In physics, an operator is almost always either a square matrix or a linear mapping from one space of functions (often on $\mathbb{R}^N$ or $\mathbb{C}^N$) to the same or other like space of functions. Operators serve as *observables* and as *time evolution operators* in Quantum Mechanics. This tag ...

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103 views

Does $[A^2,B]=0$ imply $[A,B]$=0? [closed]

The commutator $[A^2,B]$ can be written as $A[A,B]+[A,B]A$. So if $[A,B]=0$, $[A^2,B]$ is also zero. But is the converse also true? If $[A^2,B]$ is given to be zero, then is [A,B]=0? Let $C=[A,B]$. ...
3
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1answer
61 views

Squeeze operator

If $\phi(x)$ is an arbitrary normalized function, and $S$ the squeeze operator, $$ S=e^{\frac{\mu\cdot h}{2\pi}(a^{\dagger2}-a^{2})} $$ with $\mu \in \mathbb R$. How can I find the value and the ...
2
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1answer
159 views

Prove the solution of von Neumann equation will never stabilize if Hamiltonian and initial density matrix commutes

Given von Neumann equation $$\frac{d}{dt} \rho(t) = -i [H, \rho(t)] = -i e^{-iHt}[H, \rho(0)]e^{iHt}.$$ If we know that $[H, \rho(0)] \neq 0$, how do we prove in details the solution of von Neumann ...
3
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2answers
207 views

Why do the ladder operators in harmonic oscillators work?

The Hamiltonian can be diagonalized by transforming $x$ and $p$ to $a$ and $a^\dagger$. I understand how one proceeds from there to find the spectrum of $a^\dagger a$, the ground state $|0\rangle$ and ...
0
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1answer
63 views

Is there a difference between the adjoint and conjugate?

Is there a difference between the adjoint and conjugate? I have recently started some work for a quantum field theory module and I'm wondering if there is a difference between the adjoint or conjugate ...
4
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2answers
89 views

Plane wave shift in a differential operator

Does anyone can help me to prove the following equation \begin{equation} e^{-i\vec{k}\cdot\vec{x}}f(\partial_{\mu})e^{i\vec{k}\cdot\vec{x}} = f(\partial_{\mu}+ik_{\mu}) \end{equation} Where $\vec{k}\...
1
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1answer
39 views

Why the constancy of an observable w.r.t time depends on whether it commutes with $H$ or not?

I have been reading Modern Quantum Mechanics by J.J.Sakurai. Under the chapter Quantum Dynamics, the author says if an observable $A$ initially commutes with the Hamiltonian operator $H$, then it ...
0
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1answer
61 views

Question about a formula in the book by Green, Schwarz, Witten

In chapter 7 of superstring theory, it is written $$ g\langle0;k_1|\zeta\cdot\alpha_1V_0(k_2)\zeta_3\cdot\alpha_{-1}|0;k_3\rangle=g\langle0;k_1|\zeta\cdot\alpha_1e^{k_2\cdot\alpha_{-1}} e^{-k_2\cdot\...
0
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1answer
42 views

Does Heisenberg's uncertainty hold for any two quantum measurements?

Heisenberg's uncertainty principle is most commonly expressed in terms of the uncertainty in measurement of position and momentum of a particle, $$\Delta x\Delta p \geq \hbar$$and uncertainty in ...
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1answer
66 views

Taylor expansion of exponential operator

I have an operator: $$ \hat O = e^{\hat A+\hat B}$$ Is it correct to write its first order Taylor expansion by: $$\hat O = 1+\hat A+\hat B$$
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55 views

Uncertainty Principle with the corresponding operators

Why does the corresponding operator do not commute if there is uncertainty related to two observables A and B that states $\Delta A\,\Delta B > 0 $ ?
4
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1answer
190 views

Why can we not choose the stress tensor in a CFT to be identically symmetric?

