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 $\times$, $+$!
4,702
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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 ...
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1
answer
120
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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 $H$, ...
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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\...
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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.
...
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1
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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 ...
- 19
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2
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730
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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 $H_0$ ...
- 93
2
votes
2
answers
162
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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:
$$\frac{d}{dt}\rho(t) = - \...
- 29
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49
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Doubt about the diagonalization of a state
I'm studing the unitary irreducible representations of the deSitter group in two dimension.The generators of its double cover, the group $SL(2,R)$, are the following
$$
H_2 = \frac{\sigma_2}{2} \\
F_1 ...
- 171
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1
answer
47
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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\...
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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 ...
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48
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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 ...
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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 ...
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1
vote
1
answer
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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 ...
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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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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 ...
- 151
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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 ...
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46
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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 ...
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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 ...
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31
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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 ...
-1
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3
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85
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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 ...
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2
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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 ...
2
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2
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360
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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 ...
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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 ...
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1
answer
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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 ...
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2
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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 $\...
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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?
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2
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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}{...
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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 ...
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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 ...
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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\...
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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 ...
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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 “...
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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] = [...
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42
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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
$$
where all coefficients are complex ...
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1
answer
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How is a Fock state from QFT related to the wave function from quantum mechanics?
I am currently studying quantum field theory as part of my degree. I'm just lacking intuition or an understanding of some basic concepts. So please don't hesitate to correct me if i got something ...
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2
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Wigner transform of $O_1 O_2$ in terms of Wigner transforms of $O_1$ and $O_2$?
The Wigner-Weyl transform of a quantum operator $O$ is defined as
$$
W[O](q,p) = 2 \int_{-\infty}^{\infty} dy\ e^{- 2 i p y} \langle q + y | O | q - y \rangle \ dy
$$
and then given a density matrix $\...
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Newtonian generalisation of binomial theorem for commuting operators
Suppose $\hat{A}$ and $\hat{B}$ are commuting operators on some Hilbert space, $[\hat{A},\hat{B}]=0$, chosen suitably to ensure operators $1/\hat{A}$, $1/\hat{B}$ and $\hat{A}/\hat{B}$ are suitably ...
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Dirac's definition of probability in quantum mechanics
I'm currently reading "The principles of quantum mechanics" by Dirac, and I'm having some trouble understanding some of his assumptions, because in the quantum mechanics course I'm following ...
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1
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Problem with time reversal operator
I am trying to understand the time reversal operator. As I read it is defined as
$$
\mathcal{T} = \mathcal{U}\mathcal{K},
$$
where $\mathcal{U}$ is a unitary operator and $\mathcal{K}$ is a complex ...
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Deriving $\hat{p}_ x =-iℏ∂/∂x$ [duplicate]
(Taken from "The meaning of Quantum Theory, from Jim Baggott).
Starting from [x,^Px]= iℏ, if we chose the position operator to be simply 'multiplication by x', this forces the linear momentum ...
0
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1
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How to Derive the Minimum Time for $⟨ψ_0|A(t)|ψ_0⟩$ to Vanish Using Uncertainty Principle?
I am trying to work through some bookwork, and I am stuck on the titular question. The scenario is as follows:
In a generic finite dimensional Hilbert space $\mathcal{H}$, define the operator $A(t) ≡ ...
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Time reversal operator on conduction band edge function
The bottom of the conduction band of Zinc Blende structure with grand representation $T_{d}$ symmetry group have a 2-dimensional representation at $k=0$ also known as the center of the Brillouin zone ...
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0
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51
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Commutation relation of position and Hamiltonian operator [duplicate]
In the book Quantum Mechanics volume 2 by Cohen-Tannoudji, in Electric dipole approximation, it was written that $\left[\boldsymbol{Z}, H_0\right]=i \hbar \frac{\partial H_0}{\partial P_{\boldsymbol{z}...
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Is there an identity for $\hat{a}^+ \cdot \hat{a}^+$?
Is there an identity for the square of the ladder operators, $\hat{a}^+$ and $\hat{a}^-$?
4
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1
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Time-ordered matrix exponential quasi-static limit
Define a matrix differential equation $$\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}...
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3
votes
2
answers
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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:
$$\phi^{+}\left(x\right) = \int \frac{d^3p}{\...
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2
answers
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Can $\langle0|(\hat{a}-\hat{b})|0\rangle$ be written as $\langle0|\hat{a}|0\rangle-\langle0|\hat{b}|0\rangle$?
I am studying quantum mechanics and dirac notation, and I am wondering if, given the operator $\hat{A}=\hat{a}-\hat{b}$, the expectation value $\langle0|\hat{A}|0\rangle$ can be written as $\langle0|\...
4
votes
3
answers
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Where does this expression for infinitesimal rotations come from?
In Chapter 2, page 11 of Preskill's quantum computing notes, he mentions without explaining that a counterclockwise inifinitesimal rotation by $d\theta$ about the axis $\hat{n}$ is given by
$$R(d\...
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votes
3
answers
85
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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 ...
- 6,377
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2
answers
285
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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 ...
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