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 $\times$, $+$!

Filter by
Sorted by
Tagged with
0 votes
0 answers
33 views

Laplace transform: How to evaluate partial derivative in the denominator of a fraction?

I am solving a differential equation using the Laplace transform. However, to evaluate it I need to evaluate some strange terms. Specifically, I have a partial derivative in the denominator of the ...
J.Agusti's user avatar
0 votes
1 answer
120 views

Algebra of observables in Quantum Mechanics

When reading books about Quantum Mechanics, it is generally stated (in a kind of axiomatic way) that in Quantum Mechanics, the state of the system is represented by a vector in some Hilbert space $H$, ...
Weier's user avatar
  • 196
-2 votes
0 answers
20 views

Momentum Operator in Radial Quantization

I have a Hilbert space of a CFT in radial quantization. In particular, I have a Hamiltonian on a QFT/multibody system on a 2D sphere. The Hamiltonian acts as $D$ in the 3D CFT. If I have a state $|O\...
fangzhang mnm's user avatar
1 vote
0 answers
50 views

System interacting with Fermi Gas [closed]

My question denoted by a reduced dynamic for a system interacting with a reservoir. Before asking the question, for completeness I will write in detail the statement of the problem and notation. ...
ets_ets's user avatar
  • 33
-3 votes
1 answer
82 views

Variance/Standard deviation of an observable on a state that is a linear combination of eigenvectors of that observable

I know that when measuring the standard deviation of an observable the result will be zero if the system is an eigenvector of the observable on which i want to calculate the standard deviation. But ...
AlexM3020's user avatar
7 votes
2 answers
730 views

Exponential operator approximation: Suzuki-Trotter Expansion

One cannot solve the transition amplitude $\langle{x}\vert e^{-iHt}\vert{y}\rangle{}$ with $H=H_0+V$ by just applying the operators one after another on the bra/ket, because the free hamiltonian $H_0$ ...
Xhorxho's user avatar
  • 93
2 votes
2 answers
162 views

How to take derivative of density operator?

I was just trying to confirm to myself that the following density operator $$\rho(t) = e^{-iHt/\hbar} \rho(0) e^{iHt/\hbar}$$ fulfills the Liouville-von Neumann equation: $$\frac{d}{dt}\rho(t) = - \...
Physchem16's user avatar
1 vote
0 answers
49 views

Doubt about the diagonalization of a state

I'm studing the unitary irreducible representations of the deSitter group in two dimension.The generators of its double cover, the group $SL(2,R)$, are the following $$ H_2 = \frac{\sigma_2}{2} \\ F_1 ...
michael pasqui's user avatar
1 vote
1 answer
47 views

Measurement values affects probabilities in QM?

Consider a non-degenerate operator $\Omega$ with discrete eigenvalues $\omega_i$, where $i=1,2,3,...$. We can write $\Omega = \sum_i \omega_i~|\omega_i\rangle \langle \omega_i|$, where $|\omega_i\...
Deep's user avatar
  • 6,464
1 vote
1 answer
68 views

Are partition functions invariant under Bogoliubov transformations?

Consider a Hamiltonian $H(a_i, a^{\dagger}_i)$ as a function of some ladder operators $a_i, a^{\dagger}_i$. Now, consider a partition function $H(a'_i, a'^{\dagger}_i)$ where $a', a'^{\dagger}$ are ...
Dr. user44690's user avatar
0 votes
0 answers
48 views

Bra-Ket Notation vs Wavefunction Notation [duplicate]

We know that the rule for creating excited states for a Quantum Harmonic Oscillator is $|n\rangle=\frac{(a^\dagger)^n(|0\rangle)}{\sqrt{n!}}$. I wanted to derive from this the familiar rule in terms ...
V Govind's user avatar
  • 364
-3 votes
0 answers
31 views

Expectation value of vacuum state [closed]

I'm confused about the expectation value of the vacuum state b. Here's my understating: the a' and b' are defined by operator a and b which are similar to the annihilation operator. So when we act ...
MoMo's user avatar
  • 1
1 vote
1 answer
64 views

Wick rotation and Exponential mapping of an "imaginary" differential operator acting on a real-valued wavefunction

