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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How can the commutator operation not be transitive?
I noticed the following:
$$[L_{+},L^2]=0,\qquad [L_{+},L_3]\neq 0,\qquad [L^2,L_3]=0.$$
This would suggest, that $L^2,L_+$ have a common system of eigenfunctions, and so do $L^2,L_3$, but $L_+,L_3$ ...
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If all the eigenvalues of an operator are real, is the operator Hermitian?
How do I prove or disprove the following statement?
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 ...
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What is the cleverest way to calculate $[\hat{a}^{M},\hat{a}^{\dagger N}]$ when $[\hat{a},\hat{a}^{\dagger}]=1$?
What are some elegant ways to calculate
$$[\hat{a}^{M},\hat{a}^{\dagger N}]\qquad\text{with} \qquad[\hat{a},\hat{a}^{\dagger}]=1,$$
other than brute force calculation?
(EDIT) I got the same result ...
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What exactly is $\hat{\psi}^\dagger(x)$? An operator or a function?
I've recently read Cohen-Tannoudji on quantum mechanics to try to better understand Dirac notation. A homework problem is giving me some trouble though. I'm unsure if I've learned enough yet to ...
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Binomial expansion of non-commutative operators
I would like to determine the general expansion of
$$(\hat{A}+\hat{B})^n,$$
where $[\hat{A},\hat{B}]\neq 0$, i.e. $\hat{A}$ and $\hat{B}$ are two generally non-commutative operators. How could I ...
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What does vector operator for angular momentum measure?
Consider the vector operator for angular momentum $\hat L=\hat L_x \vec i +\hat L_y \vec j + \hat L_z \vec k$.
Does this mean that if we want to measure the angular momentum of a particle in state $\...
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Can a normalizable function *always* be decompose into the discrete Hydrogen spectrum?
This question has been bothering me for a while now: can one reconstruct an arbitrary (normalizable) function $\phi(\mathbf r)$ in $\mathbb R^3$, with only the (discrete) set of Hydrogen ...
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Mutual or same set of eigenfunctions if two Hermitian operators commute
If two operators commute, do they have "a mutual set of eigenfunctions", or "the same set of eigenfunctions"? My quantum chemistry book uses these as if they are interchangeable, but they do not seem ...
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The correspondence between Grassmann number and 4-spinor
In canonical quantization, we view the Dirac field $\psi$ as a $4\times1$ matrix of complex number. While in path integral quantization, we view the Dirac field $\psi$ as a Grassmann number.
For two ...
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Bounded and Unbounded (Scattering) States in Quantum Mechanics
I understand that bounded states in quantum mechanics imply that the total energy of the state, $E$, is less than the potential $V_0$ at + or - spatial infinity. Similarly, the scattering state ...
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Wick theorem and OPE
I'm trying to work out in detail how the Wick theorem is used for constructing OPEs in CFT. One of the first things which bothers me is the difference in definitions of normal ordered product and ...
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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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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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Is there a simple way of finding the eigenstates of the creation and annihilation operator in QM?
How can I find the eigenstates of creation and annihilation operator in QM?
My attempt:
Such eigenstate will obey: $$ a^{\dagger} |\psi \rangle = \alpha |\psi \rangle. $$
We can expand $|\psi \...
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Why are the inner products of the eigenfunctions of an operator with a discrete eigenvalue spectrum guaranteed to exist?
I was reading through a textbook, and the statement was made that the inner products are guaranteed to exist if the eigenvalue spectrum of the operator is discrete. I have come across no support for ...
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Can one define an acceleration operator in quantum mechanics?
It seems most books about QM only talk about position and momentum operators. But isn't it also possible to define a acceleration operator?
I thought about doing it in the following way, starting ...
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Why the Galileo transformation are written like this in Quantum Mechanics?
In Quantum Mechanics it is said that the Galileo transformation $$\hat{\mathbf{r}}\mapsto \hat{\mathbf{r}}-\mathbf{v}t\quad \text{and}\quad \hat{\mathbf{p}}\mapsto \hat{\mathbf{p}}-m\mathbf{v}\tag{1}...
