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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Is there a formalism for talking about diagonality/commutativity of operators with respect to an overcomplete basis?

Consider a density matrix of a free particle in non-relativistic quantum mechanics. Nice, quasi-classical particles will be well-approximated by a wavepacket or a mixture of wavepackets. The ...
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Why are holomorphic boundary CFT2 primary operators massless in the AdS3 bulk?

I saw a claim in this paper that holomorphic boundary CFT$_2$ primary operators correspond to massless states in the AdS$_3$ bulk. Specifically, As always, we simplify the situation by assuming ...
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158 views

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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How to replace $T$-product with retarded commutator in LSZ formula?

I am reading Itzykson and Zuber's Quantum Field Theory book, and am unable to understand a step that is made on page 246: Here, they consider the elastic scattering of particle $A$ off particle $B$: ...
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Prove the solution of von Neumann equation will never stabilize if Hamiltonian and initial density matrix commutes

Given von Neumann equation $$\frac{d}{dt} \rho(t) = -i [H, \rho(t)] = -i e^{-iHt}[H, \rho(0)]e^{iHt}.$$ If we know that $[H, \rho(0)] \neq 0$, how do we prove in details the solution of von Neumann ...
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Is there is a specific original source where the “quantum operator ordering issue” is stated?

During my research, when the quantum operator ordering ambiguity is mentioned is deemed usually in the likes of "the well-known problem of ordering in quantum mechanics". However, could anybody point ...
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Measuring position and momentum at the same time?

In a non-relativistic quantum mechanical system in an infinite potential well. I try to measure the energy and the position of the system simultaneously. Since, the respective operators do commute ...
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156 views

Hilbert space for Density Operators (instead of Banach spaces)

Is it possible to construct a well defined inner-product (and therefore orthonormality) within the set of self-adjoint trace-class linear operators? In the affirmative case, dynamics could be analyzed ...
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212 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 ...
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Normal ordering

If I understood correctly there are two terms called normal ordering: $:c c^\dagger: = c^\dagger c \hspace{.5cm}$so shifting all creation operators to the left and all annihilation operators to the ...
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57 views

How to write QM operator if I know all of it's eigenfunctions?

Suppose I have selected enough orthogonal functions in representation of operator A and I want to derive operator B which has ...
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What does QM observable operator actually DO?

I know, that each observable is represented by some linear operator, while it's eigenvalues represent probability amplitudes of possible observable values. But what if I ACT with this operator on ...
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How to write “postselection” operator?

Suppose, I wish to know an operator, which eigenvalue is 1 if state is exactly F and 0 ...
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Is mass an observable in Quantum Mechanics?

One of the postulates of QM mechanics is that any observable is described mathematically by a hermitian linear operator. I suppose that an observable means a quantity that can be measured. The mass ...
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Symmetry and Algebra

I'm trying to get a more concrete idea how symmetry is understood in quantum theories, as broad as possible. Consider a infinitesimal transformation of states in quantum physics of the form: $$ ...
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96 views

Lippmann-Schwinger solution

What's wrong with this general solution of the Lippmann-Schwinger equation: $$ |\psi_k \rangle=|\phi_k \rangle+G_k V|\psi_k \rangle $$ Taking the inner product with $\langle\phi_{k'}|$ \begin{align} ...
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Differences between probability density and expectation value of position

The expression $\int | \Psi\left(x\right)|^2dx$ gives the probability of finding a particle at a given position. If wave function gives the probabilities of positions, why do we calculate ...
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Eigenstates of a shifted harmonic oscillator

Let's say I have a quantum harmonic oscillator $H = \omega a^\dagger a$, where $a^\dagger$ is the raising operator and $a$ is the lowering operator and $H |n\rangle = \omega n |n\rangle$. Now assume ...
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Unitary transformation behind gauge transformation

It is very well-known that for bosonic operators a Gauge transformation can always be associated with it $$a\rightarrow e^{i\phi}a.$$ Obviously this is a Unitary transformation. Something like ...
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Ladder operator notation?

If I have the ladder operator $D(AB)$ which is a lowering operator, where $D(AB)=D(A)+D(B)$ does the things in the brackets e.g. AB indicate what wave function this operator applies to, i.e. is the ...
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Does the order of variables matter for a quantum Lagrangian in the path integral formula for quantum mechanics? [duplicate]

For a single particle or field, I can't see how the path-integral formulation depends on the order of terms in the Lagrangian. It seems that you integrate the classical Lagrangian to get the action on ...
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145 views

Angular momentum - maximum and minimum values for $m_{\ell}$

I want to work out the maximum and minimum values for $m_{\ell}$. I know that $\lambda \geq m_{\ell}$, therefore $m_{\ell}$ is bounded. In the lectures notes there is the following assumption: $$ ...
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52 views

Differentiation operator with respect to observable acting as a function of the observable?

In his Principles of Quantum Mechanics Dirac writes: $$\int \langle \phi \frac{d}{dq}|q'\rangle dq' \psi(q')=\int \phi(q') dq' \frac{d\psi(q')}{dq'}.$$ To me it is rather strange, and it seems as if ...
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Commutator with Pauli spin matrices and the momentum operator

How is $\left[\vec\sigma \cdot \vec p, \vec \sigma \right]$ proportional to $\vec \sigma\times \vec p$, where $\sigma$ are the Pauli spin matrices and $p$ is the momentum operator?
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Can we correctly define momentum operator only by means of position operator and their commutation relation?