The position shift operator $T^{a}$ (where $a \in \mathbb{R}$ ) takes a real valued wavefunction $\psi$ on $\mathbb{R}$ to its translation $\psi_{a}$, $T^{a} \psi(x)=\psi_{a}(x)=\psi(x+a)$. A ...
Cuntista's user avatar
  • 180
12 votes
7 answers
1k views

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 ...
user536450's user avatar
0 votes
1 answer
53 views

On the Born-Jordan quantization being an equally weighted average of all operator orderings

On my way studying quantization schemes, I came across the expression saying that the Born-Jordan quantization rule is the equally weighted average of all the operator orderings and that the Weyl's ...
user536450's user avatar
1 vote
0 answers
33 views

Questions of lower boundness of Hamiltonians in quantum theories

In general spectral analysis, we have examples of unbounded from below hamiltonians with discrete spectrum. Is it okay to say that they have no sense in physical context, because for me it looks like ...
0 votes
0 answers
46 views

On the symmetry (and anti-symmetry) of operators in QM

In quantum mechanics, the terms symmetric and antisymmetric typically refer to states; specifically, a state is said to be '(anti)symmetric' if it is an eigenstate of the exchange operator with an ...
ric.san's user avatar
  • 1,484
0 votes
1 answer
114 views

What is the difference between $UXU^{-1}$ and $UX$?

Suppose we have a state $X$ living in some vector space $S$ and a linear operator $U$ that acts on $S$. Now, my understanding is that if $X$ and $U$ are expressed in the same representation, say both ...
Hrach's user avatar
  • 258
0 votes
0 answers
31 views

Spectrum of Klein-Gordon operator in AdS Black Hole

I'm working on obtaining the spectrum of the Klein-Gordon operator in $AdS_2$ for black hole coordinates. To accomplish that, I first consider the problem in hyperbolic space $H_2$ and then Wick ...
MarcelRomp's user avatar
-1 votes
3 answers
85 views

The meaning of the equivalence of the Schroedinger and Heisenberg pictures

On my way to explore the equivalence between the Heisenberg and the Schroedinger pictures, I cannot see why textbooks quickly deduce this equivalence from the following: If $\psi$ is the state of the ...
user536450's user avatar
1 vote
2 answers
103 views

Projection operator onto support of distinct observables

Suppose $P_i$ is the projection operator onto the support of the observable $O_i$ defined on some (say, finite dimensional) Hilbert space. I'm curious as to whether we can define the projection ...
Theoreticalhelp's user avatar
2 votes
2 answers
360 views

Uncertainty on the sum of two non-commuting operators

Suppose that I have an observable $$ \hat{E} = \sin(\alpha) \hat{Q} + \cos({\alpha}) \hat{P} $$ with $\hat{Q}, \hat{P}$ being non-commuting operators satisfying $$ [\hat{Q}, \hat{P}] = i \hbar $$ It ...
Nicolas Schmid's user avatar
-1 votes
0 answers
53 views

How do quantum Fields know about changes in normal ordering?

According to Transformation of the energy-momentum tensor under conformal transformations The schwartzian term in the transformation properties arises due to the stress tensor being defined as the ...
DerHutmacher's user avatar
-3 votes
1 answer
66 views

Time-ordering and Minkowski metric's negative sign [closed]

I'm coming at the following question from a mostly lay perspective (i.e. barely-undergrad physics), so please bear with the weirdness of it if possible. I've generally been uncomfortable with the ...
allidoiswin's user avatar
3 votes
2 answers
95 views

Why does spin acting along $x$ on the spin up state yield spin down?

I'm currently struggling on how to interpret the particular result below: $$\hat{S}_{x}\ | \alpha \rangle \ = \frac{\hbar}{2}\ | \beta\rangle$$ Where $\ | \alpha \rangle \ $ is the spin up state and $\...
NormanWasHere's user avatar
1 vote
0 answers
18 views

Can we treat Gazeau-Klauder coherent states for infinite potential well as a superposition of Fock states?

If we define coherent states of infinite potential well based on Gazeau-Klauder coherent states. Can we use ladder operators and bosonic algebra for them which we use for Glauber coherent states?
Tooba's user avatar
  • 771
2 votes
2 answers
144 views

Gibbs state and creation and annihilation operators

Let's consider quantum Fermi or Bose gas. Let $a(\xi)$, $a^{\dagger}(\xi)$ are standard annihilation and annihilation operators. Hamiltonian of system is denoted as $$ \hat{H} = \int_{R^3} \frac{p^2}{...
ets_ets's user avatar
  • 33
3 votes
1 answer
136 views

Is there a standard way of calculating these transformations over creation and annihilation operators?

