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 does the Canonical Commutation Relation (CCR) tell me about the overlap between Position and Momentum bases?

I'm curious whether I can find the overlap $\langle q | p \rangle$ knowing only the following: $|q\rangle$ is an eigenvector of an operator $Q$ with eigenvalue $q$. $|p\rangle$ is an eigenvector of ...
14
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2answers
2k views

Time as a Hermitian operator in QM?

In non-relativistic QM, on one hand we have the following relations: $$\langle x | P | \psi \rangle ~=~ -i \hbar \frac{\partial}{\partial x} \psi(x),$$ $$\langle p | X | \psi \rangle ~=~ i \hbar ...
4
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2answers
251 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 ...
3
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2answers
990 views

Hermitian operator and reality of eigenvalues

Prove or disprove: The eigenvalues of an operator are all real if and only if the operator is hermitian. I know the proof in one way; that is, I know how to prove that if the operator is hermitian, ...
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4answers
5k views

Mathematical background for Quantum Mechanics [duplicate]

What are some good sources to learn the mathematical background of Quantum Mechanics? I am talking functional analysis, operator theory etc etc...
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2answers
455 views

How to prove that $i\hbar\frac{\partial}{\partial \mathbf{p}}$ is the operator of $\mathbf{x}$ in momentum space?

How to prove that $i\hbar\frac{\partial}{\partial \mathbf{p}}$ is the operator of $\mathbf{x}$ in momentum space?
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3answers
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How to tackle 'dot' product for spin matrices

I read a textbook today on quantum mechanics regarding the Pauli spin matrices for two particles, it gives the Hamiltonian as $$ H = \alpha[\sigma_z^1 + \sigma_z^2] + ...
3
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1answer
1k views

Why/How is this Wick's theorem?

Let $\phi$ be a scalar field and then I see the following expression for the square of the normal ordered version of $\phi^2(x)$. $$T(:\phi^2(x)::\phi^2(0):) ~=~ 2<0|T(\phi(x)\phi(0))|0>^2 $$ ...
1
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2answers
789 views

Prove $[A,B^n] = nB^{n-1}[A,B]$

I am trying to show that $[A,B^n] = nB^{n-1}[A,B]$ where A and B are two Hermitian operators that commute with their commutator. However, I am running into a little problem and would like a hint of ...
26
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3answers
818 views

A “Hermitian” operator with imaginary eigenvalues

Let $${\bf H}=\hat{x}^3\hat{p}+\hat{p}\hat{x}^3$$ where $\hat{p}=-id/dx$. Clearly ${\bf H}^{\dagger}={\bf H}$, because ${\bf H}={\bf T} + {\bf T}^{\dagger}$, where ${\bf T}=\hat{x}^3\hat{p}$. In this ...
15
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1answer
597 views

Discreteness of set of energy eigenvalues

Given some potential $V$, we have the eigenvalue problem $$ -\frac{\hbar^2}{2m}\Delta \psi + V\psi = E\psi $$ with the boundary condition $$ \lim_{|x|\rightarrow \infty} \psi(x) = 0 $$ If we ...
8
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564 views

How does one determine ladder operators systematically?

In textbooks, the ladder operators are always defined," and shown to 'raise' the state of a system, but they are never actually derived. Does one find them simply by trial and error? Or is there a ...
6
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2answers
1k views

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 ...
11
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2answers
2k views

Meaning of the anti-commutator term in the uncertainty principle

What is the meaning, mathematical or physical, of the anti-commutator term? $\langle ( \Delta A )^{2} \rangle \langle ( \Delta B )^{2} \rangle \geq \dfrac{1}{4} \vert \langle [ A,B ] \rangle \vert^{2} ...
3
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1answer
489 views

Why are eigenfunctions which correspond to discrete/continuous eigenvalue spectra guaranteed to be normalizable/non-normalizable?

These facts are taken for granted in a QM text I read. The purportedly guaranteed non-normalizability of eigenfunctions which correspond to a continuous eigenvalue spectrum is only partly justified by ...
10
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5answers
12k views

What is the Physical Meaning of Commutation of Two Operators?

I understand the mathematics of commutation relations and anti-commutation relations, but what does it physically mean for an observable (self-adjoint operator) to commute with another observable ...
8
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1answer
388 views

Operator Ordering Ambiguities

I have been told that $$[\hat x^2,\hat p^2]=2i\hbar (\hat x\hat p+\hat p\hat x)$$ illustrates operator ordering ambiguity. What does that mean? I tried googling but to no avail.
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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 ...
10
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2answers
675 views

Does every hermitian operator represent a measurable quantity?

In Quantum mechanics, observables are represented by hermitian operator. But does every hermitian operator represent a observable? If not , how do we know that whether a hermitian operator represent ...
6
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1answer
333 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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3answers
2k views

Matrix elements of momentum operator in position representation

I have two related questions on the representation of the momentum operator in the position basis. The action of the momentum operator on a wave function is to derive it: $$\hat{p} ...
14
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4answers
1k views

Applications of the Spectral Theorem to Quantum Mechanics

I'm currently learning some basic functional analysis. Yesterday I arrived at the spectral theorem of self-adjoint operators. I've heard that this theorem has lots of applications in Quantum ...
15
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1answer
731 views

Intuitive meaning of Hilbert Space formalism

I am totally confused about the Hilbert Space formalism of Quantum Mechanics. Can somebody please elaborate on the following points: The observables are given by self-adjoint operators on the ...
14
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6answers
1k views

Why are Only Real Things Measurable?

