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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Schrödinger equation in position representation

We start from an abstract state vector $ \newcommand{\ket}[1]{|{#1}\rangle} \ket{\Psi}$ as a description of a state of a system and the Schrödinger equation in the following form $$ ...
4
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2answers
296 views

How to promote algebraic expressions to operators in quantum mechanics?

Okay, I know that in quantum mechanics the quantum observable is obtained from the classical observable by the prescription $$ X \rightarrow x,\quad P \rightarrow -i\hbar\frac{\partial}{\partial x} ...
3
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1answer
634 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 ...
6
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3answers
367 views

Commutator with a square root

How to find the commutator $[a, \sqrt{a^\dagger a}]$? Here $a$ is a usual bosonic annihilation operator, and $[a, a^\dagger] = 1$. The first thing I tried is $$ [x,A] = [x, \sqrt{A}]\sqrt{A} + ...
2
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2answers
159 views

Are higher order mixed partial derivatives of wave function with different ordination equal?

For example, given two operators: $$A = \frac{\partial}{\partial x}+\frac{\partial}{\partial y},$$ $$B =\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2} + 1.$$ Deriving commutator ...
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1answer
2k views

Expectation value of position in infinite square well

I'm looking for some help to a question. I'm working in the infinite square well, and I have the wavefunction: $$\psi(x,t=0)=A\left( i\sqrt{2}\phi_{1}+\sqrt{3}\phi_{2} \right).$$ For every time t, ...
6
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2answers
432 views

Non-associative operators in Physics

Are non-associative operators (or other kind of elements) used in Physics? For example, in QM I'm looking for something like this: $A(BC)|\psi\rangle \ne (AB)C|\psi\rangle$ NOTE: I think that this ...
2
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1answer
149 views

Replacing an operator with its expectation value

While dealing with a circling particle in an spherical symetric potential our professor said that we can replace an operator of $z$ component of angular momentum $\hat{L}_z$ with the expectation value ...
7
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1answer
636 views

Can one define an acceleration operator in quantum mechanics?

It seems most books about QM only talk about position and momentum operators. But isn't it also possible to define a acceleration operator? I thought about doing it in the following way, starting ...
3
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0answers
80 views

Non-Hermiticity when Fourier transforming onto a finite lattice

I'm doing numerical simulations. I have the Haldane model in a honeycomb lattice where $$ H = \sum \limits_{<ij>}a^\dagger_i b_j + h.c $$ Where $i$ belongs to sublattice $A$, and $j$ to ...
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2answers
2k views

From position space to momentum space

Lets say I have a state vector $\left|\Psi(t)\right\rangle$ in a position space with an orthonormal position basis. If I now use an operator $\hat{p}$ on this basis I will get basis which corresponds ...
3
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1answer
126 views

Observables - what are they?

I often read in books that an observable is represented by an Hermitean operator. But it is deceiving as operator isn't the observable. As far as I've read the observable is denoted like $\langle ...
14
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1answer
697 views

Discreteness of set of energy eigenvalues

Given some potential $V$, we have the eigenvalue problem $$ -\frac{\hbar^2}{2m}\Delta \psi + V\psi = E\psi $$ with the boundary condition $$ \lim_{|x|\rightarrow \infty} \psi(x) = 0 $$ If we ...
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0answers
76 views

An application of Toeplitz operators

I want to find an application of the Toeplitz operators. All I need is a known problem (not an open problem) which solution use the theory of Toeplitz operators. I don't need all the details but I ...
0
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1answer
83 views

Eigenvalue $a_n$

Q1: In Zetilli's book page 166 (ch. "Postulates of QM", eq. 3.1) i encountered an expression $\hat{A}|\psi\rangle = a_n|\psi_n\rangle$. I know this is an eigenvalue equation, but i have seen another ...
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2answers
175 views

Why is this not a realisable operation on a quantum system?

Let $\rho = \begin{bmatrix}\ 1&0 \\ 0&0 \end{bmatrix}$, $\rho' = \begin{bmatrix}\ 0&0 \\ 0&1 \end{bmatrix}$, $\rho'' = \dfrac{1}{2}\begin{bmatrix}\ 1&1 \\ 1&1 \end{bmatrix}$ ...
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2answers
224 views

Quantum Mechanical Operators in the argument of an exponential

In Quantum Optics and Quantum Mechanics, the time evolution operator $$U(t,t_i) = \exp\left[\frac{-i}{\hbar}H(t-t_i)\right]$$ is used quite a lot. Suppose $t_i =0$ for simplicity, and say the ...
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0answers
43 views

Schrodinger equation in momentum space [duplicate]

