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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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 ...
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Can the momentum operator have an imaginary expectation value?
I'm making examples of wave functions to incorporate in a QM exam.
I came up with the following wave function, which gives me some troubles:
$$\psi(x,0) = \begin{cases}
A(a-x), & -a \leq x \leq a,\...
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Is the Uncertainty Principle valid for information about the past?
My layman understanding of the Uncertainty Principle is that you can't determine the both the position and momentum of a particle at the same point in time, because measuring one variable changes the ...
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Why does time evolution operator have the form $U(t) = e^{-itH}$?
Let's denote by $|\psi(t)\rangle$ some wavefunction at time $t$. Then let's define the time evolution operator $U(t_1,t_2)$ through
$$ U(t_2,t_1) |\psi(t_1)\rangle = |\psi(t_2)\rangle \tag{1}$$
and ...
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Mutual or same set of eigenfunctions if two Hermitian operators commute
If two operators commute, do they have "a mutual set of eigenfunctions", or "the same set of eigenfunctions"? My quantum chemistry book uses these as if they are interchangeable, but they do not seem ...
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Why the Galileo transformation are written like this in Quantum Mechanics?
In Quantum Mechanics it is said that the Galileo transformation $$\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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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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Why Don't the Ladder Operators Commute?
I have two problems with ladder operators.
The first is that I feel they should somehow result in measurable things. The asymmetry of applying the plus operator versus the minus operator is very ...
user24082
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Once a quantum partition function is in path integral form, does it contain any operators?
Once a quantum partition function is in path integral form, does it contain any operators?
I.e. The quantum partition function is $Z=tr(e^{-\beta H})$ where $H$ is an operator, the Hamiltonian of the ...
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Holstein-Primakoff and Dyson-Maleev representation
In Holstein-Primakoff and Dyson-Maleev representation, spin operators are represented by bosonic operators. Roughly speaking, a state with $S^z=S-m$ corresponds to a state containing $m$ bosons. In ...
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How to prove bound state's spectrum must be discrete and scattering state's spectrum must be continuous?
Consider $d$-dim Schrodinger equation without internal degree of freedom, that is, we don't consider spin etc.
How to prove that bound state's spectrum must be discrete and scattering state's ...
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Lorentz force in Dirac theory and its classical limit
It is well known that in Dirac theory the time derivative of $$P_i=p_i+A_i$$ operator (where $p_i=∂/∂_i$, $A_i$ - EM field vector potential) is an analogue of the Lorentz force:
$$\frac{dP_i}{dt} = e(...
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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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The state space is somehow defined by the observables?
In Quantum Mechanics states of a system are described by vectors in a Hilbert space called the state space while the physical quantities associated to the system are described by hermitian operators ...
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How To Use Ladder Operators?
I'm studying for a test in quantum mechanics and I'm having a hard time understanding how to use ladder operators. There are no examples in my text book, only definitions that I can't understand how ...
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Measurement of observables with continuous spectrum: State of the system afterwards
Suppose my system, described by a separable Hilbert space $H$, is in the state $\Psi$ when I measure an observable that has only continuous spectrum. What is the state of the system after 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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Velocity definition in quantum mechanics?
What is the definition of velocity in quantum mechanics? is it an operator?
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Do we need an orthonormal basis in Quantum Mechanics?
I was wondering if it is important in Quantum Mechanics to deal with operators that have an orthonormal basis of eigenstates? Imagine that we would have an operator (finite-dimensional) acting on a ...
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How to derive Uncertainty Principle relation?
How to derive Heisenberg Uncertainty Principle relation?
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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. ...
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Must observables be Hermitian only because we want real eigenvalues, or is more to that?
Because (after long university absence) I recently came across field operators again in my QFT lectures (which are not necessarily Hermitian):
What problem is there with observables represented by non-...
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What is the difference between a functional and an operator?
What is the difference between a functional and an operator? When we define an operator in physics, e.g. the momentum operator as $\hat{p} = i \frac{d}{dx}$, it is said this operator acts on the wave ...
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How does non-Hermitian quantum mechanics (PT-symmetric QM) fit in physics?
In the late nineties Bender has started a research program on what is called PT symmetric QM, or non hermitian QM, in which he has shown that if the hamiltonian enjoys a PT symmetry then the spectrum ...
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Commutator with exponential $[A, \exp(B)]$
How can I tell if $A$ and $\exp(B)$ commute?
For $[A, B]$ it's simply $AB-BA$ and for $[\exp(A), \exp(B)]$ I think it'd be $\exp(A)\exp(B) - \exp(B)\exp(A) = \exp(A+B) - \exp(B+A) = 0$. Update: it's ...
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Really how can an observable quantity be equal to an operator?
A wave-function can be written as $$\Psi = Ae^{-i(Et - px)/\hbar}$$ where $E$ & $p$ are the energy & momentum of the particle.
Now, differentiating $\Psi$ w.r.t. $x$ and $t$ respectively, we ...
user36790
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Why hermitian, after all? [duplicate]
This question is going to look a lot like a duplicate, but I've read dozens of related posts and they don't touch the subject. Here we go.
Why are observables represented by hermitian operators?
...
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Proof that energy states of a harmonic oscillator given by ladder operator include all states
In quantum mechanics, while studying the harmonic oscillator, I learnt about ladder operators. And I realised that if you are able to find or determine any energy state of the quantum harmonic ...
