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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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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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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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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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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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Quantum Mechanics - why should Probability of a degenerate eigenvalue be independent of the choice of an eigenvector in Euclidean space En?

Given $$|\psi_n> = \sum_{i=1}^{g_n}|u_{n}^{i}><u_{n}^{i}|\psi>$$ we get a projection $P_n|\psi>$. Incidentally, the square of this statement gives us the probability of finding a ...
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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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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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1answer
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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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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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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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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
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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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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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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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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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1answer
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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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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 ...
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Can embeddings be categorized into pre-embeddings and post-embeddings?

Consider a spin $1/2$ particle that moves on a two-dimensional spherical surface, and examine its orbital angular momentum in quantum mechanics. Approach 1: Take the surface as a Riemann surface, ...
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Understanding the Following Integral Notation

I'm a little confused on the notation my professor used for the following integral. \begin{equation} \int \bar{Y}_{l_f}^{m_f} \left( \dfrac{-Y_1^1 + Y_1^{-1}}{\sqrt{2}}, \dfrac{iY_1^1 + iY_1^{-1}}{\...
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Deeper understanding of the adjoint of a linear operator

My undergraduate classes in Q.M describes the adjoint of a linear operator purely as a mathematical formality. At this point, I'd like a deeper and heuristic understanding of it. My questions are ...
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The action of tensor product over N terms on a ket.

Equation (6) of the paper titled, Multi-player and Multi-choice quantum game has left me puzzled-after many hours-as to how it is being derived. My working begins from the generic form seen just after ...
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Is this general form of Von Neumann's reduction postulate correct?

I have had a look at a book on 'Quantun Measurement' by Braginsky and Khalili$^1$. In it appears an equation that I would like confirmation of. The equation seems odd, in that it sets a probability ...
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1answer
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Question about $e^{\frac{-itA}{\hbar}}(\hat{Q}+ \hat{P})e^{\frac{itA}{\hbar}}$

This arises in the context of trying to rigorously understand quantum dynamics but it's a functional analysis issue. For simplicity suppose we are in dimension $1$. Let $\hat{Q}$ be the operator ...
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Is the Unitary Group of a Hilbert Space a Lie group?

Let $H$ be an infinite-dimensional complex Hilbert space. Then the set of unitary operators on $H$ forms a group, known as the unitary group or Hilbert group. My question is, is this group a Lie ...
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1answer
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Reference request for operator theory in Quantum mechanics

I am studying Shankar's Principles of Quantum Mechanics. In the first chapter where the author introduces the necessary mathematics tool for QM, the concept of derivatives of operators with respect to ...
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Dual pairing and inner product in a Hilbert space (and in $L^2(V)$)

I put beforehand that there are some similar questions in this blog, but I nonetheless would like to pose my question as I did not find any explanatory answer. Let us consider a vector space H, ...
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Insert a Unitary operation between two others

Let's say I have two unitary operations $U_1$ and $U_2$, which together give a rotation of the following form: $$ U_1\cdot U_2 = \begin{pmatrix} e^{i\varphi} & 0 \\ 0&e^{-i\varphi} \end{...
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eigenvalue problem for a cusps model.

In applied physics we need to solve the model cusps model to get corresponding stationary solutions of the system called eigenfucnctions. These stationary solutions show the behaviour the system along ...
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First and Second Quantization of the Hamiltonian - Quantum Mechanics

I am trying to show the relation between the first and second quantization in Quantum Mechanics. I have been told that the general relationship that holds is that in the first quantization, we can ...
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Point groups where the tensor square of two-dim. irreps (over $\mathcal{O}(3)$) does not contain a two-dim. irrep in its decomposition

Which are the point groups where the tensor square of a two dimensional irreducible representation does not decompose into a sum that contains a two-dimensional irreducible representation? For ...
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Dual Space to a Lie Algebra. Problem from Takhtajan. Finding the center of a Poisson Algebra

So I am trying to solve Problem 2.19 from the book "Quantum mechanics for mathematicians" by Takhtajan. The problem is the following: Let $g$ be a finite-dimensional Lie Algebra with a Lie bracket $[,...
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1answer
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Spectral decomposition of the resolvent map

Let $P_\Omega$ be a projection valued measure and let $R_A(z)=(A-z)^{-1}$ be the resolvent map. It can be shown that $$R_A(z)=-\sum_{j=0}{\frac{A^j}{z^{j+1}}}$$ whenever this series is defined. My ...
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Retrieving a function of many complex variables from its manifold of zeros

