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Questions tagged [quantum-mechanics]

For questions on quantum mechanics, a branch of physics dealing with physical phenomena at microscopic scales.

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Borel condition vs strong continuity in unitary representations of finite dimensional Lie groups [closed]

I am trying to find a completely rigorous treatment of strongly continuous projective unitary representations of (analytic) Lie groups on separable Hilbert spaces in full generality. Up to my ...
ProphetX's user avatar
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3 votes
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124 views

Bounding the integral $\int_{-\infty}^\infty\Big|\frac{f(x)-f(0)}{x}\Big|^2\,dx$

Is there a functional inequality that allows me to bound $$\int_{-\infty}^\infty\Big|\frac{f(x)-f(0)}{x}\Big|^2\,dx$$ in terms of a Sobolev norm of $f$, say? If $f$ is equal to its Taylor series at $0$...
schrodingerscat's user avatar
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Problems in deriving the upper bound of quantum noise induced barren plateau phenomenon

I have got the main ideas of paper Noise-Induced Barren Plateaus in Variational Quantum Algorithms, but the process of proving the theorem 1 in supplementary note 2 has got me really confusing. ...
lang xian's user avatar
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72 views

How to Solve the Differential Equation Involving Pauli Matrices and Time-Dependent Terms?

I am trying to solve the following differential equation analytically`: $$ {\rm i}\,\partial_{t} \begin{pmatrix} u^{+} \\ u^{-} \end{pmatrix} = \left[\rule{0pt}{5mm}\,2\alpha \left(n - vt\right)\sigma^...
wayna's user avatar
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2 votes
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Stochastic differential equations with only time integrals

I want to reason about the stochastic differential equation $$ dX_t = A_t X_t dt $$ Where $A_t$ is a matrix valued stochastic process, and hence $X_t$ is a vector valued stochastic process. Are there ...
rufus_lawrence's user avatar
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21 views

Theoretical and mathematical explanation on calculating magnetic moment of materials in quantum mechanics [closed]

Could someone please provide a comprehensive and in detail theoretical explanation of how to calculates the magnetic moment of a material(total magnetic moment and absolute magnetic moment) of a ...
Thejan Hasaranga's user avatar
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Some details concerning projective representations in Wienberg's book

I have a question from the book "the quantum theory of fields" by S. Weinberg in page 89: How can we get $[U(\Lambda )U(\bar{\Lambda})U^{-1}(\Lambda \bar{\Lambda})]^2=1$ from the fact that ...
Mahtab's user avatar
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Differentiation under integral signs as done in basic quantum mechanics

In various text books, lectures or lecture notes on basic quantum mechanics, I've seen cases differentiating under integral signs and I am wondering why it is allowed in those situations. The typical ...
russoo's user avatar
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1 vote
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Kraus operators

Suppose we have a POVM given by the family of positive, hermitian operators $\{E_i\}_{i\in I} \in \mathcal{H}$. From the Neimark dilation theorem we know that the given POVM can be obtained from ...
ana's user avatar
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Does the trace of an operator commute with time derivatives of an operator?

I want to find the rate of entropy production in a quantum system using von Neumann entropy $$S = -tr{(\rho \ln{\rho})}$$ by taking it's time derivative. Can I take the derivative inside the trace or ...
wednesdaypotter's user avatar
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Eigenvalues of superoperators and their Choi matrices

It is well known that $\Phi$ is a completely-positive and trace-preserving (CPTP) map if and only if the corresponding Choi matrix $C_\Phi:=\sum_{i,j} E_{i,j}\otimes \Phi(E_{i,j})$ is positive semi-...
Thinkpad's user avatar
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2 answers
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Von Neumann entropy vs Shannon entropy

Let us consider a mixture of quantum states $$ \rho = \sum p_{i}\left\vert \psi_i\right\rangle \left\langle\psi_i\right\vert\quad \mbox{probability distribution}\,\,\, p_{i} $$ If the $\psi_{i}$ form ...
Jip's user avatar
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3 answers
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Completeness meaning (complete basis vs complete metric space)

Today my professor started talking about the formalism of QM. We talked about the eigenvectors of a Hermitian operator (over Hilbert space) as a "complete set". He also mentioned briefly ...
R24698's user avatar
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Quantum Computing: Quantum teleportation circuit [closed]

