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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What is the commutator of an operator and its derivative?

Is it possible to calculate in a general way the commutator of an operator $O$ which depends on some variable $x$ and the derivative of this $O$ with respect to $x$? $${O}={O}(x)\\ ...
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116 views

How do I prove the equivalence of chirality and helicity operators acting on a massless Dirac spinor?

I have massless Dirac equation and chirality and helicity operators which are given as $$ \hat {P}_{ch}\Psi = \gamma_{5}\Psi, \quad \hat {P}_{h}\Psi = \frac{(\hat {\mathbf S} \cdot \mathbf ...
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310 views

Writing an arbitrary operator in bra-ket notation

An annoying fact about my physics textbook (Griffiths' Introduction to Quantum Mechanics) is that it introduces bra-ket notation without telling us how to use it. So I have a two-part question for SE: ...
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88 views

How to find different operator representations in QM?

I read that any observable operator may be represented as: $$\Omega = \sum_n \omega _n | \omega _n \rangle \langle \omega_n |$$ Where the little omegas are the eigenvectors/eigenvalues of the ...
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800 views

Self-adjoint and unbounded operators in QM

An operator $A$ is said to be self-adjoint if $(\chi,A\psi)=(A\chi,\psi)$ for $\psi, \chi \in D_A$ and $D_A=D_{A^\dagger}$. But for the free particle momentum operator $\hat{p}$ these inner products ...
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How is the set of displacement operators best called?

Displacement operators $\hat D(x,p), \ \ x,p\in\mathbb{R},$ follow a composition rule $$D(x,p) D(x',p') = \exp\frac{i(px'-xp')}2 D(x+x',p+p').$$ Because of the extraneous phase factor, the set of all ...
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118 views

Position and momentum bases in quantum mechanics

I have seen the following two descriptions of the position basis: $$\tag{1}| x\rangle=\delta(x-x_0)$$ and also $$\tag{2}\langle x_0| x\rangle=\delta(x-x_0),$$ which (if either) of these is ...
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371 views

How can (in Dirac's terminology) the product of two “real” linear operators be “not real”?

I'm puzzled about a statement from Dirac's book, The principles of quantum mechanics, (§8, p.28): As a simple examples of this result, it should be noted that, if $\xi$ and $\eta$ are real, in ...
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453 views

The Holstein-Primakoff Representation (approximation)

I have a question regarding the Holstein-Primakoff representation. In the HP-representation we define the spin operators in terms of bosonic creation and annihilation operators. $$ S_j^+ = \sqrt{2S ...
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117 views

Normal ordering of the identity operator

I'm puzzled about what should be the normal ordering of the identity operator (or any proportional operator): looking at it from the "Fock space operators POV",the prescription is to move all the ...
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1answer
70 views

Electric field in a cylinder

We have electric charge density $\rho(r) = kr$ in a cylinder of infinite height and radius $a$. I'm asked to find the electric field. I'm doing it using two methods and I don't undesrtand why then ...
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Virasoro Operators commutation relations

For the commutation relation in quantising the bosonic string $\left[L_n,L_{m}\right]=(n-m)L_{n+m}+\frac{D}{12}n(n^2-1)\delta_{n+m,0}$ we can then calculate this for $m=-n$ in between the vacuum ...
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1answer
227 views

Intrinsic parity of particle and antiparticle with spin zero

I need to prove that the intrinsic parities of a particle and antiparticle with spin zero are the same. Can I prove that by an argument that operator of $P$-inversion commutes with charge conjugation ...
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82 views

About Divergence in polar coordinates

I've got a conductor in a cylinder shape who is rotating with angular velocity $\omega$ around its axis, that correspond to the $z$ azis I want to calculate the electric field and the density of ...
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1answer
94 views

Eigenfunction associated with the $\hat{x}$ operator

Consider the following operator $\hat{x}=i\hbar \frac{\partial}{\partial p}$. I am trying to show that the eigenfunctions of $\hat{x}$ are not square-normalizable. I am interested in doing so since ...
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Fundamental Commutation Relations in Quantum Mechanics

I am trying to compile a list of fundamental commutation relations involving position, linear momentum, total angular momentum, orbital angular momentum, and spin angular momentum. Here is what I have ...
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246 views

Shortcut to find $\hat{p}^2$ expectation value

I have been going through several calculations where I am asked to calculate $\langle p^2 \rangle$ and the task is proving to be pretty tedious. Does anyone know of a shortcut for this? Such as with ...
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264 views

Expectation of momentum in the bound state

Is it logically correct to assert that the expectation of the momentum $$\langle \hat p \rangle=0$$ for any bound state because it is bound to some finite region? What is the physical interpretation ...
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Any other bound state problems using $a_+$ and $a_-$?

