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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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 ...
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Rigged Hilbert space and QM
Are there any comprehensive texts that discuss QM using the notion of rigged Hilbert spaces? It would be nice if there were a text that went through the standard QM examples using this structure.
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Why path integral approach may suffer from operator ordering problem?
In Assa Auerbach's book (Ref. 1), he gave an argument saying that in the normal process of path integral, we lose information about ordering of operators by ignoring the discontinuous path.
What did ...
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Why/How is this Wick's theorem?
Let $\phi$ be a scalar field and then I see the following expression (1) for the square of the normal ordered version of $\phi^2(x)$.
\begin{align}
T(:\phi^2(x)::\phi^2(0):) &= 2 \langle 0|T(\...
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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 ...
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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 $$\...
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The formal solution of the time-dependent Schrödinger equation
Consider the time-dependent Schrödinger equation (or some equation in Schrödinger form) written down as
$$
\tag 1 i\hbar \partial_{t} \Psi ~=~ \hat{H} \Psi .
$$
Usually, one likes to write that it has ...
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Schrödinger wavefunctional quantum-field eigenstates
The reason that I have this problem is that I'm trying to solve problem
14.4 of Schwartz's QFT book, which've confused me for a long time.
The problem is to construct the eigenstates of a quantum ...
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What's the deal with momentum in the infinite square well?
Every now and then a question comes up about the status of the momentum operator in the infinite square well, and while we have two good answers on the topic here and here, I'm generally not satisfied ...
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Time as a Hermitian operator in quantum mechanics
In non-relativistic QM, on one hand we have the following relations:
$$\langle x | P | \psi \rangle ~=~ -i \hbar \frac{\partial}{\partial x} \psi(x),$$
$$\langle p | X | \psi \rangle ~=~ i \hbar \...
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Why do we use Hermitian operators in QM?
Position, momentum, energy and other observables yield real-valued measurements. The Hilbert-space formalism accounts for this physical fact by associating observables with Hermitian ('self-adjoint') ...
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Do derivatives of operators act on the operator itself or are they "added to the tail" 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 ...
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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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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 ...
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What's wrong with this derivation that $i\hbar = 0$?
Let $\hat{x} = x$ and $\hat{p} = -i \hbar \frac {\partial} {\partial x}$ be the position and momentum operators, respectively, and $|\psi_p\rangle$ be the eigenfunction of $\hat{p}$ and therefore $$\...
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Why are Only Real Things Measurable?
Why can't we measure imaginary numbers? I mean, we can take the projection of a complex wave to be the "viewable" part, so why are imaginary numbers given this immeasurable descriptor? Namely with ...
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Time-ordering vs normal-ordering and the two-point function/propagator
I don't understand how to calculate this generalized two-point function or propagator, used in some advanced topics in quantum field theory, a normal ordered product (denoted between $::$) is ...
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Creating a QM state of definite position in Fock space
I'm wondering if somebody could help me to finish a simple calculation. Let me first provide motivation for the question below: I would like to write a QM amplitude in the 'QFT-style', as
$$\langle \...
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Heisenberg Uncertainty Principle scientific proof
Heisenberg's uncertainty principle states that:
$$\sigma(x)\sigma( p_x )\ge \frac {\hbar}{2}.$$
What is the scientific proof of this principle?
Operators Uncertainty
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Derivative of the product of operators and Derivative of exponential
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 ...
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Is there a time operator in quantum mechanics?
The question in the title has been asked many times on this site before, of course. Here's what I found:
Time as a Hermitian operator in QM? in 2011. Answer states time is a parameter.
Is there an ...
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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 $H(...
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Quantization of a particle on a spherical surface
Suppose we have a particle of mass $m$ confined to the surface of a sphere of radius $R$. The classical Lagrangian of the system is
$$L = \frac{1}{2}mR^2 \dot{\theta}^2 + \frac{1}{2}m R^2 \sin^2 \...
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Does the canonical commutation relation fix the form of the momentum operator?
For one dimensional quantum mechanics $$[\hat{x},\hat{p}]=i\hbar. $$
Does this fix univocally the form of the $\hat{p}$ operator? My bet is no because $\hat{p}$ actually depends if we are on ...
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Hermiticity of Momentum Operator (matrix) Represented in Position Basis
I read that the momentum operator, $\hat P$ should be Hermitian (some would say by QM postulate). That makes perfect sense when it is represented in the momentum (p) basis: $\hat P =\int_{-\infty}^{\...
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How does non-commutativity lead to uncertainty?
I read that the non-commutativity of the quantum operators leads to the uncertainty principle.
What I don't understand is how both things hang together. Is it that when you measure one thing first and ...
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Half-integer eigenvalues of orbital angular momentum
Why do we exclude half-integer values of the orbital angular momentum?
