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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2answers
92 views

Proving properties of Hermitian conjugate [closed]

I have three properties: If $\hat{A}$ and $\hat{B}$ are Hermitian operators. Then $\hat{A}\hat{B}$ is Hermitian provided $\hat{A}$ and $\hat{B}$ also commute $[\hat{A},\hat{B}]=0$ If $\hat{A}$ and $\...
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1answer
71 views

Why does the time translation operator have a different sign than the time evolution operator?

The time evolution operator is $$\hat U(t)=e^{-i\hat H t}$$ where $\hat H$ is the Hamiltonian, $\hbar=1$, and the state is at time $t=0$. The time translation operator is defined similarly, but with ...
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0answers
87 views

Prove that the operator is Hermitian [closed]

I am looking to prove that the operator $\hat{O} $ is Hermitian, where, $$\hat{O} = ix\frac{d}{dy}$$ I have tried the following but I do not know if the answer is correct. An operator is Hermitian if, ...
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2answers
82 views

How to prove with and without using Einstein summation method? [closed]

To proof : [A.L,B.L] = i(AxB).L Where two vectors A & B commute with each other and with L also. L is angular momentum operator
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1answer
98 views

Properties of operators

I am having trouble understanding properties of operators For an operator $\hat{A}$ which of the following would be correct: $\int{(\hat{A}\phi)^*\psi}=\int{\hat{A}^*\phi^*\psi}$ $\int{(\hat{A}\phi)^*...
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2answers
300 views

What are Hermitian conjugates in this context?

I am having trouble understanding the definition Hermitian and Hermitian conjugate. An operator is Hermitian provided that: $\hat{O}^\dagger=\hat{O}$ The Hermitian conjugate of the differentiation ...
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0answers
48 views

Do eigenstates of the creation operator actually exist?

Regarding the eigenstate of creation operator $\hat{a}^\dagger$, the answer to this question shows that the eigenstate does not exist. However, it is stated in another answer, that the proof has ...
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3answers
76 views

Where does the expectation value of $x$ formula come from?

I want to understand precisely where the formula for the expectation value of $x$ comes from (in QM): $$\langle x\rangle=\int _{-\infty}^{\infty}\psi ^*x\psi dx $$ I know that an expectation value (in ...
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1answer
35 views

How to distribute an operator on a product of functions?

How would an operator act when it is applied on a product of functions? Let's say we have: $\hat{A}f(x)g(x)$ Is this equivalent to $\hat{A}[f(x)g(x)]$ or $[\hat{A}f(x)]g(x)$? An example would be if $\...
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The expectation value of momentum in an infinite well stationary state [duplicate]

For a particle in an infinite well potential given by: I am able to successful derive the normalized wavefunction as: $$u_n(x) =\sqrt{2/a}\sin(\frac{n\pi x}{a})$$ where this normalization has the ...
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1answer
81 views

Momentum as Generator of Translations - Classical to Quantum

To summarize my understanding of https://www.homotopico.com/honest%20physics/quantum%20mechanics/classical%20mechanics/2018/09/23/momentum-generator-translations.html, the generator of transformations ...
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54 views

Changing order of operators

I was trying to prove a specific example of Ehrenfest theorem which states: $$\frac{d}{dt}\langle p\rangle = -\left\langle\frac{\partial}{\partial{x}}H\right\rangle$$ where p is the momentum and H is ...
3
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1answer
122 views

Rotation matrices in Schwinger's oscillator model of angular momentum

I am Section 3.9 in Sakurai's Modern QM, 3rd ed (which is Section 3.8 in 2nd ed.) I am trying to obtain the given form for $\hat D(R)|jm\rangle$: I employ $\hat D^{-1}\hat D=1$ and ignore the ...
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1answer
46 views

Spin vector representation for 1/2 spin

I know that, for 1/2 spin systems, the projection of the spin vector along one of the base's axis can be represented using Pauli's matrices as $\hat{S}_i = \frac{\hbar}{2}\sigma_i$. While studying ...
2
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2answers
111 views

Weird quantum linear operator [closed]

For a problem sheet at uni, I need to find eigenvalues and normalised eigenstates of a linear operator. This operator is $\hat{Q}$ and is defined by its action on the normalised eigenstates of the ...
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0answers
58 views

Orthogonality of the annihilation operator in QFT?

In one of my QFT exercices it seems my teacher assumes $$\int dp \int dp' a^†(p)a^†(p')\exp(ipx)\exp(ip'x')=0.$$ Could someone tell me why that is?
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Wick's theorem for fermions

I'm currently studying Wick's theorem for fermions with Peskin's and Schroeder's Introduction to QFT (p.115 & p.116). Here, Wick contractions are defined as $$ \psi^\bullet (x)\bar{\psi}^\bullet(y)...
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3answers
81 views

Show that if $\psi$ is an eigenfunction for the operator $A$ and $[A, B]\psi =0$ then $\psi$ is an eigenfunction for the operator B also

It is only possible for a state to have definite values for both $A$ and $B$ if the wave function $\psi$ satisfies $[A, B]\psi =0$. This is a statement from Lectures on Quantum Mechanics by Weinberg ...
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2answers
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Does the Schrodinger equation obey the rule for differentiating a function if the function is in terms of the wavefunction?