Why is it that creation and annihilation operators ($a_+$ and $a_-$) can only be defined for the problem of quantum harmonic oscillator and nothing else? Can any other bound state problem be solves ...
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1answer
90 views

Translation Operator for Position on Momentum

Consider the translation operator $$\hat T=\exp[-ic\hat p/\hbar], $$ which acts on the position operator in the following way: $$\hat T^\dagger \hat q\hat T = \hat q+c.$$ If I take $\hat T ^\dagger ...
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200 views

Bounded and Unbounded Operator

Can someone explain with a concrete example of how can I can check whether a quantum mechanical operator is bounded or unbounded? EDIT: For example., I would like to check whether $\hat ...
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91 views

Self-adjoint and nonpositive differential operators

I recently tumbled over a statement in a geophysics paper (PDF here). They have a wave equation which they formulate as $$ \frac{1}{v_0}\frac{\partial^2}{\partial t^2} \begin{pmatrix}p \\ ...
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Translation Operator on two operators

On my last HW set, we were asked to show that the operator $$\hat T = \exp(-ic\hat p /\hbar)$$ acts as a translation operator ($\hat T^\dagger q\hat T=q+c)$. This was simple to show using commutators ...
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349 views

Self-adjoint differential operators

I'm having a hard time understanding the deal with self-adjoint differential opertors used to solve a set of two coupled 2nd order PDEs. The thing is, that the solution of the PDEs becomes ...
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211 views

Chronological and normal ordering

I've realized I'm little bit confused when I want to treat elements like this $$\left<\phi_0|T\{a_p(t)a_p^+(t')V(t_1)V(t_2)\}|\phi_0\right>$$ with $$V(t)=\dfrac12 \dfrac{1}{(2\pi ...
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Wick's theorem for calculating OPE

I am trying to understand a calculation using Wick's theorem. Let $T(z)$ be the analytic part of a stress-energy tensor, and $\phi(z)$ a free boson field. Now, ...
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259 views

Ladder Operator killing state

So in my last question, @joshphysics showed me how to prove $K_\pm$ were ladder operators. Now I need to show that there is a lowest state, i.e $$\langle m_0|K_+=K_-|m_0\rangle=0$$ I am not ...
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131 views

Showing $K_\pm$ are raising/lowering operators

In this post, I have the following operators defined: $$K_1=\frac 14(p^2-q^2)$$ $$K_2=\frac 14 (pq+qp)$$ $$J_3 = \frac 14 (p^2+q^2)$$ I am given $ J_3|m\rangle = m|m\rangle$ and asked to show that ...
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Diagonalization Of $(\sigma_x+\sigma_y)$

Can this matrix $(\sigma_x\pm\sigma_y)$ be diagonalised? Clearly, if $\sigma_x$ is diagonalized by a similarity transformation $S_1\sigma_x{S_1}^{-1}$, then $\sigma_y$ can't be diagonalized by $S_1$, ...
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1answer
88 views

Eigenvalues of Infinite Dimensional Matrix [duplicate]

If I take a infinite-dimensional square matrix, what can I say about its eigenvalue spectrum? Will they have a discrete infinity of eigenvalues or continuous infinity of them?
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1answer
351 views

Imaginary Eigenvalue Of A Hermitian Operator

The eigenfunctions of a Hermitian operator are real. But consider a function $\psi(x)=e^{-\kappa x}$, $x\in\mathbb{R}$, where $\kappa$ is a real constant. Then, $$\hat p \psi(x)=-i\hbar ...
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Ambiguity in number of basis vectors [duplicate]

The dimension of the Hilbert space is determined by the number of independent basis vectors. There is a infinite discrete energy eigenbasis $\{|n\rangle\}$ in the problem of particle in a box which ...
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1answer
216 views

Help Simplifying a Commutator Equation

For the SHO, our teacher told us to scale $$p\rightarrow \sqrt{m\omega\hbar} ~p$$ $$x\rightarrow \sqrt{\frac{\hbar}{m\omega}}~x$$ And then define the following $$K_1=\frac 14 (p^2-q^2)$$ $$K_2=\frac ...
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610 views

