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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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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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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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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3answers
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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 ...
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Reference request: representation focused view of quantum mechanics

I'm interested in learning a bit more about quantum mechanics from a more Lie algebra/representation theoretic perspective, hopefully including but not going past quantum field theory. I was ...
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Quantum Info in math departments

I'm currently a graduate student in mathematics getting my master's degree. I am interested in a bit probability and partial differential equations, and, secondarily, a bit of mathematical physics; ...
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Is this physical model exactly solvable?

There exists a popular model in the Physics of heavy quark bound systems, called the Cornell potential model, in which the inter-quark potential is modeled to vary with radial distance $r$ as $$V(r) ...
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251 views

Computing the product $(\frac{d}{dx}+x)^n(-\frac{d}{dx}+x)^n$

I want to compute the product $$ (\frac{d}{dx}+x)^n(-\frac{d}{dx}+x)^n, $$ for a natural number $n$. For $n$ equal to 0 or 1, the computation is very simple obviously, but for such a low number as 2 ...
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Hilbert space inequality in series expansion coefficients

In the context of some quantum-mechanical problem, there occurs a series development which gives the dependence of an "energy-type" eigenvalue $\varepsilon$ on some parameter $Q$. $\Psi_i$ are ...
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CNOT quantum gate using tensor products

In textbook, it states that the CNOT gate with the X gate applied on second qubit is \begin{array}{cccc} 1&0&0&0\\ 0&1&0&0\\ 0&0&0&1\\ 0&0&1&...
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Solving a differential equation for the propagation of Heller's Gaussian wave packet

I am an undergraduate student working on a research project in theoretical chemistry, but I figure this particular obstacle is primarily a mathematical one. I will begin with a pretty long context, ...
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1answer
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Common eigenfunctions of 0 eigenvalue for the angular momentum components

I need to demonstrate (considering the natural representation in $L^2(S^2)$) that if two non zero vectors in $H^2(S^2)$ satisfy $J_a\phi=0$$ \forall a \in {1,2,3}$ (where $J_a$ are the components of ...
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Examples of prime ideals for Lie algebras

I am looking for examples of prime ideals for Lie algebras. In particular, I am interested in examples involving the Lie algebra given by the commutator of endomorphisms of a complex vector space and ...
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1answer
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Does the Time Evolution Operator Satisfy the Schrodinger Equation?

I would just like to confirm my solution to the following question. I'm a bit hesitant on my solution because of a specific step. I would just like confirmation if that step, which I will point out, ...
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Clarification on notation from a probabilty/quantum mechanics question

I'm working through Stephen Barnett's book on quantum information and have come across the following question (1.5, for anyone keeping track at home) A particle counter records counts with an ...
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1answer
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Some questions on coherent states and corresponding Hilbert spaces. Mathematical formalism.

I have a few questions related to coherent states. I'm trying to understand the topic using this source. Does the set of all coherent states form a Hilbert space? Or it forms the dense subset only? ...
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Time evolution of a finite dim. quantum system part 2

This is an extension to the previous post. We will study the time-evolution of a finite dimensional quantum system. To this end, let us consider a quantum mechanical system with the Hilbert space $\...
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Time evolution of a finite dim. quantum system

We will study the time-evolution of a finite dimensional quantum system. To this end, let us consider a quantum mechanical system with the Hilbert space $\mathbb{C}^2$. We denote by $\left . \left | ...
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Von Neumann ergodic theorem for purely continuous spectrum

$\newcommand{\1}{1\negthickspace{\mathrm{I}}}$ Von Neumann ergodic theorem states that, if $U(t)$ is a one-parameter group of unitaries acting on a Hilbert space $\mathcal{H}$, we have $$ \lim_{T\to ...
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Need serious help with notation representing matrices as “scalars”

Here, $\hat{S_{j}}$ are Spin-1/2 system spin matrices. For those who don't know, it's just a 2x2 matrix. I am not familiar with representing matrices as "scalars" when doing multiplying with $e_{j}$,...
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1answer
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Inequality relating the probabilities of a quantum state to the euclidean distance of states.

My professor has provided us with the following proposition (without proof). I am trying to prove this. i'm having quite some trouble proving the first inequality, right under the first sentence. ...
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Probabilistic solution of Hydrogen atom PDE?

I know that the Feynman-Kac formula gives a representation of the solutions of PDEs of the form $$ \partial_t u = Lu + Vu$$ for some differential operator $L$ and a bounded potential $V$. From the ...
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Why does the position operator have continuous spectra?

I am new to QM and was wondering how it is deduced that the position operator has a continuous spectrum of eigenvalues and eigenvectors. In particular, is the equation $\hat{x}|x\rangle=x|x\rangle$ ...
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Trace of a matrix exponential with tensor products, and Von Neumann entropy

$\def\T{\operatorname{Tr}}$ $\def\1{\mathbb{1}}$ Let $H=H_1\otimes H_2\otimes H_3$ be a finite dimensional Hilbert space, and let $\rho_{123}$ be a self-adjoint matrix with $\rho_{123}\geq 0$ (...
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Integral representation of log of operators

$\def\1{\mathbb{1}}$ Suppose we're in a "good enough" (finite for example) space, and we have positive (semi)-definite operators $P$ and $Q$. Let $\log{(P)}$ and $\log{(P)}$ be logarithms of $P$ and ...
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Evaluation of the commutator $[\frac{\partial}{\partial x_j},\frac{x_ix_j-r^2\delta_{ij}}{r^3}]=-2x_i/r^3$.

I am trying to evaluate a commutator of the form $[\frac{\partial}{\partial x_j},\frac{x_ix_j-r^2\delta_{ij}}{r^3}]=-2x_i/r^3$. Bt acting it on a function $f(\vec{r})$, I get that the commuator is ...
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Using Sylvesters theoren in Quantum Mechanics

2-d system consists of $N$ identical cells arranged linearly in series. The transfer matrix of a single cell is a unitary Hermitian 2x2 matrix with eigen values $Exp(\pm i\times\theta)$ Need to use ...
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Is the set $\{\delta_x\}_{x \in [a, \ b]}$ a basis for the set of distributions on $C^{\infty}_c([a, \ b])$?

$\def\braket#1#2{\langle#1|#2\rangle}\def\bra#1{\langle#1}\def\ket#1{#1\rangle}$ Is the set $\{\delta_x\}_{x \in [a, b]}$ a basis for the set of distributions on $C^{\infty}_c([a, \ b])$? Below are ...
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Commutator of the Hamiltonian and Parity Operator - evaluation of derivatives

I was studying the commutator of the Hamiltonian and parity operators in the $L^2$ space from Quantum mechanics and came upon the following: To show that the two operators commute, assuming we have a ...