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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.

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Show that don't exist a linear rappresentation for the Galilei's group.

Let G Galilei's group and R a projective rappresentation, show that don't exist $\phi : G\longrightarrow \mathbb(C) $ such that $V: G \longrightarrow U(H) $, $V(g)=\phi(g) R(g) $ such that $V(g_{1}g_{...
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54 views

Schrödinger equation involving the Dirac-Delta

I am taking a course on quantum mechanics and I try to understand the time-independent Schrödinger-equation with the Delta-potential: $$\frac{-\hslash^2}{2m}\psi''(x)-V_0\delta(x)\psi(x)=E\psi(x)$$ ...
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1answer
86 views

Why can linear operators have $[A,B]=I$ in infinite dimensions when it is impossible in finite dimensions?

For any $n\times n$ matrices $A,B$, we have $\text{Tr}(AB)=\text{Tr}(BA)$. This can be easily proved by algebraic summation. Thus $\text{Tr}([A,B])=\text{Tr}(AB-BA)=0$. Also, $\text{Tr}(kI)=kn$. ...
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Is $|a_1\rangle\langle a_2|\otimes |b_1\rangle\langle b_2| = |a_1b_1\rangle \langle a_2b_2|$? Why or why not?

I'll be using the Dirac bra-ket notation. Is $|a_1\rangle\langle a_2|\otimes |b_1\rangle\langle b_2|$ the same as $(|a_1\rangle\otimes|b_1\rangle)(\langle a_2|\otimes \langle b_2|)$? I'm not sure ...
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Ehrenfest Theorem and correlation among observables at the quantum scale

I am studying quantum mechanics and I encountered the famous Ehrenfest Theorem, which states that given an observable $A$, its expectation value time evolution is governed by $\partial_t\langle A\...
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1answer
40 views

Confusion regarding $|A|^2 \int\limits_{-\infty}^\infty e^{i(p-p')x/h}=|A|^2 2\pi h\delta(p-p')$

The book "Introduction to Quantum Mechanics (Second edition)" by Griffiths says the following on pg 103: $$|A|^2 \int\limits_{-\infty}^\infty e^{i(p-p')x/h}=|A|^2 2\pi h\delta(p-p')$$ Here $\delta$...
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1answer
50 views

Lifting representation Heisenberg algebra

I (think) I've found the Heisenberg Lie algebra representation through quantization. Where we have $q \mapsto q$ and $p \mapsto -i \hbar \frac{\partial}{\partial q}$. So this is only a Lie algebra ...
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Coarsenings In Deutsch Et Al's Constructor Theory

Disclaimer: I posted a questions on constructor theory here a few days ago but received two closing votes, I guess because it consisted of several subquestions, so I deleted it and now try to focus on ...
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Hamiltonian Flows and Heisenberg picture of quantum mechanics

I am a math bachelor student studying Quantum Mechanics and I was very briefly introduced to the Heisenberg picture. (Hence many of the following may be trivial) In particular what I know is that: ...
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1answer
47 views

What are the automorphisms on the strucuture consisting of the nonzero vectors of a Hilbert space with the orthogonality relation?

Let $V$ be an infinite-dimensional complex Hilbert space. With this space we can associate a relational structure $V^+ = (V^+, \bot)$, where $V^+$ is the set of non-zero vectors in $V$, and $\bot$ ...
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85 views

When is $\exp(-iHt)$ well-defined?

If $H$ is a linear operator, what restrictions should be put on $H$ in order for $\exp(-iHt)$ to be well defined? How do you define $\exp(-iHt)$ when $H$ is infinite-dimensional? (If it is possible)
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1answer
17 views

Orthonormality of Hermite function

I was wondering if someone could tell me when the following relation holds? where $H_{n}(x)$ are Hermite polynomials and $\delta(x-x')$ is Dirac delta function: $$ \sum_{n=0}^\infty \frac{1}{\sqrt{\pi}...
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1answer
27 views

Fundamental property of Green's function is violated

$$\langle x| D|x'\rangle=D_x\langle x|x'\rangle=D_x\,\delta(x-x')$$ $DD^{-1} = I$ , Where '$I$ ' represents the Identity position representation of this equation is $\langle x|D|x'\rangle \langle ...
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How boundary conditions for a green function is same a original function

I am trying to study green's function.suppose we have a differential equation, $D_x f(x) =g(x)$ , We also provided with some boundary conditions for $f(x)$, But we supply this boundary conditions in ...
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255 views

How do I prove that the angular momentum is a Hermitian operator?

