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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3
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34 views

Rigorous definition of density of states for continuous spectrum

For operators with pure point spectra it is clear how to count number of states corresponding to a given eigenvalue - one can just calculate the dimension of eigenspaces. I am wondering how to do it ...
3
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5answers
143 views

Where does $\hat{P}\psi(x) = -i\hbar \partial_x \psi(x)$ come from?

It's a very basic question, where does the relation $$\hat{P}\psi(x) = -i\hbar \partial_x \psi(x)$$ for any square integrable $\psi(x)$ come into existence? Some texts I found states that the above ...
5
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4answers
851 views

The Momentum Operator in QM

I've seen the 'derivation' as to why momentum is an operator, but I still don't buy it. Momentum has always been just a product $m{\bf v}$. Why should it now be an operator. Why can't we just multiply ...
1
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1answer
26 views

OPE coefficients identity operator

when I have two canonically normalized operators $\phi_{1}$ and $\phi_{2}$ and I want to compute their OPE in terms of the identity operator, is there any way to actually calculate the first levels of ...
3
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1answer
408 views

How does an operator transform under time reversal?

We know that a time-reversal operator $T$ can be represented as $$T=UK$$ where $U$ is some unitary operator and $K$ is the complex conjugation operator. Then under time-reversal operation, a quantum ...
6
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1answer
185 views

Uncertainty relation for non-simultaneous observation

Heisenberg's uncertainty relation in the Robertson-Schroedinger formulation is written as, $$\sigma_A^2 \sigma_B^2 \geq |\frac{1}{2} \langle\{\hat A, \hat B\}\rangle -\langle \hat A\rangle\langle ...
0
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0answers
63 views

Parity operators and spin

Consider the following excerpt from Weinberg's Lectures on Quantum Mechanics: I follow everything up until the last statement in the excerpt. In fact, from other things I've read, it seems that one ...
1
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2answers
93 views

What is the physical states in Heisenberg picture?

The physics states in Quantum mechanics is represented by vectors in Hilbert space, however in Heisenberg's picture, the equation of motion $$ \frac{d}{dt}A_H(t) = ...
-2
votes
2answers
98 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 $C=[A,B]$. ...
-1
votes
0answers
47 views

Differentiation, Dirac operator, Hermitian [duplicate]

I am trying to prove the hermitan property of Dirac operator $$\gamma^{\mu}D_{\mu}=\gamma^{\mu}(\partial_{\mu}+iA_{\mu}).$$ The underlying space time i assumed is euclidean ...
0
votes
1answer
149 views

Expectation value of total angular momentum $\langle J \rangle$

[I am working with Griffiths Introduction to Quantum Mechanics, 3rd Edition. My problem is general but if you want to look I am reading from ch 4.1 in which the weak-field Zeeman Effect is being ...
3
votes
1answer
57 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
votes
1answer
152 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
votes
2answers
198 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
53 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 ...
3
votes
2answers
82 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 ...
1
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1answer
37 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 ...
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0answers
43 views

Why is the Hermitian conjugate of the Fourier transform of an operator not the transform of the Hermitian conjugate? [migrated]

It is defined that: \begin{align} O(\omega)&=\frac{1}{\sqrt{2\pi}}\int O(t)e^{-i\omega t} \mathrm{d}t \tag{1} \\ O^{\dagger}(\omega)&=\frac{1}{\sqrt{2\pi}}\int O^{\dagger}(t)e^{-i\omega t} ...
0
votes
1answer
58 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}} ...
9
votes
1answer
621 views

Non-separable solutions of the Schroedinger equation

I'm studying an undergraduate Quantum Mechanics course and I have some doubts about the solution of the Schroedinger equation by the separation of variables method. If we suppose that the solutions ...
0
votes
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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votes
2answers
62 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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0answers
52 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
184 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 ...
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2answers
44 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
vote
1answer
81 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 ...
0
votes
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 ...
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0answers
33 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 ...
5
votes
1answer
77 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 $\Delta x$ ...
7
votes
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
47 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
votes
1answer
59 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: ...
5
votes
1answer
65 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 ...
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vote
2answers
72 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 ...
0
votes
1answer
30 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 ...
1
vote
1answer
68 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) = ...
3
votes
1answer
44 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
votes
1answer
57 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 ...
12
votes
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
votes
2answers
119 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
votes
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
votes
2answers
96 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
votes
1answer
68 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
votes
2answers
65 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 ...
5
votes
2answers
279 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
votes
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
54 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 ...
6
votes
1answer
493 views
1
vote
1answer
283 views

Simple QFT simulation - how to do it

I would like to write a simple QFT simulation for a free scalar field with a cubic interaction term. However, I got stuck a bit. I will try to describe what I think I understand. I want to have a ...
3
votes
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 ...