Physical background: In the Bargmann-Segal representation, the states of bosonic quantum systems are holomorphic functions of $N$ complex variables where $N$ is the number of degrees of freedom. The ...
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Particle in a 3-D Box

Reading through my quantum mechanics book I've stumbled on a question any help would be great. Suppose we had a 3-D square well i.e. $$V(x,y,z)=\begin{Bmatrix} 0 \ \text{if}\ 0\leq x \leq a \ , \ 0\...
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Relationship between a metric and quantum mechanical matrices

Let $ds:=\sigma_x dx+ \sigma_y dy + \sigma_z dz$. Then squaring $ds$, we get $$ ds^2=\sigma_x^2dx^2+ \sigma_y^2dy^2+\sigma_z^2dz^2 + (\sigma_x\sigma_y+ \sigma_y\sigma_x)dxdy + (\sigma_x\sigma_z+\...
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Operator identity in Quantum mechanics

The question is : If an operator $\hat{A}$ follows the property that $\hat{A}.\hat{A}=\mathbb{I}$ , where $\mathbb{I}$ is the identity operator, then prove that: $\exp(\theta \hat{A})=\cosh (\...
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Sub-algebra such that Wigner (-Weyl) transform is a homomorphism

The Wigner distribution of an operator $A$ is given by $$ W_A(x,p) :=\frac{1}{2\pi} \int_\mathbb{R} \! dy \, \langle x+y/2| A | x-y/2 \rangle \, e^{ipy}, $$ and associates a function in phase space ...
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Why is the definition of sin and cos in terms of exponentials is similar to the definition of $L_x$ & $L_y$ in terms of raising & lowering operators?

I noted a similarity outlined below: The angular momentum operators in $x$ and $y$ direction can be written: $$L_x=\frac{1}{2}(L_++L_-)$$ $$L_y=\frac{1}{2i}(L_+-L_-)$$ $cos(x)$ and $sin(x)$ can be ...
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Expressing Equations in Lagrange Subsidiary Form

If we consider the Hamiltonian for the simple harmonic oscillator given by, $$H(p,x) = \frac{p^2}{2m}+\frac{kx^2}{2}$$ where $m$ is the mass, $k$ is the stiffness and $p$ is the momentum, then the ...
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Schrödinger's Equation with square potential

I have written some code to solve and plot the time independent Schrödinger's equation with potential x^2, which has a bound state with odd integral energy eigenvalues. My code plots the graphs up to ...
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How to generate a rotation matrix given an angular momentum matrix

In 3 dimensions, the total angular momentum (for $z$) matrix is given. It generates the rotation matrix around $z$ by $e^{-i\theta J_3/h}.$ My question is how do we actually go about doing this? I ...
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1answer
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eigenstates of linear operators and their powers

In algebra, it has come up a number of times that if $A$ is a linear operator, then, for any integer $k>0$, $A^k$ inherits the eigenvectors of $A$. This is a very straightforward proof. However, $A$...
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essentially self adjointness of Laplacian with the inverse square singular potential.

Let define $H_c: L^2(\mathbb{R}^n) \to L^2(\mathbb{R}^n)$ as a densly defined linear operator as follow: $$ H_c u= \Delta u - \frac{c}{|x|^2} u$$ In reading a paper I encountered to the following ...
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Heisenberg Picture for 1D Simple Harmonic Oscillator

The Halmiltonian for 1D simple harmonic oscillator is $$ H = \frac{1}{2m}(P^2 + m^2 \omega^2 X^2). $$ Show that in the Heisenberg picture, the sum of expectation $$ \langle X_{t+\pi/2\omega}^2 \...
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Definition of outer product

I am trying to understand the concept of outer product in quantum mechanics. I read "Quantum Computing explained" of David MacMahon. I can understand the transition in (3.12): $$(|\psi\rangle \...
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Rutherford Scattering - Annular Detector in the Far Field [closed]

I have been tasked to find the rate at which scattered electrons will be detected on an annular detector in the far-field. The exact question I'm working with is: Suppose that 1keV electrons, ...
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When is maximising a definite integral the same as maximising the integrand?

When is maximising $$\int_a^b f(x) \text{d}x$$ the same as maximising $f(x)$? Context: I was trying to find the most probable location of an electron in the ground state hydrogen atom, where the ...