It's clear for me that an angular momentum operator can only have integer values or half-integer values. However, it's not clear ...
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What does the ordering of creation/annihilation operators mean?
When a system is expressed in terms of creation and annihilation operators for bosonic/fermionic modes, what exactly is the physical meaning of the order in which the operators act?
For example, for ...
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A curious issue about Dyson-Schwinger equation (DSE): why does it work so well?
This question comes out of my other question Time ordering and time derivative in path integral formalism and operator formalism, especially from the discussion with drake. The original post is ...
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Proving that $i\hbar\frac{\partial}{\partial \mathbf{p}}$ is the operator of $\mathbf{x}$ in momentum space
How can I prove that $i\hbar\frac{\partial}{\partial \mathbf{p}}$ is the operator of $\mathbf{x}$ in momentum space?
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Normal ordering of the commutator between annihilation and creation operator
According to the commutation relation of annihilation and creation operators,
$$[a,a^{\dagger}]=1. \tag{1}$$
I would like to calculate the vacuum expectation value of the normal order of this ...
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Operator Ordering Ambiguities
I have been told that $$[\hat x^2,\hat p^2]=2i\hbar (\hat x\hat p+\hat p\hat x)$$ illustrates operator ordering ambiguity.
What does that mean?
I tried googling but to no avail.
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Does every hermitian operator represent a measurable quantity?
In Quantum mechanics, observables are represented by hermitian operator. But does every hermitian operator represent a observable? If not , how do we know that whether a hermitian operator represent ...
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How exactly is "normal-ordering an operator" defined?
(In this question, I'm only talking about the second-quantization version of normal ordering, not the CFT version.)
Most sources (e.g. Wikipedia) very quickly define normal-ordering as "reordering ...
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Why is normal ordering a valid operation?
Why is normal ordering even a valid operation in the first place? I mean it can give us some nice results, but why can we do the ordering for the operators like that?
Is its definition motivated by ...
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Why is the "canonical momentum" for the Dirac equation not defined in terms of the "gauge covariant derivative"?
The canonical momentum is always used to add an EM field to the Schrödinger/Pauli/Dirac equations. Why does one not use the gauge covariant derivative? As far as I can see, the difference is a factor <...
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Not all self-adjoint operators are observables?
The WP article on the density matrix has this remark:
It is now generally accepted that the description of quantum mechanics in which all self-adjoint operators represent observables is untenable.[...
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Evolution operator for time-dependent Hamiltonian
When I studied QM I'm only working with time independent Hamiltonians. In this case the unitary evolution operator has the form $$\hat{U}=e^{-\frac{i}{\hbar}Ht}$$ that follows from this equation
$$
i\...
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What Hermitian operators can be observables?
We can construct a Hermitian operator $O$ in the following general way:
find a complete set of projectors $P_\lambda$ which commute,
assign to each projector a unique real number $\lambda\in\mathbb R$...
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Are the path integral formalism and the operator formalism inequivalent?
Abstract
The definition of the propagator $\Delta(x)$ in the path integral formalism (PI) is different from the definition in the operator formalism (OF). In general the definitions agree, but it is ...
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Difficulties with bra-ket notation
I have started to study quantum mechanics. I know linear algebra,functional analysis, calculus, and so on, but at this moment I have a problem in Dirac bra-ket formalism. Namely, I have problem with "...
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Position operator in QFT
My Professor in QFT did a move which I cannot follow:
Given the state $$\hat\phi|0\rangle = \int \frac{d^3p}{(2\pi)^3 2 E_p} a^\dagger_p e^{- i p_\mu x^\mu}|0\rangle,$$ he wanted to show that this ...
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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 ...
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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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How to construct the radial component of the momentum operator?
I'm having trouble doing it. I know so far that if we have two Hermitian operators $A$ and $B$ that do not commute, and suppose we wish to find the quantum mechanical Hermitian operator for the ...
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Which derivative with respect to time is which in the Heisenberg picture of quantum mechanics?
For an observable $A$ and a Hamiltonian $H$, Wikipedia gives the time evolution equation for $A(t) = e^{iHt/\hbar} A e^{-iHt/\hbar}$ in the Heisenberg picture as
$$\frac{d}{dt} A(t) = \frac{i}{\hbar} ...
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Commutator $[\hat{p},F(\hat{x})]$ of Momentum $\hat{p}$ with a Position dependent function $F(\hat{x})$?
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 ...
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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 (...
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Is there any theorem that suggests that QM+SR has to be an operator theory?
UPDATE
To make my question more precise, I'll define what I mean by an operator theory:
An operator theory is a theory in which the dynamical objects are operators, i.e., the equations of motion are ...
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How are anyons possible?
If $|\psi\rangle$ 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 ...