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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Weyl Ordering Rule

While studying Path Integrals in Quantum Mechanics I have found that [Srednicki: Eqn. no. 6.6] the quantum Hamiltonian $\hat{H}(\hat{P},\hat{Q})$ can be given in terms of the classical Hamiltonian ...
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Implications of unbounded operators in quantum mechanics

Quantum mechanical observables of a system are represented by self - adjoint operators in a separable complex Hilbert space $\mathcal{H}$. Now I understand a lot of operators ...
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Can eigenstates of a Hilbert space be thought of as delta functions?

Say we have an observable that describes a Hilbert space and that observable acts on state kets. Lets take the position observable for example. Then $\langle y|x\rangle = \delta(y - x)$. But can the ...
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What does it mean to apply an operator to a state?

Let's say I have an operator $\hat{A}$ and a state $|\psi\rangle$. What exactly is the state $\hat{A}|\psi\rangle$? Is it just another different state that I am describing using my $\hat{A}$ and ...
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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 ...
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What is the physical meaning of weak expectation values?

In the two-state formalism of Yakir Aharanov, the weak expectation value of an operator $A$ is $\frac{\langle \chi | A | \psi \rangle}{\langle \chi | \psi \rangle}$. This can have bizarre properties. ...
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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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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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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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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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Is the momentum operator diagonal in position representation?

The matrix elements of the momentum operator in position representation are: $$\langle x | \hat{p} | x' \rangle = -i \hbar \frac{\partial \delta(x-x')}{\partial x}$$ Does this imply that $\langle x ...
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What is $\hat{p}|x\rangle$?

Trying to solve a QM harmonic oscillator problem I found out I need to calculate $\hat{p}|x\rangle$, where $\hat{p}$ is the momentum operator and $|x\rangle$ is an eigenstate of the position operator, ...
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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} ...
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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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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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Operator norm of creation and annihilation operators

Are the creation and the annihilation operators $a(f)$ and $a^{\dagger}(f)$ for the bosonic Fock space bounded? What is their norm? So far I did not have found any note about this in the linked ...
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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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When you apply the spin operator, what exactly is does it tell you?

The example I'm trying to understand is: $ \hat{S}_{x} \begin{pmatrix} \frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}} \end{pmatrix} = 1/2 \begin{pmatrix} \frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}} ...
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The cleverest way to calculate $\left[\hat{a}^{M},\hat{a}^{\dagger N}\right]$ with $\left[\hat{a},\hat{a}^{\dagger}\right]=1$

Who can provide me some elegant solution for $$\left[\hat{a}^{M},\hat{a}^{\dagger N}\right]\qquad\text{with} \qquad\left[\hat{a},\hat{a}^{\dagger}\right]~=~1$$ other than brute force calculation? ...
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Some questions on observables in QM

1-In QM every observable is described mathematically by a linear Hermitian operator. Does that mean every Hermitian linear operator can represent an observable? 2-What are the criteria to say whether ...
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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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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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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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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 ...
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Basic Question - Green's Functions in Quantum Mechanics

I am trying to learn about Green's functions as part of my graduate studies and have a rather basic question about them: In my maths textbooks and a lot of places online, the basic Greens function G ...
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Simultaneously commuting set

How does one determine the members of an simultaneously commuting set (of operators)? For example, I have read that for orbital angular momentum, the set is {$H,L^2,L_z$}. How does one know that these ...
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How do we measure $i[\hat\phi(x),\hat\phi(y)]$ in QFT?

What operational procedure is required to measure $i[\hat\phi(x),\hat\phi(y)]$ in an interacting (or non-interacting) QFT? [assume smearing by test-functions, or give an answer in Fourier space, for ...
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How to evaluate spin operators in second quantization for spin symmetry-broken Slater determinants?