Suppose we have families $\{a_{x}^{*}\}_{x\in \mathbb{Z}}$ and $\{a_{x}\}_{x\in \mathbb{Z}}$ of fermionic creation and annihilation operators, respectively. By construction, they satisfy canonical ...
IamWill's user avatar
  • 545
2 votes
1 answer
54 views

Why is the usage of just linear, unitary operator that isn't necessarily self-adjoint in the middle of a bra-ket a problem?

In the question Define antilinear or antiunitary operator $Θ$ acting on the ket state and on the bra state consistently? an answer mentions that the bra-ket $⟨β|A|α⟩$ in a proper way iff $A$ is ...
lake_apricot's user avatar
0 votes
0 answers
16 views

How can I understand the meaning of a Hermitian eigenvalue problem? [duplicate]

Suppose we have the $\textit{mode functions}$ given as $\textbf{u}_m(\textbf{x})$ which are defined by the following eigenvalue equation: \begin{equation} \nabla^2\textbf{u}_m(\textbf{x}) = -k_m^2\...
Rasmus Andersen's user avatar
0 votes
0 answers
23 views

Examples of operators like multi body fermions in finite potential well with total energy being sum of energies, and not entirely homogeneous

I'm looking for examples of systems with operators like the 2 fermion system in finite potential well where the total energy $\lambda(k) = \lambda_1(k) + \lambda_2(k)$ where $k$ is the depth of the ...
Razor's user avatar
  • 455
1 vote
0 answers
58 views

Physical meaning of the domain of a quantum operator

It's generally interesting to ask what's the physical interpretation of the various mathematical entities one employs in physics. When doing quantum mechanics one often leaves aside all the “...
Leonardo Rossi's user avatar
1 vote
1 answer
38 views

Commutation relation between pairs of conjugated variables

Suppose I have four operators $\hat{q}_1$, $\hat{p}_1$, $\hat{q}_2$ and $\hat{p}_2$ such that $[\hat{q}_1, \hat{p}_1] = [\hat{q}_2, \hat{p}_2] = i$ $[\hat{q}_1, \hat{q}_2] = [\hat{p}_1, \hat{p}_2] = [...
Maxime's user avatar
  • 11
0 votes
0 answers
42 views

Wigner's theorem for 2-state system

I am trying to see how Wigner's theorem can be proven in a 2-state system. Let, $$ |\psi\rangle= a |0\rangle+b|1\rangle, \quad|\phi\rangle= c |0\rangle+d|1\rangle $$ where all coefficients are complex ...
pepper's user avatar
  • 1
2 votes
1 answer
74 views

How is a Fock state from QFT related to the wave function from quantum mechanics?

I am currently studying quantum field theory as part of my degree. I'm just lacking intuition or an understanding of some basic concepts. So please don't hesitate to correct me if i got something ...
Benny's user avatar
  • 21
1 vote
2 answers
78 views

Wigner transform of $O_1 O_2$ in terms of Wigner transforms of $O_1$ and $O_2$?

The Wigner-Weyl transform of a quantum operator $O$ is defined as $$ W[O](q,p) = 2 \int_{-\infty}^{\infty} dy\ e^{- 2 i p y} \langle q + y | O | q - y \rangle \ dy $$ and then given a density matrix $\...
QuantumEyedea's user avatar
0 votes
0 answers
85 views

Newtonian generalisation of binomial theorem for commuting operators

Suppose $\hat{A}$ and $\hat{B}$ are commuting operators on some Hilbert space, $[\hat{A},\hat{B}]=0$, chosen suitably to ensure operators $1/\hat{A}$, $1/\hat{B}$ and $\hat{A}/\hat{B}$ are suitably ...
Theoreticalhelp's user avatar
8 votes
3 answers
1k views

Dirac's definition of probability in quantum mechanics

I'm currently reading "The principles of quantum mechanics" by Dirac, and I'm having some trouble understanding some of his assumptions, because in the quantum mechanics course I'm following ...
Fede's user avatar
  • 415
0 votes
1 answer
47 views

