Questions tagged [operators]

In physics, an operator is almost always either a square matrix or a linear mapping between two function spaces (defined on, say, $\mathbb R^n$). Operators serve as *observables* and as *time evolution operators* in Quantum Mechanics. This tag will most often find valid use in quantum mechanics; don't use this tag just because your equations contain "everyday operations" like $\times$, $+$!

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37 views

Proof with complete set of eigenvectors [closed]

I have to prove, that if $\hat{\textbf{A}}$ and $\hat{\textbf{B}}$ are self-adjoint operators, and each one of them has its complete set of eigenvectors, and if $\hat{\textbf{A}}\hat{\textbf{B}} = \...
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How does this simplification in expectation value algebra work?

We have a Hamiltonian of form: $$\hat{H} = \hat{H}_0 + \hat{H}_1$$ Where $\hat{H}_1$ is a time dependent perturbation which can be written as: $$\hat{H}_1(t) = - \hat{A}F(t)$$ Now $B$ is another ...
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34 views

What does the probability of finding the system in a random state mean?

In my quantum mechanics course there are lots of problems about finding the probability of a system in some random state given the initial one which involves using the inner product of those two ...
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4answers
63 views

Symmetry transformations: a doubt about the relations that we assume true

When we deal with symmetry transformations in quantum mechanics we assume true that, If before the symmetry transformation we have this $ \hat A | \phi_n \rangle = a_n|\phi_n \rangle,$ and after ...
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Dimension of space spanned by angular momentum operator eigenvectors

I read that the dimension of the space spanned by $$ J^2|j,m\rangle=\hbar^2 j(j+1)|j,m\rangle, \qquad J_z |j,m\rangle=\hbar m|j,m\rangle ,\tag{1}$$ the eigenvectors $|j,m\rangle$ is $2j+1$, since $m$...
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28 views

What the correct way to write the discrete kinetic energy operator?

For a bit of context, I am making simulations of a quantum algorithm that is meant to variationally find the ground state of a quantum harmonic oscillator potential. In one dimension, we know that $\...
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20 views

Convert a Lindbladian time evolution operator to the Kraus operator sum representation

I try to understand how I can convert a Lindbladian time evolution operator to the corresponding Kraus operator sum. Let's assume we have a time independent Hamiltonian $H$ and a set of time ...
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1answer
45 views

Vector operators in spherical basis

This question is about the components of a vector operator in the spherical basis. In 3D real Euclidean space, a vector $\mathbf{v}$ can be expanded in the standard Cartesian components as $$ \mathbf{...
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1answer
44 views

Expected value of operator or expected value of observable?

A question about terminology. I have seen both $\langle p\rangle$ and $\langle\hat{p}\rangle$ to calculate the expected value of momentum (same thing with position, energy etc.). The first one would ...
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Good quantum numbers in solids

We know that good quantum numbers are associated with operators that commute with the Hamiltonian of the system. For example consider an Hydrogen atom without spin, we know that the good quantum ...
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1answer
37 views

Schrödinger equation for charged particle in potential

This might be a silly question, but I don't think it is trivial. I am trying to solve an example for my class. In it the Schrödinger equation for a charged particle in a vector potential is given: $$i\...
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1answer
51 views

How can you subtract a value from an operator/matrix?

I'm currently following Quantum Computation and Quantum Information by Nielsen & Chuang. I'm struggling to understand the derivation of The Heisenberg Uncertainty Principle in Box 2.4 page 89. I ...
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1answer
31 views

Rotation operator in the matrix representation in the standard basis

I recently been introduced to rotation operators in Quantum-Mechanics. Can someone please provide an explanation on what is means to represent a operator for the rotation of a quantum-mechanical ...
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2answers
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How to find matrix representation of an operator in new basis

I have recently begun to learn QM and I cannot solve this task: Let's say I have operator $\hat H = \begin{bmatrix}\epsilon & \upsilon \\ \upsilon & \epsilon \end{bmatrix} ,(\upsilon \in \...
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1answer
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Commutator of the number and momentum operator [closed]

Does the number and momentum operator commute? $[\hat p,\hat N]=0$?
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Operator acting on bras

I need some help. Suppose, $\hat{\textbf{A}}$ and $\hat{\textbf{B}}$ are operators and $|\psi\rangle$ is any state, so that $$ \hat{\textbf{A}}|\psi\rangle=a|\psi\rangle. $$ And I wonder if this ...
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Physical meaning of the operator $\exp(-a {\hat{p}}^2)$

I am curious about the physical meaning of the operator $\exp(-a {\hat{p}}^2)$ with $a$ being a positive constant. With respect to the coordinate basis, I find that $\langle x |\exp(-a {\hat{p}}^2)|x' ...
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Rigorous definition of position operator in 3D