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Non-separable solutions of the Schroedinger equation
I'm studying an undergraduate Quantum Mechanics course and I have some doubts about the solution of the Schroedinger equation by the separation of variables method.
If we suppose that the solutions ...
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Why are the inner products of the eigenfunctions of an operator with a discrete eigenvalue spectrum guaranteed to exist?
I was reading through a textbook, and the statement was made that the inner products are guaranteed to exist if the eigenvalue spectrum of the operator is discrete. I have come across no support for ...
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The position-representation matrix elements of the propagator for a particle in a ring
I have a question about obtaining matrix elements of time evolution operator. I have the following Hamiltonian for a particle in a ring with magnetic field
$$H=\dfrac {\hbar ^{2}} {2mR^{2}}\left[ -i\...
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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
$$U(t,0) = \sum_n(\dfrac{-i}{\hbar})^n\...
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"An operator is hermitian". Implications?
Alastair Rae states that there are 4 postulates of Quantum Mechanics in his text on the subject matter. The first part of his second postulate can be stated as:
Every dynamical variable may be ...
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Position representation of spin states and spin operators
How can we represent a spin states
$ \lvert S_x:+\rangle, \lvert S_y:+\rangle,\lvert S_z:+\rangle ,\lvert S_x:-\rangle, \lvert S_y:-\rangle $ and $\lvert S_z:-\rangle$ in position representation ...
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Hermitian adjoint of 4-gradient in Dirac equation
I'm having issues deriving the Dirac adjoint equation, $$\overline{\psi}(i\gamma^{\mu}\partial_{\mu}+m)=0.\tag{1}$$
I started by taking the Hermitian adjoint of all components of the original Dirac ...
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Are angles ($\theta$ and $\phi$) in spherical coordinates treated as operators in quantum mechanics?
Position is specifically considered as an operator in quantum mechanics. I want to know if $\theta$ and $\phi$ are explicitly considered as operators in quantum mechanics for solutions to 3D ...
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Composition of squeeze operators?
I'm wondering if it exists a composition law for the squeezing operation ? I guess so for geometric reason, since they are (generalized, and the phase is annoying of course) hyperbolic rotations of ...
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The uncertainty in angular momentum
It is known that the different spatial components of the angular momentum don't commute with each other.
$$
[L_x,L_y] \propto L_z \\
[L_y,L_z] \propto L_x \\
[L_z,L_x] \propto L_y
$$
Also it is known ...
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Proof for $\langle q| p \rangle = e^{iqp}$
What would be the proof for $\langle q| p \rangle = e^{iqp}$?
Is it derived from canonical commutation relation?
($|q \rangle $ represents the position eigenstate, while $|p \rangle$ represents the ...
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The Momentum Operator in QM
I've seen the 'derivation' as to why momentum is an operator, but I still don't buy it. Momentum has always been just a product $m{\bf v}$. Why should it now be an operator. Why can't we just multiply ...
user24082
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Wick Order and Radial Ordering in CFT
I am not so much familiar with the computations tools of conformal field theory, and I just run into an exercise asking to demonstrate the following formula (related to the bosonic field case):
$$\...
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Correlation function of single annihilation/creation operator vanishes
I could not find anything on that on google, or here on physics stack exchange, which surprises me. My problem is, that I do not see, why exactly
$$\left<a\right> = \left<a^\dagger\right> =...
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Why are Hermitian operators linked to observables?
In Quantum Mechanics, why is it that a self-adjoint operator is linked to an observable? What makes it measureable? And why isn't a non-Hermitian operator linked to an observable?
Also, what type of ...
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General expectation value
I have a basic question related to finding expectation values of an operator $\hat{Q}$
We know that the expectation of $\hat{Q}$ (in the position space) is given by
$$\langle Q \rangle=\int {\Psi^* ...
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What are the Time Operators in Quantum Mechanics? [duplicate]
I don't understand at all what the time operators are in quantum mechanics. I thought that given a wave function, because it's a function of time, we could simple put in any time in the future to find ...
user24082
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Recovering QM from QFT
Reading through David Tong lecture notes on QFT.
On pages 43-44, he recovers QM from QFT. See below link:
QFT notes by Tong
First the momentum and position operators are defined in terms of "...
user56963
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Matrix representation for fermionic annihilation operator
My guess it should look something like this:
$ c_\sigma = (\left|0\right>\left<\uparrow\right|+\left|\downarrow\right>\left<\downarrow\uparrow\right|)\delta_{\sigma,\uparrow}+(\left|0\...
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Deriving a QM expectation value for a square of momentum $\langle p^2 \rangle$
I already derived a QM expectation value for ordinary momentum which is:
$$
\langle p \rangle= \int\limits_{-\infty}^{\infty} \overline{\Psi} \left(- i\hbar\frac{d}{dx}\right) \Psi \, d x
$$
And I ...
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Why there is no operator for time in QM? [duplicate]
Is there one central reason why there is no "Time" operator in QM?
I know this question has been asked before, but I thought I would try to stimulate some fresh thinking.
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Relationship between normal-ordered vacuum state and parity operator
In the paper "Operator ordering in quantum optics
theory and the development of Dirac’s
symbolic method" by Hong-yi Fan, as referenced in this question, the authors mention the property
$$:A:...
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