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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16 views

Does the abstract wavefunction change in this following example?

Suppose, we have a basis $|u\rangle$, described by the function $u=g(x)$. We can normalize this basis, using our standard $|x\rangle$ basis using the following : $$\hat{I}=\int |x\rangle\langle x|dx=\...
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Form of the interaction representation time evolution operator

I am a bit confused about the interaction representation picture. Consider the time independent Hamiltonian $H = H_0 + V$. My question concerns the interaction representation time evolution operator: $...
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34 views

Are the Orbital Angular Momentum Operators Linear?

I'm a bit confused whether or not I can distribute out the operators in the definition of the raising and lowering operators. For example, given: $$L_{+}|l \space m\rangle = a |l\space m+1\rangle$$ $$...
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Translation operator and position operator

I am a little bit confused about the translation and position operator and hope for some clarification. Let $\hat{x}$ be the position operator, which satisfies $\hat{x} \vert x \rangle = x \vert x \...
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Two-particle operators in QFT and the factor 1/2

I am learning about QFT through the book Quantum Field Theory for the Gifted Amateur and I am having trouble understanding the factor 1/2 in the definition of two particle field operators. In the book ...
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Wick contraction for fermions of different types

When we have an interaction hamiltonian with fermion field of two different types ($\Psi_a$ $\overline{\Psi_b}$) and a scalar field $\Phi_c$. How does one do the wick contraction? eg. Leptoquark decay ...
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59 views

Inserting a position operator in the path integral in QFT

With the usual path integral description, we have the formula $$\langle q''t''|q't'\rangle =\int\mathcal{D}q \exp{(iS)}$$ where $S=\int_{t'}^{t''}L(q,\dot{q})$ is the action evaluated for $t\in (t',t''...
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Prove that $i\hbar\dot{\hat{\Omega}}=[\hat{\Omega},\hat{H}]$ , where $\hat{\Omega}(\hat{x},\hat{p})$ is Taylor expandable

How do I show that $i\hbar\dot{\hat{\Omega}}=[\hat{\Omega},\hat{H}]$ , where $\hat{\Omega}(\hat{x},\hat{p})$ is Taylor expandable, $\hat{H}(\hat{x},\hat{p})$ is the Hamiltonian of the system and $\hat{...
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69 views

Is there a (semiclassical) electric field operator?

So I come from a chemistry background, where the electronic structure of atoms and molecules is central. For practical purposes, we usually work with a charge density operator $$ \hat{\rho}(r) = q \...
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I need help with operators [closed]

So, my main question is just simply, what are physics operators and what are the most common ones?
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26 views

Ehrenfest theorem: On which classical circle can we find the electrons in an homogenous magnetic field?

In the French wiki article about the Ehrenfest theorem I found these formulas. $${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\langle {\hat {x}}\rangle ={\frac {1}{m}}\langle {\hat {p}}\rangle }...
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Translation matrix for multiple spins

I need to do a translation, but not a translation in the classical definition, I need to, in a system with a operator, for example, $$ \sigma^z_j=1\otimes \cdots \otimes 1\otimes \sigma^z\otimes 1 \...
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Ladder Operators and Angular momentum

The angular momentum operator $L_z$ can be expressed in terms of $L_z = xp_y - yp_x$ where $ p = \hat{p} = -i\sqrt{hwm/2}(a^{\dagger} - a)$ and $x = \hat{x}\sqrt{\hbar/2mw}(a^{\dagger} + a)$. We want ...
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Expectation values in path integral formalism

In quantum field theory, it is often assumed that the expectation value $\langle A\rangle$ of an operator $A$ can be written in the path integral formalism in the following way: $$ \langle A\rangle = \...
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Many-body Bose-Hubbard Interaction Energy

I am trying to understand the derivation in this link. They start with a slightly modified Hamiltonian (with a gauge field) (I call $H_j$ the tunneling, since my question is regarding the interaction ...
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How can you calculate a reduced matrix element $\langle 0|\varepsilon|0\rangle $? [closed]

How can you calculate a reduced matrix element $\langle 0|\varepsilon|0\rangle $ ? $\varepsilon$ is a polarization vector. I also want to know the lists on the webpage that says some reduced matrix ...
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What is the actual use of Hilbert spaces in quantum mechanics?

I'm slowly learning the quirks of quantum mechanics. One thing tripping me up is... while (I think) I grasp the concept, most texts and sources speak of how Hilbert spaces/linear algebra are so useful ...
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Is there a well-defined association between abstract linear operators in Fock space and normal ordered polynomials of fermionic operators?