In 1D, let $$\mathcal{D}:=\{f\in L^2(\mathbb{R};\mathbb{C})\bigg|\int_{\mathbb{R}}|x^2f(x)^2|\text{ d}x)<+\infty \}.$$ Then, $\mathcal{D}$ is a dense subspace of $L^2(\mathbb{R},\mathbb{C})$ and ...
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Time derivative in Schrödinger equation

In quantum mechanics, a system is descibed by an element $|\psi\rangle\in\mathcal{H}$, where $\mathcal{H}$ is a Hilbert space. Then on $\mathcal{H}$ (or on a dense subspace of $\mathcal{H}$), we can ...
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Calculate $\langle 2s |V| 2p,m= 0\rangle =3ea_0 |E|$ in linear Stark effect with operator properity

From Sakurai Eq 5.1.65 and Eq 5.2.19, one would be able to obtain the matrix element for degenerate perturbation theory of linear Stark effect under the potential $V=-ez|E|$. However, I'm wondering ...
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Commutator between Angular Momentum J and Cross Product

I'm trying to show that a cross product $C$ of two vectorial operators (let's say $A$ and $B$) it's a vector by it's own, which means, I want to show $$\left [J_i,C_j \right ] = i\hbar \epsilon_{...
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Does no-level-crossing theorem (aka avoided crossing) always hold in perturbation theory?

In perturbation, J.J. Sakurai Modern Quantum Mechanics Second Edition page 310 stated a no-level-crossing theorem stated that "a pair of energy levels connected by perturbation do not cross as ...
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Annihilation operator in the Heisenberg picture

I study the monograph An Introduction to the Standard Model of Particle Physics written by W. N. Cottingham and D. A. Greenwood. I can't understand the equation about the annihilation operator in the ...
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1answer
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Division of two operators (or polonomials of operators) in quantum mechanics

Consider a function of an operator $\hat{A}$, which is like follows $$ f\left(\hat{A}\right) = \frac{a + i b\hat{A}-c\hat{A}^2}{3-\hat{A}} $$ where $a$, $b$ and $c$ are complex numbers. My question ...
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How do you write commutators in polar coordinates?

In quantum field theory we have the commutators of fields must be zero outside the light cone. $[\phi(x),\phi(y)]=0$ if $|x-y|^2<0$ How can one write this in polar coordinates or a general ...
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Stone's theorem on one-parameter unitary groups and new self-adjoint operators

I have been following the proof of the Stone's theorem on one-parameter unitary groups. The question is if the current list of self-adjoint operators used in quantum mechanics, including position, ...
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1answer
42 views

How to calculate $\left[ \vec{L}^2, x_i \right]$

I've been asked to prove $\left[ \vec{L}^2, x_i \right] = -2i\hbar \varepsilon_{ijk}L_j x_k -2\hbar^2 x_i $ and I don't seem to get it correctly. I propose $\left[ \vec{L}^2, x_i \right] = \left[ L_l ...
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Radial Commutativity in AdS/CFT

I am reading Daniel Harlow's Tasi lecture notes on the emergence of bulk physics in AdS/CFT, and the question that I have is regarding the radial commutativity puzzle given in these notes. First, I ...
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Minimal coupling to electric dipole form - II

In addition to link, may I ask you for details of the Hamiltonian transformation. Knowing that: $[\textbf{x},\textbf{p}]=i\hbar$, $[\textbf{x},\textbf{p}^2]=2i\hbar\textbf{p}$ and $e^{-i\alpha A}Be^{...
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Dirac expression derivation

In Quantum Mechanics, 2nd Edition by Davies & Betts on page 78 it states that there is a symmetry implied by the following Hermitian operator equation: $${\displaystyle \int \phi^{*}(A \psi)d \,\...
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How to operationally realize the following type of equations of motion?

It is well known that for a free particle, described by $H=\hat{p}^2/2m$, $\hat{p}_{x}(t)=$ constant (similarly for other components of momentum). Meanwhile, $\hat{x}(t)$ is not a constant, being ...
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Generalized momenta and quantum mechanics

In my introductory quantum mechanics book it was stated that the operator $-i\hbar\vec{\nabla}$ represents the momentum $\vec{p}=m\vec{v}$ of a particle. In the book "Physics of atoms and molecules" ...
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Representations of Conformal Group

I want to work out the Representations of the Conformal Group. I work with Francesco's Conformal Field Theory. He stats in equation 4.30 that $$e^{i x^\rho P_\rho}K_\mu e^{-i x^\rho P_\rho}= K_\mu +...
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What does the operator's explicit dependence or independence on time actually mean in Quantum mechanics?