The stress tensor for a conformal field theory (or any quantum field theory) can be derived from the action $S$ by the functional derivative $$T^{\mu \nu} ~=~ -\frac{2}{\sqrt{|g|}}\frac{\delta S}{\...
1
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2answers
46 views

Annihilation operator in harmonic oscillator

In Wikipedia's QHO page there is a moment when the following is stated: I don't know why "the ground state in the position representation is determined by $a|0\rangle=0$". I would say that the ...
1
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1answer
87 views

Transforming to a rotating frame in the $x$-basis

I was reading this paper on analytically Solvable driven time-dependent two level quantum systems. The Hamiltonian considered in the paper is the following: $$H=\sigma_z\cdot J(t)/2)+\sigma_x\cdot h/2$...
0
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1answer
27 views

Heisenberg Representation of Quantum Computers explain observable transformations

The Heisenberg Representation of Quantum Computers (Daniel Gottesman) http://arxiv.org/abs/quant-ph/9807006 Suppose we have a quantum computer in the state $|\psi\rangle$, and we apply the ...
1
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0answers
39 views

Show that partial derivative with respect to time is anti-hermitian [closed]

I have a definition that $$<s_1,s_2> = \int_{-\infty}^{+\infty}s_1^*(t)s_2(t) dt$$ I need to show that $\partial_t$ operator which is just the partial derivative with respect to time is anti-...
5
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1answer
81 views

Heisenberg's uncertainty principle derivation in a ring [duplicate]

The standard derivation But now suppose the space is a ring of length $L$, it seems the derivation could work out exactly the same and we get $$\Delta p \Delta x \geq \hbar/2.$$ But since $\Delta x$ ...
8
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2answers
1k views

Why are eigenfunctions which correspond to discrete/continuous eigenvalue spectra guaranteed to be normalizable/non-normalizable?

These facts are taken for granted in a QM text I read. The purportedly guaranteed non-normalizability of eigenfunctions which correspond to a continuous eigenvalue spectrum is only partly justified by ...
1
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1answer
51 views

Writing operator evolution as a quantum dynamical map

In the Heisenberg picture we have the evolution of the operator in time given by: $$A(t)=U^+A(0)U$$ I was looking into the theory of open quantum systems where we introduce the concept of a quantum ...
0
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1answer
69 views

Representing operators in the Glauber-Sudarshan P-representation

If $| \alpha >$ represents a coherent state (the normalized right eigenstate of the destruction operator $a$ in Quantum Mechanics; $\alpha$ is a complex number), then it is known that: \begin{...
5
votes
1answer
70 views

Time-ordering of fermion operators

If $A$ and $B$ are fermion operators then the time ordering is defined as \begin{eqnarray} T(AB) = \left\{ \begin{array}{rl} AB, & \mbox{if $B$ precedes $A$}\\ -BA, & \mbox{if $A$ precedes $B$...
1
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2answers
74 views

Calculating the expectation value of spin [closed]

Consider the state-space with a base formed by the eigenstates of the operator $\hat{S}_z$. For the state $|\phi\rangle=\frac{1}{\sqrt2}|+\rangle_z-\frac{1}{\sqrt2}|-\rangle_z$, what is the value of $\...
0
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1answer
32 views

Uncoupled and coupled bases for electrons in hydrogen atom?

I'm given that for an electron in a hydrogen atom, $L=2$ and $S=1/2$ (quantum numbers associated with $L^2$ and $S^2$). I'm also given that for the uncoupled representation, the basis function is $|L,...
1
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1answer
75 views

Time-ordered product of two normal-ordered products of fields

Suppose you have a scalar field theory with field operators $\phi(x)=\phi(x)_+ + \phi(x)_- $ that can be decomposed into terms of annihilation and destruction operators. Let $$ D(x-y) = <0|T(\phi(...
3
votes
1answer
46 views

Momentum operator in effective mass approximation

When we calculate the band structure of some solid then we often find that in the bottom of the conduction band the dispersion looks approximately quadratic with some new effective mass: $$E(k) = \...
0
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1answer
59 views

Quantum Mechanics: Rotation operators

How do I know what direction of the rotation operator to use on the initial state of a spin-1/2 particle? For example, a spin-1/2 particle initially in the $\lvert y \rangle$ state enters a SGz ...
13
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3answers
5k views

Why do we use Hermitian operators in QM?