Confirm that the operator $$\hat I_z= \left(\frac hi\right)\frac{d}{dφ},$$ where $\varphi$ is an angle, is Hermitian.
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52 views

Position representation of an operator

$$\langle x| M|x'\rangle=M(x)\langle x|x'\rangle=M(x)\,\delta(x-x')$$ I know this is true for if $M$ is a momentum operator or position operator, is this is true for a general operator $M $? $\...
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Green function derived from the idea of position representation in quantum mechanics

I am trying to study green's function by using a lecture by NPTEL its link is https://www.youtube.com/watch?v=ZJ7v6VZQ32k&t=439s I don't get this step by him at the minute of ten. $D_x G(x,x^{'}...
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2answers
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Does adding a constant to the potential of Schödinger's equation always make the solution pick up a factor $e^{-iV_0t/\hbar}$?

I've been going over problem 1.8 of [Griffiths, Introduction to Quantum Mechanics, 2nd edition], where it asks to prove the following: Say $\Psi(x,t)$ satisfies the time-dependent Schrödinger equation ...
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32 views

Is this the correct definition of a discretization of a functional?

Physicists define path integrals by limits of discretizations, instead of using measure theory and Wiener measures. Now, the issue is that most texts do not give clear definitions, so it seems almost ...
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Quantum representation of a system of identical particles

I'm studying mathematics and I began a course in quantum statistics, in which I got to the discussion related to indistinguishibility of particles. My professor's notes are not very clear and ...
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1answer
35 views

Probability difference scattering potentials

Let $V_1(x)$ and $V_2(x)$ be two real potential functions of one space dimension, and let $m$ be a positive constant. Suppose $V_1(x)\le V_2(x) \le 0$ for all $x$ and that $V_1(x) = V_2(x) = 0$ for ...
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prequantization euclidiean space

In general the prequantum hilbert space of some manifold $M$ will be the sections of the complex line bundle. I ommitted a lot of details of course. Locally, a section of a complex line bundle ...
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169 views

Meaning of the notation $L^2(\mathbb{R}^3)$ and its generalization

I'm not sure whether this question really belongs to this website. In quantum physics texts, and physics stackechange website, I have often seen the notation $L^2(\mathbb{R}^3)$. My glossary of ...
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14 views

Discrete spectrum of potential with negative integral

I have a Schrödinger operator $H=-\frac{d^{2}}{dx^{2}}+V$. Let's assume that $V$ is compactly supported and that $\int_{-\infty}^{\infty}V(x)dx<0$. Does this guarantee that $\sigma_{p}(H)\subset [-\...
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1answer
1k 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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1answer
27 views

How to find the value with which I need to divide a unitary matrix such that its first component lies within the range of cosine?

In quantum mechanics, we generally deal with unitary matrices. In the IBM cloud computers and Qiskit, the general $2\times 2$ unitary is defined as $$U(\theta, \phi, \lambda) = \begin{pmatrix} \...
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2answers
100 views

Mathematically rigorous Quantum Mechanics

I am a student of mathematics attending a course in Quantum Mechanics. This course is held by a physicist, and it is really confusing for me to follow his reasonments. With this, I do not mean to be ...
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1answer
37 views

Is there a construction of the Wiener measure by discretization and limits which parallels the Physics ideas?

In Physics one constructs the path integral by a limiting process together with a discretization procedure. Now, in order to better paralell with the Wiener measure, consider this in Euclidean ...
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Showing that any linear operator can be written as a sum of Hermitian matrices [duplicate]

Let $(V, \mathbb{C})$ be a complex - valued vector space. Let $A$ be any linear operator acting on this vector space. Suppose that $B = \{|v\rangle_{k}\}_{k=1}^{n}$ is a basis set for $(V, \mathbb{C})$...
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1answer
21 views

How to properly define the “discretization of a functional”?

In the derivation of the path integral formulation of quantum mechanics, most Physics books end up finding the following (or similar) expression: $$K(q',t';q,t)=\lim_{N\to \infty}\int\left[\prod_{k=1}...
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1answer
36 views

Momentum operator in the position basis

J.J Sakurai shows in the section of ' Momentum operator in the position basis' as $P$$\lvert\alpha\rangle$=$\int dx^{'}\lvert\ x{'}\rangle\Bigl(-i{h\over 2\pi}$ $\partial\over\partial x{'}$$ \...
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Oscillations of an Energy Eigenstate

Energy eigenstates of a 1-dimensional particle are given by solutions to differential equations of the form $$ \left(-\frac{\hbar^2}{2m} \frac{d^2}{dx^2} + V(x) \right) \psi(x) = E\psi(x) $$ where $V$ ...
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Different behavior of the Klein-Gordon equation