Suppose we have the following Slater determinant: \begin{equation} | \Psi \rangle = \prod \limits_{i,i'} a^+_{i\alpha} a^+_{i'\beta} | \rangle \end{equation} where $a^+_{i\alpha}$ creates an electron ...
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Eigenstate of field operator in QFT

Why don't people discuss the eigenstate of the field operator? For example, the real scalar field the field operator is Hermitian, so its eigenstate is an observable quantity.
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Why don't we use Hamilton-Jacobi method in QM?

In classical mechanics, we usually try to find a set of coordinates by Hamilton-Jacobi method to transform the Hamiltonian to zero such that the coordinates are conservations. However, we never try ...
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Constructing the space of quantum states

I want to learn how to construct spaces of quantum states of systems. As an exercize, I tried to build the space of states and to find hamiltonian spectrum of the quantum system whose Hamiltonian is ...
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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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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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Complex conjugate of momentum operator

Consider momentum operator representation in position space. $$\hat{p}=-i\frac{\partial}{\partial x} \,\ \text{and its eigen functions are } e^{ipx} \,\text{and} \,\ e^{-ipx}.$$ ...
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How does the momentum operator act on state kets?

I have been going through some problems in Sakurai's Modern QM and at one point have to calculate $\langle \alpha|\hat{p}|\alpha\rangle$ where all we know about the state $|\alpha\rangle$ is that ...
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Identify the coefficients of Operator Product Expansion (OPE)

Sorry I have a stupid question in Polchinski's string theory book vol 1, p46. For a holomorphic function $T(z)$ with a general operator $\mathcal{A}$, there is a Laurent expansion $$T(z) A(0,0) \sim ...
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How can the product of two real linear operators be not real?

I'm puzzled about a statement from Diracs 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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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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Heisenberg picture of QM as a result of Hamilton formalism

Let's have formula of full time-derivative of physical value in Poisson's formalism: $$\tag{1} \frac{df}{dt} = -[H, f]_{P. br.} + \frac{\partial f}{\partial t}, $$ where $[A, B]_{P. br.}$ is Poisson's ...
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Proof of Canonical Commutation Relation (CCR)

I am not sure how $QP-PQ =i\hbar$ where $P$ represent momentum and $Q$ represent position. $Q$ and $P$ are matrices. The question would be, how can $Q$ and $P$ be formulated as a matrix? Also, what is ...
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Unitary Operator as a complex valued function

A book on Quantum Mechanics by Schwinger states, "A unitary operator can be considered to be a complex valued function of a Hermitian operator." Please give a hint on how to prove this assertion.
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How to calculate the quantum expectation of frequency of a particle?

I know how to calculate the expectation of < $\Psi$|A|$\Psi$ > where the operator A is the eigenfunction of energy, momentum or position, but I'm not sure how to perform this for a pure frequency. ...
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Quantum Mechanics: Creation and Annihilation operators

Is an eigenvector/eigenstate of the creation operator an eigenvector/eigenstate of the annihilation operator too? Why?
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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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Evaluation of expectation values

I will denote operators with hats. Suppose we got an operator of the form $i[\hat p, \tan^{-1}(e^{\hat x})]$ and we want to calculate the amplitude for a transition from a state $|p_i\rangle$ to the ...
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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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An alternative definition of the creation and annihilation operators?

Suppose we have a system of bosons represented by their occupation numbers $$\tag{1} | n_1, n_2, ..., n_\alpha, ... \rangle$$ Then we can define creation and annihilation operators $$\tag{2} ...
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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} &= ...
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How to prove that the symmetrisation Operator is hermitian?

Let $\mathcal{H}_N$ be the $N$ particle Hilbert space. So a quantum state $\left| \Psi \right>$ may be representated by $$\left| \Psi \right> = \left| k_1 \right>^{(1)}\left| k_2 ...
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Index Manipulation and Angular Momentum Commutator Relations

I have been trying for hours and cannot figure it out. I am not asking anyone to do it for me, but to understand how to proceed. We have the relations $$[L_i,p_j] ~=~ i\hbar\; \epsilon_{ijk}p_k,$$ ...