Does the Hamiltonian operator act like a derivative when acting on a functional in terms of wavefunctions? For example, does $$H\psi^2=2\psi H\psi$$ hold true? More generally, if the functional, $F(\...
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How do ladder operators in harmonic oscillator problem manage to accomplish this?

When we try to solve the harmonic oscillator problem by projecting the time independent equation onto position basis,we obtain solutions which do not vanish at infinity, then we ignore these solutions ...
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30 views

Angular momentum operator: converting derivatives to spherical polar [closed]

Given, for example, $$ L_x=-i\hbar\left( y\dfrac{\partial}{\partial z}-z\dfrac{\partial}{\partial y} \right) ~~,$$ I want to convert the expression to polar coordinates. I have \begin{align} x&...
3
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1answer
140 views

Dirac-delta-functions as eigenbasis of the position operator - pure nonsense? Or can more be said?

I remember overthinking equations like \begin{equation} \mathbf{1}=\int dx\ |x\rangle\langle x|\tag{1} \end{equation} and \begin{equation} X=\int dx\ |x\rangle\langle x|x\tag{2} \end{equation} when I ...
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1answer
146 views

Conjugate complex of linear operators in quantum mechanics

I'm pretty new to quantum mechanics (I would like to understand it broadly as an hobbyist). I'm trying to reading Principles of Quantum Mechanics by Dirac. I've found difficult to understand a ...
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29 views

Four-momentum and potential cross product identity

In my script about the dirac equation, i come across the following equation identity: $$ \vec p \times \vec A = -e\vec A\times\vec p + \hbar/i e (\nabla\times\vec A) .$$ $p$ momentum vector, $A$ ...
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2answers
52 views

Localised wavepacket (basic question from Srednicki chap. 5)

I'm currently working through Srednicki and I am confused by a couple of lines in chap. 5 (the LSZ reduction formula). He wants a wavepacket localised around $\mathbf{k = k}_1$ and $\mathbf{x} = 0$ at ...
3
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2answers
203 views

Fundamental Understanding of Hamiltonians

First of all, my major is CS for several months I have been exploring the area Quantum Computing, therefore my background in Quantum Mechanics is a bit lacking. I know that a Hamiltonian is a self-...
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49 views

Background field method for QED

I want to evaluate the 1-loop beta function for massless QED using background field method. This is my trying. First we separate the gauge field into $$A_\mu(x)=\bar{A}_\mu(x)+\delta A_\mu(x)$$ \begin{...
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2answers
52 views

Help proving bound on POVM measurement probabilities

I am trying to follow Nielsen and Chuang's 1 proof that the difference in measurement probabilities is bounded by the difference between two unitary operators applied to a given state. Can someone ...
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1answer
87 views

Determinant of differential operator $( \partial^2 + m^2)$

For a scalar field in QFT the generating functional is given as: $$ Z[J] = \int \left[ d\phi \right] \exp{\left( i\ S[\phi] + i \int d^4 x\ \phi (x) J(x) \right)} $$ with $ S = \frac{1}{2} \int d^4 x\ ...
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2answers
85 views

Exponential of spin operator

I was working with time evolution of a spin state under constant magnetic field and I stumbled upon this operator: $$\exp\left[{\frac{i \omega _0 t}{2} S_x}\right]$$ where $S_x$ (also called $\sigma ...
2
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2answers
128 views

Disentangling exponential of number operator and creation and annihilation operators

Is there a way to disentangle the exponential of the sum of the number, the annihilation and the creation operator? For example, $$e^{\alpha N + \beta a + \gamma a^\dagger } = e^{G a^\dagger}e^{A N}e^{...
1
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1answer
48 views

Hermiticity of the Hamiltonian operator with probability conservation

I am following MIT lessons on quantum physics (Prof. Zwiebach): Part I, Lecture 6, at https://ocw.mit.edu/courses/physics/8-04-quantum-physics-i-spring-2016/lecture-notes/ Video lecture: https://youtu....
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0answers
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If the multiplication of two operators is hermitian, then will they commute? [migrated]

It is proven that if two operators $\hat{X}$ and $\hat{Y}$ commute, then the multiplication of them will be hermitian, i.e. if $\hat{X}\hat{Y}=\hat{Y}\hat{X}$, then $\left(\hat{X}\hat{Y}\right)^\...
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2answers
68 views

Position matrix representation in QM [closed]

For quantum harmonic oscillator, if wave function is in a superposition of two wave functions, $$ \psi(x)=(1/\sqrt2)\psi_n(x)+(1/\sqrt2)\psi_m(x) $$ and position operator is represented as a matrix, ...
6
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1answer
142 views

Why do self-adjoint operators have to be densely defined?