Suppose I have a fermionic Fock space $H$ of dimension $2^n$. If I fix an operator $O$ acting on $H$ that commutes with the number operator $N$, I typically make an assumption internally that such an ...
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42 views

An identity about exponential of operators

Suppose we have a finite set $\mathcal{S}$ and operators (actually, matrices): $$b_{x-\frac{1}{2},x} = a_{x} + a_{x}^{\dagger} \quad \mbox{and} \quad b_{x,x+\frac{1}{2}} = i(a_{x}-a^{\dagger}_{x})$$ ...
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Expectation value of position operator for driven quantum harmonic oscillator

I'm having quantum harmonic oscillator which is in ground state at $t=0$. Now it is subjected to arbitrary force $F(t)$. How can I calculate $\langle \hat{x} \rangle$ and $\langle \hat{p} \rangle$? I ...
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Rotation of spin operator itself

Consider for example the rotation of the $x$ component of spin by $\pi$ about the $z$ axis. This flips the spin giving $$e^{i\pi S_z}S_xe^{-i\pi S_z}=-S_x.$$ However, can this be proved directly using ...
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Is the momentum operator in position basis a first-order approximation?

In my course we have just derived the momentum operator in the position basis to be the first-order derivative of the position. This was achieved by using a first-order taylor expansion: $$ \psi(\...
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Mathematical expression of the coordinate exchange operator

So I know that the total exchange operator is the product of the coordinate, spin and isospin exchange operators, as, \begin{equation} P_{12} = P_{12}^r P_{12}^\sigma P_{12}^\tau \end{equation} The ...
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32 views

Parity transformation on the adjoint spinor

In eq. (3.128) of page 66 of "An Introduction to QFT", by Peskin & Schroeder, a step involves: $$P\,\overline{\psi}\left(t,\textbf{x}\right)P^{-1}=P\,\psi^{\dagger}\left(t,\textbf{x}\...
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When is the Liouvillian superoperator not diagonalisable?

An $n \times n$ matrix, $L$, is diagonalisable if it has $n$ linearly independent eigenvectors. I've recently been working with open quantum systems and come across the non-hermitian, Liouvillian ...
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64 views

Definition of invariance of a QM operator under a transformation

In the Sakurai book "Modern quantum mechanics" (pg. 263) an operator $S$ is said to be invariant under a unitary transformation $T$ if: $$T^\dagger S T = S.$$ Where that definition come from?...
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Quadratic Expansion With Operators [duplicate]

I was looking at the hamiltonian of a particle confined to the $x$-$y$ plane when it has mass $m$ and charge $q$ coupled to the electromagnetic field. My question is actually a very simple one. During ...
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Lorentz transformation of operator

I am reading the Zee book on group and at pg.442 when explaining the Poincare' algebra an the infinity dimensional representation, it states that since $P_{\mu}P^{\mu}$ is a Lorentz invariant then it ...
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Find operator that produce a given amplitude

While I was studying spinor helicity formalism, in many articles, it gives the operators that produce a given amplitude. Is there a generic method of finding operators that produce a given amplitude? ...
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56 views

Can the Hermitian operator be related to state space to describe physical phenomena?

Can space-time, in which phenomena occur, and the space of states in which phenomena are described by means of the Hermitian operator be related? I suspect it is because the hermetic operator is built ...
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Sign of the momentum operator in QM [duplicate]

What is the physical reason or concept underlying the minus sign in the definition of the momentum operator in QM: $$\hat{p}_i = -i \hbar\frac{\partial}{\partial x_i}~? $$
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Why $⟨a│H'│a⟩=0$, when $H'$ has zero diagonal components?

On 'David J.Griffiths, Introduction to Quantum Mechanics' Chapter 11.1.1 equations (11.16) When $H'$ has zero diagonal component, $⟨a│H'│a⟩=0$ However, I think that there is needed conditon '...
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Proof of the Glauber relation $e^{A}e^{B} = e^{A+B}e^{(1/2)[A,B]}$ [duplicate]

If $A$ and $B$ are two operators that commute with their commutators, \begin{align} e^{A}e^{B} &= e^{A+B}e^{(1/2)[A,B]} \end{align} That well-known statement is known as Glauber's Formula. I am ...
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46 views

Heisenberg Equation for Raising and Lowering Operators of the Harmonic Oscillator

For a Schrödinger operator $O$ that is independent of time, the Heisenberg equation is $$\dot O_H(t) = i[H, O_H(t)], \tag{1}$$ with solution given by $$O_H(t) = e^{iHt}O_S e^{-iHt}. \tag{2}$$ I want ...
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76 views

Commutation between hermitian operator $H$ and the operator $U(m,n) = |\varphi_m\rangle \langle \varphi_n|$

I was trying to do this exercise from Cohen-Tannoudji Quantum Mechanics book: $|\varphi_n\rangle$ are the eigenstates of a Hermitian operator $H$ ( $H$ is, for example, the Hamiltonian of some ...
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What does cohomology of $Q_B$ mean in BRST quantization in Polchinski?