Given the following quantum teleportation circuit. It is required to calculate $\psi_i$ for $i=\{1,...,6\}$. My answer for $\psi_3 = [\alpha/2,\beta/2,\beta/2,\alpha/2,\alpha/2,-\beta/2,-\beta/2,\...
angel25's user avatar
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1 answer
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Understanding the Relationship Between the Principal Symbol of $-\Delta$ and $\sqrt{-\Delta}$ and Geodesic Flow in Hamiltonian Systems

In the context of Hamiltonian systems in symplectic and Riemannian geometry, consider the following fact: Let $(M,g)$ be a Riemannian manifold and $(M,\omega,H)$ a Hamiltonian system with $$H(q,p)=\...
ayphyros's user avatar
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Why does it seem like two parameters $k_1$ and $k_2$ are needed to match $e^{-r}$ and $k_2\sin(k_1\,r)$ as well as their derivatives $\frac{d}{d\,r}$?

The Spherical Bessel functions that solve the Spherical Helmholtz equation in the Spherical Coordinate system come in four kinds, the Spherical Bessel Functions of the first kind, the Spherical Bessel ...
Stephen Elliott's user avatar
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Quantum Ergodic Theorem: why $\sqrt{-\Delta}$ is used instead of $-\Delta$?

I'm studying the proof of Quantum Ergodic Theorems in the book Partial Differential Equations II: Qualitative Studies of Linear Equations (3rd edition) by Michael E. Taylor. The book includes the ...
ayphyros's user avatar
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Apparent or real contradiction is in Eberlein's paper?

The questions I pose here concerns the paper "The Spin Model of Euclidean 3-Space" by W. F. Eberlein (The American Mathematical Monthly, Vol. 69, No. 7 (Aug. - Sep., 1962), pp. 587-598) (...
mma's user avatar
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Are there any useful convexity properties of quantum dynamical semigroups?

I'm am wondering if there are any useful properties of quantum dynamical semigroups I can exploit for convex/concave optimization with respect to the semigroup parameter. A proper definition of a ...
nlupugla's user avatar
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Restrictions on a set to be the spectrum of a 1D (discrete) Schrödinger operator.

What restrictions are there on a compact set $E\subset\mathbb{R}$ for $E$ to be the spectrum of a bounded (discrete) Schrödinger operator on $l^2(\mathbb{Z})$? Is there a known necessary and ...
Mathmo's user avatar
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2 votes
1 answer
58 views

A question on geometric quantization

Assume $(M,\omega)$ is a symplectic manifold. Consider $H$ to be the space of complex wavefunctions on $M$, $\{ \psi: M\to \mathbb{C}\}$ with scalar product given by $\langle \psi |\phi \rangle =\int_{...
Mahtab's user avatar
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Are quasi-sets (and therefore Schrödinger logic(s)) studied by mathematicians or are they purely in the domain of philosophers?

Context: I'm a fan of different kinds of logic. I'm conflicted about whether different logics actually exist beyond, say, a philosophical oddity. The Question: Are quasi-sets (and therefore ...
Shaun's user avatar
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Spectral analysis for harmonic oscillator operator?

Let $L=-\frac{d^2}{dx^2}+x^2, x\in\mathbb R$, the one-dimensional harmonic oscillator; this is an unbounded self-adjoint operator acting in $L^2(\mathbb R)$. I am looking for a reference that deals ...
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Adding a linear term to the quantum harmonic oscillator.

The problem asks to find the Eigen-states and Energies- for a harmonic oscillator in the potential $V(x) = ax^2+bx$ I first start with the Schrödinger equation: \begin{equation} -\frac{\hbar^2}{2m}\...
haifisch123's user avatar
1 vote
1 answer
46 views

Trouble with 3D Fourier transform of a cross product expression

I don't understand a Fourier transform identity that has been quoted with no source in several papers (relevant to quantum mechanics): $$ \vec f(\vec r_i)=\int \frac{\vec r_i - \vec r_j}{|\vec r_i - \...
user2188518's user avatar
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1 answer
23 views

Solving second order constant coefficient differential equation with complex parameter

For context, I'm trying to solve the time-independent Schrodinger equation for an imaginary potential, that is, $$ V = iV_0, 0 \leq x \leq 2L $$ The equation is: $$ \frac{d^2\psi}{dx^2} = -\frac{2m(...
Deeb's user avatar
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Bounding $L^2$ norm independently of $\hbar$ constant