Position, momentum, energy and other observables yield real-valued measurements. The Hilbert-space formalism accounts for this physical fact by associating observables with Hermitian ('self-adjoint') ...
4
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2answers
137 views

Are the path integral formalism and the operator formalism inequivalent?

Abstract The definition of the propagator $\Delta(x)$ in the path integral formalism (PI) is different from the definition in the operator formalism (OF). In general the definitions agree, but it is ...
0
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2answers
45 views

Order of operators and numbers inside a bracket

I had an argument with my professor. Let $H$ be an operator (e.g. hamiltonian). Let capital $X$ denote the position operator. Let $f$ and $g$ be functions of $X$ that do NOT commute with $H$. Now ...
0
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2answers
100 views

Unitary operators evolving the set of Pauli matrices

Consider the Heisenberg picture of Quantum Mechanics. For a two state system we have the Pauli matrices evolving according to the relation $$\sigma_i(t)=U^+\sigma_i(0)U$$ where $U=e^{-iHt/\hbar}$ and $...
0
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1answer
73 views

Exact solution of Qubit Decoherence using Transfer Matrix

I'm going through a particular paper on decoherence: Exact Solution of Qubit Decoherence models by a transfer matrix method I'm having trouble understanding a particular step in the mathematics ...
0
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2answers
67 views

How to recognize a Complete Set of Commuting Operators (CSCO)

A question about 'completeness'. These two operators are commuting, but I want to know more about their completeness. How do you know if {H}, {B}, {H,B} and/or {$H^2$,B} are forming (a) Complete Set(...
5
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2answers
285 views

Tricky operator identity: $[L^2,[L^2,\vec{r}]]=2 \hbar ^2 \{ L^2, \vec{r}\}$?

This operator identity showed up in a course I was taking, and it was given without proof. $$[L^2,[L^2,\vec{r}]]=2 \hbar ^2 \{ L^2, \vec{r}\}$$ The curly brackets denote the anticommutator, $AB+BA$. ...
17
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3answers
3k views

Equivalence of canonical quantization and path integral quantization

Consider the real scalar field $\phi(x,t)$ on 1+1 dimensional space-time with some action, for instance $$ S[\phi] = \frac{1}{4\pi\nu} \int dx\,dt\, (v(\partial_x \phi)^2 - \partial_x\phi\partial_t \...
4
votes
0answers
56 views

Elegant method to show $[L^2,[L^2,\vec{r}\,]\,] = 2\hbar^2\{L^2, \vec{r}\}.$ [duplicate]

Show that $[L^2,[L^2,\vec{r}\,]\,] = 2\hbar^2\{L^2, \vec{r}\},$ where $\vec{r} = x\, {\hat x} + y\, {\hat y} + z\, {\hat z}.$ "Edit: $\{A,B\} = AB + BA$ is the anti-commutator." I am able to solve ...
3
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0answers
20 views

Gaussian Minimizes Uncertainty - Statement Qualification [duplicate]

On the last page of this paper, the following statements are made (I'll jump right to around the point of interest): Example: Consider $A=p_x$, $B=x$. Then $$\langle A\rangle=\langle B\...
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votes
2answers
91 views

Is the Wave Function a Unitary Operator? [closed]

A unitary operator can be represented as an exponential $$e^{iA}$$ and as we represent the wave function in general as $$e^{i k x}.$$ Does that mean that the wavefunction is unitary as the exponent is ...
4
votes
2answers
106 views

Is the eigenvalue of Hamiltonian invariant under linear transformation of momentum operator?