Consider the equation of motion for the expectation value of an operator $A$ $$\frac{d\langle A\rangle}{dt} = \frac{1}{i\hbar}\langle [A,H]\rangle + \left \langle \frac{\partial A}{\partial t} \right \...
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Correlation function of single annihilation/creation operator vanishes

I could not find anything on that on google, or here on physics stack exchange, which surprises me. My problem is, that I do not see, why exactly $<a> = <a^{\dagger}> = 0$ where <...> ...
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Trick with “functional” derivative to evaluate commutators between diagonal hamiltonian and creation fermionic operator

I found a theorem that states that if $A$ and $B$ are 2 endomorphism that satisfies $[A,[A,B]]=[B,[A,B]]=0$ then $[A,F(B)]=[A,B]F'(B)=[A,B]\frac{\partial F(B)}{\partial B}$. Now i'm trying to apply ...
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Norm of a specific operator

I am working on a problem for my Quantum Mechanics class. Given the operator $\operatorname{M}: \, L^2(0,1) \rightarrow L^2(0,1): \, f(x) \mapsto (\operatorname{M}f)(x) = m(x)f(x)$ I shall prove, ...
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Is there any operator for the scattering problem of an elastic wave in Hilbert space?

From an algebraic perspective, there is a common shape for wave propagation that includes operators in Hilbert space. I want to know is there any operator in Hilbert space that describes the elastic ...
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1answer
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Quantum measurement:can we recover evolution operator from measurement?

Suppose there is a quantum system in state $|a\rangle$$\in$ H,an instrument designed to measure some physical quantity, which itself is in state $|b\rangle$ $\in$ H'. If we consider the bigger system "...
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1answer
57 views

Show that a wavefunction is an eigenfunction of the lowering operator

Given $\psi_{\lambda}=e^{\lambda a_{+}}\psi_{0}$, show that $a_{-}\psi_{\lambda}$ is an eigenfunction of $\psi_{\lambda}$ with eigenvalue $\lambda$. In this case $a_{\pm}=\frac{1}{\sqrt{2\hbar m \...
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Simplifying Exponentials of Matrix Operators [closed]

I have been given the following question and would really appreciate any insight. I apologise for some of the symbols as I can't find a way to compile them properly. Assume the environment is given ...
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3answers
121 views

Is there an identity for anti-commutator $\{ A B, \, C D \}$ in terms of commutators $[\, , \,]$ only?

I'm looking for an identity that could express the anti-commutator $$\tag{1} \{ A B , \, C D \} \equiv A B C D + C D A B $$ expressed as a combination of commutators only: $[A,\, C]$, $[A, \, D]$, etc....
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Anti-commutator for annihilation and creation operators: ordering of indices

I'm trying to prove that $\{\tilde a_i,\tilde a_j^{\dagger} \}=\delta_{ij}$, by defining $\tilde a_i=\sum_j \bar U_{ji}a_j$. U is an unitary matrix and $a_i$ refers to an element of the operator $a$. ...
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Dirac matrix algebra from bosonic creation/anihilation operators?

Using some generic fermionic creation/anihilation operators $a_i$ and $a_i^{\dagger}$ ($i, j = 1, 2, 3, \dots, N$) such that \begin{align} \{ a_i, \, a_j^{\dagger} \} &= \delta_{ij}, \tag{1} \\[...
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QFT in Curved Spacetime - Commutation Relations for Annihilation Operators

I am currently learning about QFT in curved spacetime using these notes. Given a subspace $S_p$ of positive frequency solutions to the Klein-Gordon equation, we defined the corresponding ...
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1answer
60 views

Derivation of Single Particle Operator in second quantization?

See, I have looked through a bunch of scripts about second quantization on the internet, but everywhere at some point something weird is happening so I get stuck over and over and over again, which is ...
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4answers
194 views

The exponential of an operator

I have a problem to have an intuitive idea about these operators: $\hat{D_x}(x)=e^{-i\frac{x}{\hbar}\hat{p_{x}}}$: the spatial displacement operator, moves the wave function $\psi$ along the x ...
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2answers
88 views

Trace of the Operator

I want to ask a question about the fundamental knowledge of trace of the an operator. The operator $A$ is $$A = v (G_r-G_a)$$ where v is the velocity operator of the Hamiltonian ($v=dH/dk$); $G_r$ and ...
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1answer
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Fermion operators: why $\rm SU(2)$ symmetry and not $\rm U(2)$ symmetry?

Let us consider operators $c_{\uparrow}$ and $c_{\downarrow}$ which destroy a fermion with spin up and a fermion with spin down, respectively. These operators can be found, for example, in the Hubbard ...
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52 views

Understanding bra-ket outer product infinite sum

I have an elementary question on clarifying the following expression: $ \sum_{n=0}^{\infty} |n+2\rangle \langle n| $ Can the term $|n+2\rangle \langle n|$ be observed as the outer product of the ...

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