Let the dimension be $d$ and $k\in\mathbb R^d$ be a constant I am trying to bound the $L^2_p$ norm of $$\hbar^{d-1}\Big(g(\hbar(p-k))-g(\hbar(p+k))\Big)e^{-i\hbar k\cdot p}$$ in a way that is ...
schrodingerscat's user avatar
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$C^*$ algebra and deformation quantization

I heard that (from ref ) Classical observables : The set of observables $\mathcal{O}$ of a classical systems are exactly the self-adjoint elements of a separable commutative unital $C^*$-algebra. ...
phy_math's user avatar
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proving complex Integral relation from perturbation theory MQ

Someone can help me to prove this identity, it comes from a normalization in MQ. From perturbation theory time dependent we have $$H(t)=H_0+H’(t)$$ $$|Ψ>=c_a(t)e^{-iE_at/\hbar}|Ψ_a>+c_b(t)e^{-...
Gabriele Nicoletti's user avatar
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0 answers
79 views

Disagreement between two methods of computing the determinant of a differential operator

I want to compute the determinant of the following operator over the interval $[0,L]$: \begin{equation} A=-\frac{d^2}{dx^2}+\omega^2. \end{equation} I imposed the boundary conditions $\phi(0)=\phi(L)=...
Ervand's user avatar
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2 votes
0 answers
36 views

Is the following infinitesimal change in von Neumann entropy correct?

Let $\rho_t$ be the time-dependent density matrix and $S(\rho_t)=-\operatorname{tr}(\rho_t \ln\rho_t)$ be the von Neumann entropy. When $\rho_t$ obeys the following stochastic equation $d\rho_t =f(\...
Kochan's user avatar
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0 answers
65 views

Projective representations of $U(1)$

Let $G$ be a Lie group and $T:G\to GL(H)/\mathbb{C}^{\times}$ be a projective unitary representation in a Hilbert space $H$: i.e. for all $f,g\in G$, $$T(f)T(g)=e^{iC(f,g)}T(fg),\tag{*}$$ where $C$ is ...
Mahtab's user avatar
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0 votes
1 answer
34 views

Spectrum of a positive definite differential operator

When solving the two body problem in three dimensions in my QM lectures, you end up, essentially, with the Legendre associated equation. In terms of the angular momentum operator: $$L^2Y_l^m(\theta, \...
Lagrangiano's user avatar
2 votes
0 answers
59 views

Motivating spinors via the Dirac equation

I'm trying to motivate spin through Dirac's equation. So far, here is what I understand: Upon trying to take the "square root" of the space-time Laplacian (i.e. find an operator such that $D ...
Integral fan's user avatar
2 votes
1 answer
48 views

Looking for an argument why $e^{-tK}$ is completely positive for $K(A):=\frac{1}{2}(VA+AV^*)$

A completely positive map $\Phi : M_d(\mathbb{C})\longrightarrow M_d(\mathbb{C})$ is a map such that for any $n \in \mathbb{N}$ the map $$\Phi \otimes Id_n : M_d(\mathbb{C})\otimes M_n(\mathbb{C}) \...
CoffeeArabica's user avatar
2 votes
1 answer
92 views

About the spectrum family of the multiplication operator

Let $<,>$ be the inner product of $L^2(\mathbb{R})$. For a measurable function $F:\mathbb{R}^d\to\mathbb{R}$, we define a multiplication operator $M_F:L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)$, ...
neconoco's user avatar
2 votes
0 answers
44 views

How to lower bound the quantum conditional entropy?

I am trying to lower bound the quantum conditional entropy $H(X|Y)$ when $X$ and $Y$ are two quantum systems. Classically, it can be done as follows: $$ H(X|Y) = \sum_{y}P_Y(y) H(X|Y=y) \geq \sum_{Y \...
Jaswanthi Mandalapu ee19d700's user avatar
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1 answer
48 views

Is the decomposition $|0\rangle\langle0| \otimes \rho =( |0\rangle\otimes I_n)( I_A\otimes \rho)( \langle0|\otimes I_n) $ correct?