Commutators involving functions

I am looking for the commutator: $$[e^{aq},p]$$ My approach is to Taylor expand the function: $$[\sum_n \frac{1}{n!}(aq)^n,p]$$ I know that $[q^n,p]=ni\hbar q^{n-1}$ So how do I account for $n$ ...
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1answer
120 views

Simple Commutator question

For some reason this is really tripping me up: $$[q_rq_sp_r,q_sp_rq_s]$$ Where $r$ and $s$ are different. Is this just zero because $p_r$ on $q_s =0$. I am trying to simplify this and I feel like 0 ...
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1answer
228 views

Mathematical motivation of OPE?

In Peskin & Schroeder (and also Cheng which I have skimmed through) they motivate the Operator Product Expansion with a lot of words. Is there any way to motivate it mathematically, e.g. Taylor ...
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513 views

Example of application of creation/annihilation operators in matrix form

I was wondering how it would sound like the creation/annihilation of particles that we usually do in the context of Dirac formalism, with matrices and vectors. As a reminder we know that: ...
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1answer
54 views

Implementing a transformation as $UaU$ and not $UaU^{-1}$?

I know one associates to each symmetry transformation a unitary/antiunitary operater...etc. But equation 3.123 in Peskin and Schroeder (PS) says that parity is implemented as $(\mathbf{p}$ is the ...
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Representation of operators in quantum mechanics

For which systems we represent the Hamiltonian as a differential operator and for which system we represent it by a matrix? Can the momentum be represented by a matrix operator?
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1answer
201 views

Continuity domain for momentum operator

I know this is essentially a mathematic question, but I received no answer on math SE. Moreover it has a direct application in physics, so I thought to ask this here too. The momentum operator in one ...
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1answer
397 views

Question about creation operator and its Hermitian adjoint

The definition of creation operator for bosonic system is $$a^{\dagger}|... n_i ...\rangle = \sqrt{n_i + 1} |... n_i + 1 ...\rangle $$ If I take Hermitian adjoint of this I will get ...
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1answer
114 views

Linear Operators and their representations

I am currently learning Quantum mechanics on a slightly advanced level. I am curious in knowing if there are Linear Operators (Linear Maps) in the Hilbert Space (finite dimensional ones) that don't ...
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1answer
710 views

Showing that an operator is Hermitian

Consider the operator $$T=pq^3+q^3p=-i\frac{d}{dq}q^3-iq^3\frac{d}{dq}$$ defined to act on the Hilbert Space $H=L^2(\mathbb{R},dq)$ with the common dense domain $S(\mathbb{R})$. Here $S(\mathbb{R})$ ...
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1answer
195 views

Lorentz transformation implemented by a non-unitary operator.

One often come across in QFT sentences like the following, for instance: ...under a Lorentz transformation $\Lambda$ implemented by the unitary operator $U(\Lambda)$, a Dirac field transforms ...
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Matrix element of an operator

Suppose we want to calculate the following matrix element. $$ \langle \alpha |\hat{o}|\beta\rangle $$ Where $\alpha$ and $\beta$ are two arbitrary basis states in a many-particle basis set, and ...
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Action of Parity operator on Impulse representation

Is my derivation of the action of the parity operator $\mathbb{P}$ on the $|p\rangle$ representation correct? $$\left( \mathbb{P}\tilde\psi \right)(p)= - \tilde\psi (p).$$ Obtained from $$\left( ...
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266 views

Commutator not transitive

I noticed the following: $$[L_{+},L^2]=0,\qquad [L_{+},L_3]\neq 0,\qquad [L^2,L_3]=0.$$ This would suggest, that $L^2,L_+$ have a common system of eigenfunctions, and so do $L^2,L_3$, but $L_+,L_3$ ...
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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, $AB+BA$. ...
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294 views

Countable Matrix Representation

In my quantum mechanics class, my professor explained that the Hamiltonian along with position and momentum operators can be represented by matrices of countable dimension. This is especially usefull ...
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209 views

Algebraic formulation of QFT and unbounded operators

In AQFT one specifies the structure of the observables as a $C^*$-algebra. This seems to excludes algebras that don't have a norm, such as the Heisenberg algebra. Fortunately for this case one turns ...