I have been watching the Schiller lectures on QM and have been going through ‘quantum mechanics and quantum field theory’ by Dimock. Both seem to ensure operators are densely defined, especially if ...
1
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1answer
50 views

How does the derivative of an operator valued function act on vectors?

Let $A(x)$ be an (linear) operator valued function a real parameter $x$. Is it true that: $$\left[\frac{d}{dx}A(x)\right]\left|\psi\right> = \frac{d}{dx}\left[A(x)\left|\psi\right>\right],$$ ...
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2answers
353 views

Conceptual question on eigenvectors in quantum mechanics

Page number 1 on Quantum Many-Particle Systems by John W. Negele and Henri Orland says the following about quantum mechanical position eigenvector $|r\rangle$ & momentum eigenvector $|p\rangle$ in ...
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2answers
85 views

Hermicity of Lorentz group generators

In Ashok Das Lectures on QFT book, pg. 135-136, he stated the following hermicity properties for the Lorentz group generators: $$ {J_i}^\dagger=-J_i\,,\quad{K_i}^\dagger=K_i \tag{4.45}\label{4.45} $$ ...
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1answer
168 views

How should I solve the Schrödinger equation by diagonalization in QFT?

My professor recently send me off to solve the Anderson Impurity Model (AIM), which is expressed in terms of creation and annihilation operators: $$ \widehat{H} = \sum_\sigma \varepsilon_d d_\sigma^\...
4
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2answers
74 views

Quantum expressions involving Dirac delta function

I want to find the following quantum expression: $$ \langle x|PX|x'\rangle.$$ A. If I use $X|x'\rangle = x'|x'\rangle $, I will get: $$ \langle x|PX|x'\rangle = \langle x|Px'|x'\rangle = x'\langle x|...
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1answer
34 views

How is a tensor operator defined in terms of commutators?

If $J_i$ represent the angular momentum operators, then a scalar operator $S$ (rank-0 tensor) is defined as an operator which satisfies $$[S,J_i]=0$$ for $i=1,2,3$. $A_i$ is a vector (rank-1 tensor) ...
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2answers
173 views

How is the term 'current' defined in QFT?

Reading papers and books about QFT, the term current is often mentioned with examples like the quark current in QCD or the electromagnetic current in QED. I was wondering, if there is a precise ...
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0answers
28 views

Ordering of operators in 2nd Quantization of the Tight Binding Model

I am looking at the position space representation of the Tight Binding Hamiltonian (without the spin component and ignoring any constants) as given by: $$\hat H = \sum_{i}\sum_{\delta}(\hat c_{i}^{\...
2
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2answers
58 views

The unit momentum operator $\hat{\textbf{p}} ={\textbf{p} \over |\textbf{p}|}$

In studying the Dirac equation, I often come across the unit momentum operator $\hat{\textbf{p}}$ which is defined as $$\hat{\textbf{p}} ={\textbf{p} \over |\textbf{p}|},$$where $\textbf{p}$ is ...
2
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1answer
73 views

$\phi^4$-theory, S-matrix Feynman diagram to first order from Peskin and Schroeder

This relates to page 111 in Peskin and Schroeder. We have the $\phi^4$ S-matrix for a 2-particle to 2-particle scattering reaction: $$-i\frac{\lambda}{4!}\int d^4x \langle p_1p_2|\mathcal T\left(\phi(...
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0answers
54 views

What does it “physically” mean, in terms of uncertainty and measurements, for a commutator to be different than zero in quantum mechanics?

Let's consider the commutator $[L_i,L_j] = i \hbar L_k$ of the angular momentum. The consequence of this equation is that two components of the angular momentum cannot be simultaneously measured. I ...
1
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1answer
84 views

Transforming Observables, Misunderstanding Griffiths, Intro. to QM, or a Different Definition

In Griffiths' Intro. to QM 3rd, Sec. 6.2, transforming an observable $Q$ by the translation operator $T$ is found to be $$ Q' = T^\dagger Q\ T $$ the same for the parity operator $\Pi$ instead of $T$ ...
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2answers
52 views

Building eigenfunctions from eigenkets

Suppose I have a wavefunction into which I insert the completeness relation of some discrete basis as $$ \psi_\alpha(x)=\langle x|\alpha\rangle=\sum_k \langle x|a_k\rangle\langle a_k|\alpha\rangle=\...
3
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2answers
185 views

Any quantum state is an eigenstate of some complete observable?

$\newcommand{\ket}[1]{\left|#1\right>}$ Quantum Mechanics and Experience by David Albert states (p. 63): $$\ket{A} = \tfrac{1}{\sqrt{2}} \ket{\text{black}}_1 \ket{\text{white}}_2 - \tfrac{1}{\...
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0answers
25 views

Commutation relation of ladder operators for multiple fields

I'm studying the Klein-Gordon free theory for 2 real fields. The commutation relation for the ladder operators are \begin{matrix} [\hat{a}_p,\hat{a}_q]=[\hat{a}^\dagger_p,\hat{a}\dagger_q]=0 & \;\;...

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