$ %\newcommand{\ketbra}[2]{\mathinner{|{#1}\rangle\,\langle{#2}|}} \newcommand{\ketbra}[1]{\mathinner{|{#1}\rangle\,\langle{#1}|}} \newcommand{\bra}[1]{\langle{#1}|} \newcommand{\ket}[1]{|{#1}\rangle}$...
Coco's user avatar
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1 vote
1 answer
49 views

Why is the modulus of a function taken before it is squared?

I am a studying quantum mechanics, and frequently I see that the modulus of a function has been taken before it is squared. For example, in a problem I was working on, the following was in the ...
cookiecainsy's user avatar
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1 answer
31 views

Commutator on a finite-dimensional $\mathcal{H}$ is scalar multiple of an identity

Suppose $A$ and $B$ are operators on a finite-dimensional Hilbert space and suppose that $[A, B] = c I$ for some constant c. Show that $c = 0$. I have tried approaching the problem using matrices $A, ...
Tomáš Macháček's user avatar
1 vote
1 answer
63 views

Moller Operator and the Determination of Bound States in Quantum Scattering Theory

I am trying to understand the Moller operator in quantum scattering theory. An important result concerning this operator is the following useful property that can be used to determine bound states. ...
Debbie's user avatar
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2 votes
0 answers
34 views

Nuclear spaces: the Schwartz class

Good morning, I'm studying quantum mechanics as a mathematician. I read that the Schwartz class $$\begin{equation*} \mathcal{S}(\mathbb{R}) := \{ \varphi \in \mathcal{C}^\infty (\mathbb{R}, \...
Marco Lugarà's user avatar
1 vote
0 answers
31 views

Hardy Hille type eigenfunction exapansion

I am trying a figure out Eigen-function expansion of the following kind. $$ \exp\left[{-\frac{1}{4\alpha} \left(x^2 + y^2\right)}\right] I_{l + \frac{1}{2}} \left(\frac{x y}{2 \alpha}\right) = \sum_{n}...
Purnendu's user avatar
2 votes
1 answer
111 views

Mathematically motivated derivation of Feynman path integral

In Hall's book Quantum Theory for Mathematicians he gives a wonderful derivation of the Feynman path integral. His formal derivation is as follows. Let $\psi \in L^2(\mathbb{R}^n)$ and $\hat{H}$ the ...
CBBAM's user avatar
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0 answers
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Continuity on countably-normed Hilbert spaces

i was studying some Quantum Mechanics from this doctorate's work http://galaxy.cs.lamar.edu/~rafaelm/webdis.pdf and ata certain point, in Proposition 2 pag. 166 he means to prove the continuity of an ...
Marco Lugarà's user avatar
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0 answers
35 views

Shooting method for a couple of non linear ODE representing Helium atom

In quantum mechanics we can maybe express the s-states (spherically symmetric wave functions) of the Helium atom as two wave function depending on the spherical radius variable $r$ ($0 < r < + \...
Fefetltl's user avatar
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0 answers
31 views

Deriving Components of a Quaternion-Based Wave Function

I am currently exploring an intriguing topic related to quaternion-based wave functions and have encountered a mathematical challenge that I hope to get some insights on. The concept is detailed in an ...
p yz's user avatar
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0 votes
0 answers
26 views

Lossless steering Order of Magnitude estimation

This is an equation from Rau's book "Quantum theory-an Information Processing Approach". It's leading to "lossless steering" of a particle if you insert an infinite number of ...
N1otAn1otherN1ame's user avatar
1 vote
0 answers
45 views

Sum of Spherical Harmonics and Rotational Invariance

PROBLEM Suppose that $$ \sum_{m=-l}^{l} c_m Y_m^l(\theta, \phi) Y_m^l(\theta', \phi')^* $$ is rotationally invariant, then how can we show that the $c_m$'s must be all equal? ATTEMPT AT A SOLUTION I ...
Matteo Menghini's user avatar
2 votes
1 answer
42 views

Help With Showing Local Integrability of Inverse Fourier Transform (Quantum Theory for Mathematicians Chapter 4 Problem 1 Part b)

I am working on part (b) of exercise 1 in Chapter 4 of Brian C. Hall's book. It states: $\textbf{Exercise 1}$: A $\textbf{locally integrable}$ function $\psi(x,t)$ satisfies the free Schr$\ddot{\...
Derek H.'s user avatar
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