By Fourier transform, it is well known that the Klein-Gordon, wave and Schrödinger flows are $e^{it\sqrt{1-\Delta}}$, $e^{it \sqrt{-\Delta}}$ and $e^{it\Delta}$, respectively. I saw a phrase "the ...
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47 views

Maximising linear function over a specific convex set of density matrices

All matrices being discussed in this question are density matrices, so they have the following properties: Hermitian Positive Semidifinite Trace = 1 We are currently in the space of all 4*4 density ...
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24 views

quantization clarification

I'm wondering something small about the link between representations and quantization. For quantization you start with some phase space (symplectic manifold) $M$, and you have a classical observable $...
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1answer
34 views

Determine the matrix representation for an operator written as an outer product

Suppose $|v_{i} \rangle$ is an orthonormal basis for an inner product space $V$. What is the matrix representation for the operator $|v_{j}\rangle \langle v_{k}|$, with respect to the $|v_{i}\rangle$ ...
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508 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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how can i determine the equations of the instantaneous helical axis relative

A point O of a rigid solid describes a plane O1X1Y1 a radius b of angular velocity ω const n. The body rotates around O by executing a regular precession motion (φ' = ν = const, θ = θ0 = const.). The ...
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1answer
50 views

Proving a vector state is pure

Let $\mathcal{M}$ be a until $*$-algebra of 3x3 complex matrices. We have the general form of a vector state $\omega_{\psi} : \mathcal{M} \to \mathbb{C}$ over $\mathbb{C}^3$ as given by $$\omega_{\psi}...
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1answer
59 views

Technical operator theory question on Albeverio's “Solvable Models in quantum mechanics”

I'm currently studying S. Albeverio's book "Solvable models in quantum mechanics" where some technical things are used that I don't fully understand. It is a general technical operator theory question,...
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Proofing equation containing time-ordering operator

Preparing for a presentation at university (I'm a Bachelor physics student) I have come across the formula below containg the time-ordering operator $T$. Although i have now understood the action of ...
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172 views

Absolute convergence of Dyson series for evolution operator

In this paper https://arxiv.org/abs/math-ph/9901018 the author (on page 4) mentions : "...a standard argument for the absolute convergence of the Dyson series for [the evolution operator] $\hat{U}_{\...
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Why is the set of 2-norm preserving matrices richer than set of 4-norm preserving matrices

First, I seek a general characterization of 2-norm and 4-norm preserving matrices. Second, I seek to understand, using this characterization, why preserving the 2-norm can be described to have a far ...
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1answer
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Why is the spectrum of the hamiltonian for an infinite square well just a point spectrum?

Consider the Hamiltonian $H = -\Delta + V$ where $V$ is the potential conrresponding to an infinite square well: $$V(x) = \begin{cases}0,&\text{if } 0, \leq x \leq L;\\\infty,&\text{otherwise}...
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Geometric phase in magnetic field

$\newcommand{\ket}[1]{\left|#1\right>}$ $\newcommand{\bra}[1]{\left<#1\right|}$ $\newcommand{\dv}[2]{\frac{d #1}{d#2}}$ $\newcommand{\braket}[2]{\left<#1\middle|#2\right>}$ Following ...
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2answers
41 views

Commuting Operators Have the Same Eigenvectors, but not Eigenvalues.

The following problem is out of my quantum mechanics textbook. Assume that two operators $H$ and $\Gamma$ commute. Show that if $|\psi\rangle$ is a non-degenerate eigenvector of $H$, that is, $H|\...
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2answers
33 views

Does the Trotter Product Formula apply to the Simple Harmonic Oscillator?

We can write the Hamiltonian for the simple harmonic oscillator as $\hat{H} = \hat{T} + \hat{V}$ where $\hat{T} = \dfrac{-\hbar^2}{2 m} \dfrac{\partial^2}{\partial x^2}$ and $\hat{V} = \dfrac{1}{2} m \...
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75 views

Application of the operator $\exp\left( \alpha \frac{\partial^2}{\partial q^2}\right)$

I need to apply the operator $$\exp\left( \alpha \frac{\partial^2}{\partial q^2}\right) \tag{1} \label{1}$$ To the function $$M(x) N(y +C_{1}p)\mathcal{F}[f(q)](p) \tag{2} \label{2}$$ where $M(x)$ ...
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63 views

Why does the “shooting” or “wag the dog” method give a bound state?

I am using numerical methods to solve Schrodingers equation. I have identified an interval for E (energy) in which one solution tends to infinity, but on the other side the other solution tends to ...
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28 views

exponentiating a matrix with complex elements

$$\exp(i\pi/4* \begin{matrix} 0 & 0 & 0 & 1 \\ 0 & 0 & -1 & 0 \\ 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ \end{matrix} )$$ pretty ...