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Energy is actually the momentum in the direction of time?
By comparatively examining the operators
a student concludes that `Energy is actually the momentum in the direction of time.' Is this student right? Could he be wrong?
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Is there a general meaning for a Quantum Field?
In the traditional presentation, Quantum Fields are usually presented as operator valued fields defined on spacetime, in the sense that $\varphi : M\to \mathcal{L}(\mathcal{H})$ for some Hilbert space ...
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Curved spacetime as a coherent state in string theory
I have a question about Polchinski's string theory book, volume I, p 108. When we write the Polyakov action in curved spacetime, it is said
$$ S_{\sigma} = \frac{1}{4\pi\alpha'} \int_M d^2 \sigma g^...
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Explaining why $\mathrm{ d/d}x$ is not Hermitian, but $\mathrm{i~ d/d}x$ is Hermitian
There is the standard argument, using the definition of the inner product; that $\langle f|A|g\rangle =\langle g|A|f\rangle ^{*}$ for a Hermitian operator $A$, given any wave vectors $|f\rangle,~ |g\...
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Resolution of the identity of operator with mixed spectrum
In most quantum mechanics text books, the resolution of the identity or completeness relation is stated in the following (or similar) form
$$ \mathbb I_\mathcal H = \sum\limits_n |\lambda_n\rangle \...
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Is there a generalised Wigner-Eckart theorem?
The Wigner-Eckart theorem gives you the matrix element of a tensor transforming according to a representation of $\mathfrak{su}(2)$, when sandwiched between vectors transforming according to another (...
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What do field operators in QFT act on?
I have been self-studying physics and QFT for a quite a while now and there are a couple of basic ideas of QFT that I just can't find the answer to no matter how hard I try. I know this might sound ...
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What is the value of a quantum field?
As far as I'm aware (please correct me if I'm wrong) quantum fields are simply operators, constructed from a linear combination of creation and annihilation operators, which are defined at every point ...
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Is it always possible to express an operator in terms of creation/annihilation operators?
I'm referring to "Path integral approach to birth-death processes on a lattice", L. Peliti, J. Physique 46, 1469-1483 (1985), available at: http://people.na.infn.it/~peliti/path.pdf
The ...
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How to know if a set of commuting observables is complete?
We define a complete set of commuting observables as a set of observables $\{A_1,\ldots, A_n\}$ such that:
$\left[A_i, A_j\right]=0$, for every $1\leq i,~j \leq n$;
If $a_1,\ldots, a_n$ are ...
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Mathematical motivation of OPE?
In Peskin & Schroeder (and also Cheng which I have skimmed through) they motivate the Operator Product Expansion with a lot of words.
Is there any way to motivate it mathematically, e.g. Taylor ...
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Why does the expectation value in quantum mechanics correspond to the classically measured value?
I understand that we can use the Heisenberg picture to show, for a Hamiltonian of the form
$$
\hat{H}=\frac{\hat{P}^{2}}{2m}+\hat{V}(\hat{X})
$$
the Ehrenfest theorem:
$$
m\partial_{t}\langle \hat{X}\...
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How does a linear operator act on a bra?
I'm studying QM from Shankar. He introduces linear operators and says that an operator is an instruction for transforming one ket into another. But then a few lines below he says operators can also ...
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The position-representation matrix elements of the propagator for a particle in a ring
I have a question about obtaining matrix elements of time evolution operator. I have the following Hamiltonian for a particle in a ring with magnetic field
$$H=\dfrac {\hbar ^{2}} {2mR^{2}}\left[ -i\...
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Time-ordering vs normal-ordering and the two-point function/propagator
I don't understand how to calculate this generalized two-point function or propagator, used in some advanced topics in quantum field theory, a normal ordered product (denoted between $::$) is ...
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Are eigenstates of the position operator continuous?
The way I've understood it is that eigenfunction of an operator are the different states which the actual wavefunction can take when the property/observable corresponding to the given operator is ...
