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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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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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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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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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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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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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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123 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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64 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 $$\hat T =exp(-ic\hat p /\hbar)$$ Acted as a translation operator ($\hat T^\dagger q\hat T=q+c)$. This was simple to show using commutators ...
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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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118 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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108 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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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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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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220 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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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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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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95 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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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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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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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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171 views

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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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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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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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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187 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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108 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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81 views

Does the success of canonical quantization guarantee the path integral works too?

If the canonical quantization approach to a field theory is successful, is it a good indicator that the path integral will work as well? Furthermore, can the success of a particular quantization ...
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67 views

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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Space and time translation of quantum field

In book I am studying, the authors want to calculate the spectral representation of the g-lesser (and g-greater) functions and find a general expression for these. In order to do so they start with ...
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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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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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188 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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104 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 ...
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Diagonalize a dot product with Pauli matrices

How can I diagonalize the following operator? $$\lambda \hat{\vec{\sigma}}\cdot\vec{r}$$ where $\lambda$ is a real constant, $\hat{\vec{\sigma}}=(\hat{\sigma_{x}},\hat{\sigma_{y}},\hat{\sigma_{z}}) ...
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Change of QM Momentum operator under coordinate transformation

Can any one please let me know what is the general procedure to construct the momentum operator under some coordinate transformation? For example, I understand that if ...
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100 views

What is the Weyl algebra of a confined bosonic particle?

The abstract Weyl Algebra $W_n$ is the *-algebra generated by a family of elements $U(u),V(v)$ with $u,v\in\mathbb{R}^n$ such that (Weyl relations) $$U(u)V(v)=V(v)U(u)e^{i u\cdot v}\ \ Commutation\ ...
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Is kinetic energy in QM a state-property or is it distributed?

Suppose we have a quantum mechanical system, which is well described by its wave function in r-representation $\Psi$. We are interested in the properties of an observable, say the kinetic energy $T$. ...
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Commutator of $L^2$ and $X^2$, $P^2$

In our quantum mechanics script, it states that $[L^2, X^2] = 0$ and $[L^2, P^2] = 0$, therefore for the following Hamiltonan $$H = \frac{P^2}{2m} + V(X^2)$$ it is that $[H, L^2] = 0$ therefore $H$ ...
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76 views

Harmonic oscillator

Let $|0\rangle,...$ be the states of the harmonic oscillator. Then a squeezed state was defined as $|\xi\rangle =S(\xi)|0\rangle $, where $S(\xi):=e^{\frac{1}{2}( \xi (a^{ \dagger ^2}-a^2))}$, where ...
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Do eigenvectors of quantum operators span the whole Hilbert Space?

I am trying to solve an exercise in Shankar's QM book (concretely 4.2.1), and I am asked the probability of each possible value for the operator $L_x$ when the particle is in a certain eigenstate of ...
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Evaluating commutator of $[\operatorname{sign}(X),\, \operatorname{sign}(P)]$

I wish to evaluate the following commutator: $[\operatorname{sign}(X),\, \operatorname{sign}(P)]$. Is there a general method for evaluating $[\operatorname{f}(X), \operatorname{f}(P)]$? I thought of a ...
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191 views

Quantum mechanics problem? [closed]

I had a test on Quantum mechanics a few days ago, and there was a problem which I had no clue how to solve. Could you please explain me? The problem is: Let's look at the $\hat H=E_0[|1 \rangle ...
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112 views

Time Dependent Position Operator

How does one find the time dependent position expectation value for a wave function? I thought we could simple take the time dependent wave and apply the position operator like normal, but this gave ...
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63 views

Commutators with function

I have following exercise: If $[C,D]$ is a c-number and $f(x)$ is a well-behaved function (i.e. all derivatives exist and are finite), show that: $$[C, f(D)]=[C,D]f'(D)$$ where $f'(D) = ...
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195 views

Are all scattering states un-normalizable?

I am an undergraduate studying quantum physics with the book of Griffiths. in 1-D problems, it said a free particle has un-normalizable states but normalizable states can be obtained by sum up the ...
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When should one apply the unitary time evolution operator?

When is it appropriate to use $\hat U$, the unitary time evolution operator? For example, say I had a system in a certain potential that is changed to a different one at time $t = 0$. Would it be ...