In "J.M. Ziman. Electrons and Phonons: The Theory of Transport Phenomena in Solids" the author formally introduces the position (displacement) operator and then defines the momentum operator with the ...
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Proof for a time-ordering equation in Negele & Orland (1998)

Let $T$ be the time-ordering operator which orders operators $A_1(t_1), A_2(t_2), \ldots$ such that the time parameter decreases from left to right: $$T[A_1(t_1) A_2(t_2)] = A_2(t_2) A_1(t_1) \text{ ...
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Observables in Quantum Mechanics

Studying on own quantum mechanics I came across: Preceeding text: A basic postulate of quantum mechanics tells us how to set up the operator corresponding to a given observable. Observables, ...
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Is quantum field operator $\psi$ same as quantum field $\psi$?

So in QFT, quantum field operator $\psi$ is there. $\psi$ seems to take the role of wavefunction in QM, which now acts upon vacuum state. Then, in lagrangian of various quantum field theories, $\psi$ ...
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Dilation operator in CFT viewed as 'hamiltonian'?

From the commutation relations for the conformal Lie algebra, we may infer that the dilation operator plays the same role as the Hamiltonian in CFTs. The appropriate commutation relations are ...
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Self-adjointness

I know I have posted this question before some time ago. But no one could help so I decided to put my problem in another background. The Schrödinger equation of a free scalar field is given by ...
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59 views

How to get the time derivative of an expectation value in quantum mechanics?

The textbook computes the time derivative of an expectation value as follows: $$\frac{d}{dt}\langle Q\rangle=\frac{d}{dt}\langle \Psi|\hat Q\Psi\rangle=\langle \frac{\partial\Psi}{\partial t}|\hat ...
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Time-ordering in QFT

In Srednicki QFT page 37. In the derivation of LSZ reduction formula, he introduces the time-order operator $T$, so no time-dependent creation/annihilation operators are left in the transition ...
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Angular momentum for 3D harmonic oscillator in two different bases

I know that the energy eigenstates of the 3D quantum harmonic oscillator can be characterized by three quantum numbers: $$ | n_1,n_2,n_3\rangle$$ or, if solved in the spherical coordinate system: ...
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About the definition of expectation value in quantum mechanics

In quantum mechanics, the expectation value of a observable $A$ is defined as $$\int\Psi^*\hat A\Psi$$ But in probability theory the expectation is a property of a random variable, with respect to a ...
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Are operators in quantum mechanics linear transformations?

Observables in quantum mechanics correspond to self-adjoint linear operators. If $\psi$ is an eigenvector of $\hat A$, then $\hat A\psi=\alpha\psi$ where $\alpha$ is the eigenvalue of $\psi$. ...
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Proof for the completeness of eigenfunctions of a self-adjoint operator

I always heard the eigenfunctions of a self-adjoint operator form a complete basis. Where can I find a proof in infinite dimension space? Presumably readable for physicists.
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The Eigenstate Existence Problem in Dirac's 'Principles of Quantum Mechanics'

In Chapter II of Dirac's Principles of Quantum Mechanics, Dirac explains that in general it is very difficult to know whether, for a given real linear operator, that any eigenvalues/eigenvectors exist ...
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What is the analogy of $|x\rangle$ in quantum field theory?

Let me start from path integral formulation in quantum mechanics and quantum field theory. In QM, we have $$ U(x_b,x_a;T) = \langle x_b | U(T) |x_a \rangle= \int \mathcal{D}q e^{iS} \tag{1} $$ ...
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51 views

What are the proper domains of the position and squared angular momentum operator?

I am looking at the position operator on a compact set $K \subset \mathbb{R}^n$ and the squared angular momentum operator (so essentially the Laplace-Beltrami operator where I just look at the angular ...
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Is the potential in Schrödinger equation an operator?

In the Schrödinger equation in the position representation $$ i\hbar\frac{\partial}{\partial t}\Psi(x,t) ~=~[\frac{-\hbar^2}{2m}\nabla^2+V(x,t)]\Psi(x,t), $$ is the potential $V(x,t)$ an operator ...
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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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305 views

Basis in quantum mechanics

My quantum mechanics textbook (Primer of Quantum Mechanics, by Marvin Chester) says that both the momentum space and the position space are basis spaces. It also says that the momentum space is ...
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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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Question about equation 2.27 from Pachos's Introduction to topological quantum computing

http://quince.leeds.ac.uk/~phyjkp/Files/IntroTQC.pdf above is the PDF that is hosted on his website. The equation is on page 22 (pg 30 in the pdf). In chapter 2. It is the second equation of the ...
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171 views

Position Representation in Quantum Mechanics

How does the 3d position operator look like in position representation? I know that in 1d the position operator $\hat{x}$ is just multiplication by $x$.
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63 views

Eigenstates of an observable

Can we use eigenstates of ANY observable as base of the Hilbert space? If we can, is this equal to the statement that those eigenstates are orthogonal to each other and normalizable?
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Why does the probability of obtaining a value of a measurement follow from Dirac's general assumption?

In Dirac's The Principle of Quantum Mechanics he makes the general assumption that "if the measurement of the observable $\xi$ for the system in the state corresponding to $|x\rangle$ is made a large ...
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Quantum Mechanics: Time dependence of an expectation value

In Griffiths's Introduction to Quantum Mechanics, he says that the time dependence of an expectation value is $$\frac{d\langle Q\rangle}{dt}=\frac{i}{\hbar}\langle [H,Q]\rangle+\langle \frac{\partial ...
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Operator Product Expansion in Massless 2D QED

In Peskin & Schroeder chapter 19 page 656, where the axial current anomaly of massless 2D QED is discussed, the authors go from: $$ ...
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Explanation of Dirac's proof of arbitrary ket being expressible with eigenkets of observable

In P.A.M. Dirac's The Principles of Quantum Mechanics, Chapter 10 (Observables), pp. 40, at the end of the chapter there is a proof that I don't understand at all. Here is a pdf link to the book ...