Problem with time reversal operator

I am trying to understand the time reversal operator. As I read it is defined as $$ \mathcal{T} = \mathcal{U}\mathcal{K}, $$ where $\mathcal{U}$ is a unitary operator and $\mathcal{K}$ is a complex ...
blahblah's user avatar
  • 105
-2 votes
0 answers
58 views

Deriving $\hat{p}_ x =-iℏ∂/∂x$ [duplicate]

(Taken from "The meaning of Quantum Theory, from Jim Baggott). Starting from [x,^Px]= iℏ, if we chose the position operator to be simply 'multiplication by x', this forces the linear momentum ...
Jon Andoni Gomez's user avatar
0 votes
1 answer
94 views

How to Derive the Minimum Time for $⟨ψ_0|A(t)|ψ_0⟩$ to Vanish Using Uncertainty Principle?

I am trying to work through some bookwork, and I am stuck on the titular question. The scenario is as follows: In a generic finite dimensional Hilbert space $\mathcal{H}$, define the operator $A(t) ≡ ...
2307's user avatar
  • 103
1 vote
0 answers
36 views

Time reversal operator on conduction band edge function

The bottom of the conduction band of Zinc Blende structure with grand representation $T_{d}$ symmetry group have a 2-dimensional representation at $k=0$ also known as the center of the Brillouin zone ...
EL.K. MORAD's user avatar
0 votes
0 answers
51 views

Commutation relation of position and Hamiltonian operator [duplicate]

In the book Quantum Mechanics volume 2 by Cohen-Tannoudji, in Electric dipole approximation, it was written that $\left[\boldsymbol{Z}, H_0\right]=i \hbar \frac{\partial H_0}{\partial P_{\boldsymbol{z}...
Lusypher's user avatar
  • 185
1 vote
1 answer
68 views

Is there an identity for $\hat{a}^+ \cdot \hat{a}^+$?

Is there an identity for the square of the ladder operators, $\hat{a}^+$ and $\hat{a}^-$?
Barney_Dinosaur's user avatar
4 votes
1 answer
156 views

Time-ordered matrix exponential quasi-static limit

Define a matrix differential equation $$\dot{X}=A(t)X(t),$$ where $X=[x1,x2,...]^T$ is a 1D vector and $A(t)$ is a complex-valued time-dependent matrix. This system can be solved by $$X(t)= \mathcal{T}...
J.Agusti's user avatar
3 votes
2 answers
246 views

Wick contraction between two scalar fields

I have a short question about Wick contraction. It is given that $$\phi\left(x\right) = \phi^{+}\left(x\right) + \phi^{-}\left(x\right)\tag{1}$$ where: $$\phi^{+}\left(x\right) = \int \frac{d^3p}{\...
Jochem4T's user avatar
  • 217
3 votes
2 answers
121 views

Can $\langle0|(\hat{a}-\hat{b})|0\rangle$ be written as $\langle0|\hat{a}|0\rangle-\langle0|\hat{b}|0\rangle$?

I am studying quantum mechanics and dirac notation, and I am wondering if, given the operator $\hat{A}=\hat{a}-\hat{b}$, the expectation value $\langle0|\hat{A}|0\rangle$ can be written as $\langle0|\...
Barney_Dinosaur's user avatar
4 votes
3 answers
1k views

Where does this expression for infinitesimal rotations come from?

In Chapter 2, page 11 of Preskill's quantum computing notes, he mentions without explaining that a counterclockwise inifinitesimal rotation by $d\theta$ about the axis $\hat{n}$ is given by $$R(d\...
Eulerian's user avatar
  • 310
0 votes
3 answers
85 views

Derivation of kinetic energy operator QM [closed]

In griffiths introduction to quantum mechanics, he starts off by writing $$\langle x\rangle = \int x|\psi(x,t)|^2 dx$$ $$\langle x \rangle = \int \psi^* [x] \psi(x,t) dx$$ He then says that the ...
jensen paull's user avatar
  • 6,377
2 votes
2 answers
285 views

Eigenfunctions of Momentum Operator [closed]

Suppose we have the 1-d wave function $\psi(x)=A\sin\left(\frac{p_0x}{\hbar}\right)$ and we want to know wheter this is an eigenfunction for $\hat{p}=-i\hbar\dfrac{d}{dx}$ The argument usually goes ...
Johann Wagner's user avatar

1
2 3 4 5
95