It is given The dynamics of a particle moving one-dimensionally in a potential V(x) is governed by the Hamiltonian $H_0 = p^2 /2m + V(x) $, where $p = -i\hbar d/dx$ is the momentum operator. ...
0
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0answers
85 views

Eigenfunctions of translation operator

I had an HW assignment in which we were asked to find the eigenfunctions of the translation operator which is defined as follows: $$\hat{D}(a)=e^{-(i/\hbar)a\hat{P}}$$ where $\hat{P}$ is the momentum ...
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1answer
47 views

Deriving eigen values of $\hat{N}$

So let's say we have an operator $\hat{a}$ (ladder operator), where $\left[\hat{a},\hat{a}^\dagger\right] = 1$, and $\hat{a}^2 |\phi\rangle = 0$. How do I show that the eigenvalues of $\hat{N}=\hat{...
0
votes
2answers
469 views

Expectation values of commutator and anti-commutator (momentum and position)

What are the expectation values of commutator and anti-commutator for momentum and position operators? In the case of commutator: $$\langle[x,p]\rangle=\langle i\hbar\rangle=~?$$ In the case of anti-...
1
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0answers
15 views

Relating $C_j(t-t') = \left<\hat{B}_j(t)\hat{B}_j(t')\right>$ to $\left<\hat{B}_j(t)^2\right>$

I'm trying to relate a known quantum correlation function $C_j(t-t') = \left<\hat{B}_j(t)\hat{B}_j(t')\right>$ (which is not real!) of a (time dependent, but this is not super relevant) quantum ...
0
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1answer
34 views

Can a sum of Pauli matrices be a real value?

In the page 2 of Quantum Annealing for Constrained Optimization, the authors introduced a constraint term under the Constrained quantum annealing (CQA) section. The ultimate goal is to work out a ...
3
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1answer
48 views

Generators of a certain symmetry in Quantum Mechanics

In Classical Mechanics to describe symmetries like translations and rotations we use diffeomorphisms on the configuration manifold. In Quantum Mechanics we use unitary operators in state space. We ...
5
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2answers
839 views

Why is the “canonical momentum” for the Dirac equation not defined in terms of the “gauge covariant derivative”?

The canonical momentum is always used to add an EM field to the Schrödinger/Pauli/Dirac equations. Why does one not use the gauge covariant derivative? As far as I can see, the difference is a factor <...
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2answers
49 views

Quantization of the Hamiltonian of a particle in a uniform magnetic field

If a particle of mass $m$ and charge $q$ is subject to a uniform magnetic field and if we have a vector potential $\mathbf{A}$ then we know that classically the dynamics of the particle will be ...
8
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2answers
3k views

Weyl Ordering Rule

While studying Path Integrals in Quantum Mechanics I have found that [Srednicki: Eqn. no. 6.6] the quantum Hamiltonian $\hat{H}(\hat{P},\hat{Q})$ can be given in terms of the classical Hamiltonian $H(...
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0answers
39 views

Lippmann-Schwinger equation and time dependence

Consider the Lippmann-Schwinger equation (LSE) $$ |\psi\rangle = |\phi\rangle + \hat{G}_0(\epsilon) \hat{V} |\psi\rangle \tag{1}$$ where $\hat{G}_0(\epsilon) = \frac{1}{\epsilon - \hat{H}_0 + i\eta}$...
1
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1answer
109 views

Three dimensional isotropic harmonic oscilator Hamiltonian

Let us consider the Hamiltonian for the isotropic three dimensional harmonic oscilator: $$H = \dfrac{\mathbf{P}^2}{2m}+\dfrac{m\omega^2\mathbf{R}^2}{2},$$ where $\mathbf{P}$ and $\mathbf{R}$ are the ...
0
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0answers
21 views

Field Coherent State relationship to annihilation operator

I am trying to show that $|\Psi_{\lambda\bar{n}}> = \sum c_{\lambda\bar{n}m}\exp(-i(n+\frac{1}{2})\omega_\lambda t)|n_\lambda>$, where $c_{\lambda\bar{n}m} = \sqrt{\frac{\bar{n}^{n_\lambda}e^{-\...