Why can't we measure imaginary numbers? I mean, we can take the projection of a complex wave to be the "viewable" part, so why are imaginary numbers given this immeasurable descriptor? Namely with ...
16
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1answer
921 views

Is the Uncertainty Principle valid for information about the past?

My layman understanding of the Uncertainty Principle is that you can't determine the both the position and momentum of a particle at the same point in time, because measuring one variable changes the ...
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3answers
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Why do we use operators in quantum mechanics?

In classical mechanics, physical quantities, such as, e.g. the coordinates of position, velocity, momentum, energy, etc, are real numbers, but in quantum mechanics they become operators. Why is this ...
9
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1answer
492 views

How are anyons possible?

If $|ψ\rangle$ is the state of a system of two indistinguishable particles, then we have an exchange operator P which switches the states of the two particles. Since the two particles are ...
7
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1answer
139 views

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 ...
6
votes
2answers
609 views

Eigenvalues of a quantum field?

Fields in classical mechanics are observables. For example, I can measure the value of the electric field at some (x,t). In quantum field theory, the classical field is promoted to an operator-valued ...
3
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6answers
1k views

Is H=H* sloppy notation or really just incorrect, for Hermitian operators?

I saw it in this pdf, where they state that $P=P^\dagger$ and thus $P$ is hermitian. I find this notation confusing, because an operator A is Hermitian if $\langle \Psi | A \Psi \rangle=\langle A ...
8
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1answer
819 views

“An operator is hermitian”. Implications?

Alastair Rae states that there are 4 postulates of Quantum Mechanics in his text on the subject matter. The first part of his second postulate can be stated as: Every dynamical variable may be ...
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0answers
158 views

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( ...
4
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2answers
336 views

What is the most general expression for the coordinate representation of momentum operator?

I have a question about deriving the coordinate representation of momentum operator from the commutation relation, $[x,p]= i$. One derivation (ref W. Greiner's Quantum Mechanics: An Introduction, 4th ...
4
votes
1answer
205 views

Linearizing Quantum Operators

I was reading an article on harmonic generation and came across the following way of decomposing the photon field operator. $$ \hat{A}={\langle}\hat{A}{\rangle}I+ \Delta\hat{a}$$ The right hand side ...
2
votes
2answers
992 views

Derivative of the product of operators

I'm asked to show that $\frac{d(\hat{A}\hat{B})}{d\lambda} = \frac{d\hat{A}}{d\lambda}\hat{B} + \hat{A}\frac{d\hat{b}}{d\lambda}$ With $\lambda$ a continuous parameter Should I use the definition ...
5
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3answers
241 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 ...
4
votes
1answer
289 views

Wick Order and Radial Ordering in CFT

I am not so much familiar with the computations tools of conformal field theory, and I just run into an exercise asking to demonstrate the following formula (related to the bosonic field case): ...
3
votes
1answer
151 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 ...
2
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2answers
75 views

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 ...
0
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1answer
938 views

Rotation matrix is always a unitary operator

Can someone explain why the rotation matrix is a unitary, specifically orthogonal, operator?
4
votes
2answers
2k views

Derivatives of operators

How do derivatives of operators work? Do they act on the terms in the derivative or do they just get "added to the tail"? Is there a conceptual way to understand this? For example: say you had the ...
1
vote
3answers
251 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
1k views

Commutator of Momentum with a Position dependent function

I heard from my GSI that the commutator of momentum with a position dependent quantity is always $-i\hbar$ times the derivative of the position dependent quantity. Can someone point me towards a ...
1
vote
3answers
425 views

Simple Quantum Mechanics Question about The Commutator of Translation Operators

Say there is $\hat{J} = \exp[-i \hat{p} l/ \hbar]$ and $\hat{U}= \exp[-i\hat{H}t/ \hbar]$, where $\hat{H}$ is time-independent. Can we say anything about $[\hat{J},\hat{U}]$? Is it zero? How do we ...
35
votes
2answers
488 views

Physical interpretation of different selfadjoint extensions

Given a symmetric (densely defined) operator in a Hilbert space, there might be quite a lot of selfadjoint extensions to it. This might be the case for a Schrödinger operator with a "bad" potential. ...
15
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4answers
620 views

Density matrix formalism

The density matrix $\hat{\rho}$ is often introduced in textbooks as a mathematical convenience that allows us to describe quantum systems in which there is some level of missing information. ...
11
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4answers
408 views

Applying an operator to a function vs. a (ket) vector

I have a question regarding the effect of quantum mechanical operators. The definition that I'm familiar with says that an operator $A$ acts on a vector from a Hilbert space, $|\psi\rangle$, and the ...
8
votes
1answer
346 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 ...
8
votes
2answers
566 views

Regularisation of infinite-dimensional determinants

Can a regularisation of the determinant be used to find the eigenvalues of the Hamiltonian in the normal infinite dimensional setting of QM? Edit: I failed to make myself clear. In finite ...
7
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1answer
3k views

Evolution operator for time-dependent Hamiltonian

When i studyed QM I'm only working with non time-dependent Hamiltonians. In this case unitary evolution operator has the form $$\hat{U}=e^{-\frac{i}{\hbar}Ht}$$ that follows from this equation $$ ...