I have a problem this is: When I solve the Schrodinger equation in momentum space, I had done as this: $\begin{array}{l} i\hbar \frac{{\partial \Psi }}{{\partial t}} = - \frac{\hbar ...
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2answers
122 views

Proof $\left[ {\hat H,{{\hat p}_i}} \right] = - \frac{\hbar }{i}\frac{{\partial \hat H}}{{\partial {{\hat q}_i}}}$ [closed]

I have a problem with the Hamiltonian, I don't think anything to solve it!! So could you give me some hints! Knowing that: $$\left[ {{{\hat p}_i},{{\hat q}_k}} \right] = \frac{\hbar }{i}{\delta ...
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1answer
442 views

Some Dirac notation explanations

Equation for an expectation value $\langle x \rangle$ is known to me: \begin{align} \langle x \rangle = \int\limits_{-\infty}^{\infty} \overline{\psi}x\psi\, d x \end{align} By the definition we ...
2
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2answers
516 views

How do we know that $\psi$ is the eigenfunction of an operator $\hat{H}$ with eigenvalue $W$?

I am kind of new to this eigenvalue, eigenfunction and operator things, but I have come across this quote many times: $\psi$ is the eigenfunction of an operator $\hat{H}$ with eigenvalue $W$. ...
2
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1answer
88 views

Statistical sum of physical quantities in a quantum system

Let $C = A + B$ (statistical sum, so $\mathbb{E}[C] = \mathbb{E}[A] + \mathbb{E}[B]$), and let $p(A = a) = 1$. Are the following true? $\mathbb{E}[C^2] = a^2 + 2a\mathbb{E}[B] + \mathbb{E}[B^2]$ ...
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1answer
143 views

Energy eigenvalues of a Q.H.Oscillator with $[\hat{H},\hat{a}] = -\hbar \omega \hat{a}$ and $[\hat{H},\hat{a}^\dagger] = \hbar \omega \hat{a}^\dagger$

I just finished deriving the commutators: \begin{align} [\hat{H}, \hat{a}] &= -\hbar \omega \hat{a}\\ [\hat{H}, \hat{a}^\dagger] &= \hbar \omega \hat{a}^\dagger\\ \end{align} On the ...
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129 views

A particlar normal ordering problem [duplicate]

Say we have an expression of the form: $$ \left<0\right|:\phi(x)^2: : \phi(y)^2:\left|0\right>, $$ where $\phi$ is some scalar field. I have heard the claim several times, that in evaluating ...
3
votes
2answers
516 views

Coherent State, Unitary Operators, Harmonic Oscillator

Consider the operator: $$O = e^{\theta(a^\dagger b - b^\dagger a)}$$ where $\theta$ is a constant. $O$ is a unitary operator. $a$, $a^\dagger$, $b$, and $b^\dagger$ are ladder operators for two ...
4
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2answers
471 views

Proof for commutator relation $[\hat{H},\hat{a}] = - \hbar \omega \hat{a}$

I know how to derive below equations found on wikipedia and have done it myselt too: \begin{align} \hat{H} &= \hbar \omega \left(\hat{a}^\dagger\hat{a} + \frac{1}{2}\right)\\ \hat{H} &= ...
2
votes
1answer
198 views

The matrix element of a normal-ordered operator

Eq (1.137) in Negele and Orland gives the following identity for a normal-ordered operator $A(a_i^\dagger,a_i)$: $$\langle \phi|A(a_i^\dagger,a_i)|\phi'\rangle=A(\phi_i^*,\phi'_i)e^{\sum ...
5
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2answers
1k views

Translator Operator

In Modern Quantum Mechanics by Sakurai, at page 46 while deriving commutator of translator operator with position operator, he uses $$\left| x+dx\right\rangle \simeq \left| x \right\rangle.$$ But for ...
3
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2answers
188 views

Phys.org Spectral geometry to unite relativity and quantum mechanics, restate in laymens terms?

Lingua Franca links relativity and quantum theories with spectral geometry Could someone give me a short synopsis of this article in laymens terms? What implications does this have in the physics ...
3
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1answer
585 views

How to define the mirror symmetry operator for Kane-Mele model?

