a separable Hilbert space on $\mathbb{C}$. $P_{\psi}:=\langle\psi,\,\rangle\psi$, where $\psi\in H$ is such that $\|\psi\|=1$. I have to prove that $P_{\psi}$ is an ...
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2answers
140 views

Trace in non-orthogonal basis

In Dirac notation we can define the trace of an operator in Hilbert space $\rho$ as the follows, $Tr(\rho)=\sum\limits_{|s\rangle \in B} \langle s| \rho |s\rangle$ where B is some orthonormal ...
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1answer
116 views

Hermitian transformation

I am studying Quantum Mechanics, and the book by Griffths introduces some concepts that I have never come across in my Math courses. I will try to summarize my questions, and hopefully someone will be ...
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2answers
104 views

Problem with a wavefunction in Quantum Mechanics (math) (Book solution possibly wrong?)

Well there is a problem in my book which lists this problem: Calculate the probability that a particle will be found at $0.49L$ and $0.51L$ in a box of length $L$ when it has (a) $n = 1$. Take the ...
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1answer
489 views

Integrals involving Hermite Polynomials

Could you please tell me, How to evaluate this integral which involve hermite polynomials, $\int_{-\infty}^\infty e^{-ax^2}x^{2q}H_m(x)H_n(x)\,dx=?$ where $H_n$ is the $n$-th Hermite polynomial ...
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1answer
216 views

Evaluate an Integral involving Gaussian divided by square root of a quartic polynomial

Could you please tell me, How to evaluate the integral, $\int_{-\infty}^\infty\int_{-\infty}^\infty\dfrac{e^{-a(x^2+y^2)}}{\sqrt{k^2+\beta^2(x^2-y^2)^2}}dx~dy$ I already have obtained a series ...
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1answer
108 views

Derive an algorithm to determine convex combinations

Problem statement Given is the density matrix of a spin-1/2 system which was set up in a state of superposition $$ \varrho = \begin{pmatrix} \frac{3}{4} & \frac{1}{4} \\ \frac{1}{4} & ...
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1answer
113 views

Calculate spin wave function given probabilities of its alignment along 2 axes

Problem: An $e^{-}$ exists in such a state that the probability of its spin aligning across the $x_{(+)}$ axis is $P_{x+}=1/2$ and across the $y_{(+)}$ axis is $P_{y+}=1/2$ as well. What is the spin ...
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1answer
62 views

Quantum Fourier Transform and roots of unity.

I need to find $QFT_{6}$ for the state quantum state $\frac{1}{\sqrt2}(|0\rangle + |3\rangle)$. I received a very sufficient answer recently on simplifying nth roots of unity, but I am having a lot of ...
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1answer
45 views

Probability distribution of two party quantum states

I am going through a blog post written by Thomas Vidick. It states following three assumptions by Bell. Measurement independence (“free will”): the state $\lambda$ is independent of ${x,y}$ (since ...
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1answer
64 views

What is the notation for separable states or independent variables?

Is there any specific notation that two quantum states are separable or that two random variables are independent?
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33 views

Expecation for tensor products

We are told that $$Z|0 \rangle = | 0\rangle \\ Z | 1\rangle =-|1\rangle \\ X|0\rangle =|1\rangle \\ X|1\rangle =|0\rangle$$ and we have the state $$|\psi\rangle =|0\rangle |1\rangle +|1\rangle ...
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47 views

Manipulating derivatives after substitution: $\xi=\gamma x$

I am following a quantum mechanics text book which uses a simple looking substitution in a derivative. The substitution is $$\xi=\gamma x\tag1$$ It then says that ...
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1answer
42 views

Unwanted $i$ floating around when trying to calculate $\langle p\rangle$

$\def\sp#1{\left\langle#1\right\rangle}$I am given $$ \Psi(x,0)=A_0 \exp\left(-\frac{x^2}{2\sigma_0^2}\right) \cdot \exp\left(\frac{i}{\hbar}p_0x\right)\tag1$$ where $A_0=(\pi ...
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1answer
680 views

Commutator relationship proof $[A,B^2] = 2B[A,B]$

I'm trying to find the condition necessary for this commutator relationship equality: $$[A,B^2]=2B[A,B]$$ So far I've done this: \begin{align*} [A,B^2] & = B[A,B] + [A,B]B \\ ...
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1answer
93 views

substitution in a non linear differential equation and to get a nicer form

well I had this equation at the begining $$ i \frac{\partial u}{\partial{z}} + \frac{1}{2 k_0} \frac{\partial^2 u}{\partial x^2} +\frac{1}{2}k_0 n_1 F(z) x^2 u-\frac{i[g(z) -\alpha(z)]}{2}u + k_0 ...
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1answer
95 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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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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1answer
25 views

Position operator is self adjoint

Let $H=L^2(\Bbb{R})$ with the linear (unbounded) operator $P(f)(x)=x\cdot f(x)$ for each $x\in\mathbb{R}$. Have a look at the following domain: $$D(P)=\{f\in ...
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27 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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22 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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48 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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25 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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52 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
49 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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40 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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52 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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45 views

Why such strange separation of variables?

In this article the authors give the following expansion of wavefunction of three-body system (equation $(16)$ in text): $$\Psi(\textbf{x},\textbf{y})=\sum_{q=\lambda}^l\psi_q(\textbf x^2,\textbf ...
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34 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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80 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 ...