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The Physical Meaning of Projectors in Quantum Mechanics
Let $O$ be a single-particle observable for a system, and $|L\rangle$ and $|R\rangle$ two orthonormal eigenstates of $O$. You may imagine that the system consists in two photons, and $|L\rangle$ and $|...
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Must bounded operators have normalisable eigenfunctions and discrete eigenvalues?
When we have bound states, to my knowledge, we have states that are normalisable and a discrete energy spectrum. However, in the case of scattering states that have a continuous energy spectrum, the ...
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How is the hamiltonian a hermitian operator?
My book about quantum mechanics states that the hamiltonian, defined as $$H=i\hbar\frac{\partial}{\partial t}$$ is a hermitian operator. But i don't really see how I have to interpret this. First of ...
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Does an Operator that neither commutes with $\hat{X}$ or $\hat{P}$, nor can be expressed as a "function" of $\hat{X}$ and $\hat{P}$ make sense?
When you come from classical hamiltonian mechanics (which is based on the phase space), observables are introduced as functions $f$ on the phase space $(q, p)$. There can't be a classical observable ...
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What's a particle anyway?
I always thought that a particle is an eigenvector of $P^2=H^2-\boldsymbol P^2$ with an isolated eigenvalue. In other words, a necessary condition for $\varphi$ to be a particle is that
$$
P^2|\varphi\...
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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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What is the commutator of an operator and its derivative?
Is it possible to calculate in a general way the commutator of an operator $O$ which depends on some variable $x$ and the derivative of this $O$ with respect to $x$?
$${O}={O}(x)\\
\left[\partial_x{O}(...
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Math. Axioms of Quantum Mechanics: $C^*$- vs. $W^*$-Algebras
In the general formulation of Quantum Mechanics, a physical system is described by a $C^*$-algebra $\mathcal{A}$, where the observables correspond to the self-adjoint elements of the algebra, and the ...
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Is a partial trace cyclic?
I want to know if a partial trace keeps the cyclic property of the trace.
The partial trace is defined as
$$ tr_B: \mathcal{B}_1(\mathcal{H}_A\otimes \mathcal{H}_B) \longrightarrow \mathcal{B}_1(\...
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Schrödinger equation in position representation
We start from an abstract state vector $
\newcommand{\ket}[1]{|{#1}\rangle} \ket{\Psi}$ as a description of a state of a system and the Schrödinger equation in the following form
$$
\...
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What's the relation between path integral and Dyson series?
If one solves the Schrodinger equation
$$i\hbar\partial_tU(t,0) = H U(t,0)$$
for time evolution operator $U(t,0)$, one can get the following Dyson series
$$U(t,0) = \sum_n(\dfrac{-i}{\hbar})^n\...
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Radial quantization and infrared divergences
I am reading Ginsparg lectures "Applied CFT" https://arxiv.org/abs/hep-th/9108028 which is not my first material on the subject. He tries to motivates radial quantization on the reason that ...
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Shape of the state space under different tensor products
I am currently studying generalized probabilistic theories. Let me roughly recall how such a theory looks like (you can skip this and go to "My question" if you are familiar with this).
Recall: In a ...
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On the asymptotics of interacting correlation functions
Consider an interacting QFT (for example, in the context of the Wightman axioms). Let $G_2(x)$ be the two-point function of some field $\phi(x)$:
$$
G_2(x)=\langle \phi(x)\phi(0)\rangle
$$
Question: ...
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Explaining causal completion axiom in Haag-Kastler axioms?
There are several variants of the Haag-Kastler axioms for algebraic quantum field theory. Usually one associates an algebra $\mathcal{A}(O)$ to each open region $O$ of spacetime. An often-suggested ...
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Why do eigenvalues correspond to observable quantities?
It makes sense to me that we can find some operator that gives us eigenfunctions that correspond to definite values for some desired observable. However, I do not see how the eigenvalues happen to ...
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Why are expectation values of an observable important in QM?
I've been reading that expectation values of an observable is all what we can get and are the key quantities of the theory, but performing the same experiment many times would generate a distribution ...
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