Let us take the famous Kane-Mele(KM) model as our starting point. Due to the time-reversal(TR), 2-fold rotational(or 2D space inversion), 3-fold rotational and mirror symmetries of the honeycomb ...
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1answer
385 views

Matrix representation for fermionic annihilation operator

My guess it should look something like this: $ c_\sigma = ...
1
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0answers
132 views

Time ordering and Fermions

Having time ordering operator for fermions, should it reverse sign if it swaps operators with opposite spin variable? In other words should $T[c_{t_1,\uparrow}c_{t_2,\downarrow}^\dagger]$ return ...
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0answers
140 views

QFT basics for Klein-Gordon fields

I am teaching myself QFT from Peskin for next years maths course and I have two questions: What is a c-number? Is it a complex number, and if so why does it mean, ...
3
votes
2answers
459 views

Quantum commutator

I'm given this commutator: $$\left[PXP,P\right]$$ Being $P\psi=-i\hbar\partial_x\psi$, and $X\psi=x\psi$ I've solved it in two ways, the first one is just aplying the commutator to some function ...
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1answer
920 views

Hermitian Adjoint of differential operator

I came across this equation (identity) (Eq. 4 in this paper): $\int(-i d\psi/dx)^*\psi dx = \int \psi^*(-i d\psi/dx) dx + id(\psi^*\psi)/dx\mid_{-\infty}^{+\infty}$ I have trouble proving it. I ...
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3answers
681 views

Associating a Unitary operator to proper Lorentz transformations?

If one reads eg page 32 of Srednicki where he says: In quantum theory, symmetries are represented by unitary (or antiunitary) operators. This means that we associate a unitary operator U(Λ) ...
2
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1answer
497 views

Quantum mechanical analogue of conjugate momentum

In classical mechanics, we define the concept of canonical momentum conjugate to a given generalised position coordinate. This quantity is the partial derivative of the Lagrangian of the system, with ...
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1answer
122 views

Operators in quantum mechanics

According to the Quantum Mechanics, can we write $\langle q|p\rangle = e^{ipq}$? If so then how? And if we transfer to integrate formulation then how it will look like?
5
votes
1answer
186 views

Physics Applications of Fredholm Theory:

I find Fredholm theory beautiful, especially the Liouville-Neumann series for solving Fredholm integral equations of the second kind. There seems to be a consensus that these equations are quite ...
3
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1answer
245 views

Quantum graph theory: complex spectra

In quantum graph theory, what are the properties of a given graph to own complex conjugated complex eigenvalues, either finite or infinite? Spectral graph theory is as far as I know a not completely ...
3
votes
1answer
117 views

Spectral properties of CFT

What are the general spectral properties of CFT? I mean what is the "spectrum"/eigenvalues of CFT in 2d and d>2 spacetime dimensions? I understand the "spectrum" and "Fock space" realization of Dirac ...
3
votes
1answer
235 views

The issue on existence of inverse operations of $a$ and $a^{\dagger}$

I have asked a question at math.stackexchange that have a physical meaning. My assumption: Suppose $a$ and $a^\dagger$ is Hermitian adjoint operators and $[a,a^\dagger]=1$. I want to prove that ...
11
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3answers
2k views

How to tackle 'dot' product for spin matrices

I read a textbook today on quantum mechanics regarding the Pauli spin matrices for two particles, it gives the Hamiltonian as $$ H = \alpha[\sigma_z^1 + \sigma_z^2] + ...
4
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5answers
627 views

Math of eigenvalue problem in quantum mechanics

I learned the eigenvalue problem in linear algebra before and I just find that the quantum mechanics happen to associate the Schrodinger equation with the eigenvalue problem. In linear algebra, we ...
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1answer
119 views

Notational techniques for dealing with creation operators on Fock space

This question is trying to see if anyone has some simple notation (or tricks) for dealing with operators acting on coherent states in a Fock space. I use bosons for concreteness; what I'm interested ...
3
votes
1answer
255 views

Show that for QM operator A: $\int_{-\infty}^{\infty}\psi A^{\dagger}A\psi dx = \int_{-\infty}^{\infty}(A\psi)^*(A\psi)dx $

I need to show for $$A = \frac{d}{dx} + \tanh x, \qquad A^{\dagger} = - \frac{d}{dx} + \tanh x,$$ that $$\int_{-\infty}^{\infty}\psi^* A^{\dagger}A\psi dx = ...
2
votes
1answer
184 views

Coordinate representation of quantum ladder operator?

I can't seem to figure out how to derive the coordinate representation of the $a_+$ ladder operator in quantum mechanics. I know that $a_-$ is $\sqrt{\frac{1}{2mwh}} (mwx + i\dot{p}) $ in which where ...
8
votes
1answer
289 views

String theory - OPE and primary operators

First, a disclaimer: I am new to Physics SE, and I am primarily a mathematician, not a physicist. I apologise in advance for the possibly poor quality of the question, any and thank you for your ...
5
votes
3answers
437 views

Why is this identity an if, rather than if and only if?

A recent question (Product of exponential of operators) asked who to proved that the exponentials of operators multiply in same manner as those of scalars if and only if the commutator of the ...
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104 views

Explicit evaluation of a radially ordered product

I am trying to understand the application of the operator product expansion to calculate the radially ordered product in the complex plain of $T_{zz}(z)\partial_w X^{\rho}(w)$ which should result in ...