While proving no-ghost theorem ($4.4$ Polchinski) the term cohomology of $Q_B$ is used quite a lot of time. From what I understand this has to be a set since "cohomology of $Q_B$" is ...
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Do fermionic creation/annihilation anticommutation relations fix the creation and annihilation operators?

If you define operators $a, a^\dagger$ which satisfy (e.g.) the relations $\{a,a\}=\{a^\dagger,a^\dagger\}=0$ and $\{a,a^\dagger\}=1$. Will this uniquely define the operators such that $a |0\rangle \...
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53 views

Intuition for Spin operator in arbitrary direction

I understand why the Spin operators in $x$, $y$ and $z$ direction are given by : $\begin{align*} S_x = \begin{pmatrix} 0 &\hbar/2\\ \hbar/2 & 0 \end{pmatrix} S_y = \begin{pmatrix} 0 & -i\...
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Derivation of $T(z)(TT)(w)$ in CFT

I am trying to derive eq. (6.213) in Di Francesco's CFT book, $$T(z)(TT)(w) \sim \frac{3c}{(z-w)^6}+\frac{(8+c)T(w)}{(z-w)^4}+\frac{3 \partial T(w)}{(z-w)^3}+\frac{4 (TT)(w)}{(z-w)^2}+\frac{\partial(...
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1answer
64 views

Confusion about interpretation of expectation values in quantum mechanics

Given a state $|\psi \rangle$ one can form the expectation value of an observable $O$ as: $$ \langle \psi|O|\psi \rangle. $$ For the case $O = H$, where $H$ is the Hamiltonian of the quantum system, ...
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Compatible observables for a quantum harmonic oscillator 3D

In quantum mechanics, from what I understand from the theory, if any operators commute then they have a complete set of simultaneous eigenvectors. This means that any vector of space can be expressed ...
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Confusion about in and out states, interacting Hilbert space etc, referring to Weinberg QFT

There are many posts related to this issue on this site, but I have found none that answer my specific questions about this matter. I review my understanding of Weinbergs approach. There are probably ...
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First quantization Entropy of coherent state

I'm trying to follow the computations of example 5.1 in this paper. To begin with they have a symplectic Hilber space $(\mathcal{K},\tau,\sigma)$, where $(\mathcal{K},\tau)$ is a separable Hilbert ...
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Kinetic Energy of wavefunction

Suppose we have been provided the form of some wavefunction on a graph, but not the exact mathematical expression of the wavefunction $\langle x|\psi\rangle $. Now I'm asked to find the average ...
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How does commutator of $Q_B$ with change in $H$ results in moving around in gauge space

In ch-$4$ Polchinksi states following: In order to move around in space of gauge choices, the BRST charge must remain conserved. Thus it must commute with change in the Hamiltonian. Commutation with ...
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1answer
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Counting sign definiteness in BRST cohomology of string

In Polchinksi ch-$4$ following manipulation is done: $$|\psi_1\rangle=(e\cdot\alpha_{-1}+\beta b_{-1}+\gamma c_{-1})|0,\textbf{k}\rangle .\tag{4.3.25}$$ $$\langle\psi_1|\psi_1\rangle=\Big(e^*\cdot e+(\...
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What's the (intuitive) difference between a primary and a quasi-primary operator in a CFT?

I've come across many different definitions of primaries and quasi-primaries. Some references define them based on their transformation law \begin{align}\hat\phi'(x')=\left|\frac{\partial x'}{\partial ...
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82 views

Is this identity true for operators?

We have the classical identiy: $$ \boldsymbol{L}\cdot\left(\boldsymbol{p}\times\boldsymbol{L}\right) = 0$$ But I was wondering if that is also right in QM, considering $L$ and $P$ do not generally ...
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123 views

Shared eigenstates of angular momentum

The angular momentum operators $L_x$, $L_y$ and $L_z$ don't commute with each other but they do all commute with the operator $L^2 = L_x^2+L_y^2+L_z^2$. I know that if two matrices $A$ and $B$ commute ...
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74 views

Commutators in Poincare algebra

Consider the method of induced representations for the Poincare algebra, i.e. given a field $\phi$ (which need not be a scalar field despite its notation), we have the commutator $$[J^{\mu\nu},\phi(0)]...

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