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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Simple question about the identity operator in quantum mechanics
This might be a very trivial question, but I'll ask it anyway: consider a self-adjoint operator $B$, such that for each mode $a_1,...,a_n$ of a quantum bosonic system we have $B a_i B^\dagger = a_i$. ...
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Mathematical tools in quantum mechanics [closed]
Prove that unitary transformation transforms one complete set of basis vectors into another.
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Functional analysis question about operator on quantum wave functions
If I have two time-independent wave functions $\psi_{t_{1}}$ and $\psi_{t_{2}}$ and define an operator $\hat{U}$ such that $$\psi_{t_{2}} = \hat{U}_{t_{1},t_{2}}(\psi_{t_{1}})$$ and $$\psi_{t_{2}}(x) =...
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Where does the complex conjugate term generally come from in a Hamiltonian?
I find myself stumbling across Hamiltonians which go like
$$
\hat{H}\sim\alpha\hat{a}+\alpha^*\hat{a}^\dagger
$$
How does this form of Hamiltonian actually come about?
To my knowledge, the Hamiltonian ...
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Determining Bound States from Møller Operator
Hello I came across an interesting property of the Møller operator, which I summarize below:
The Møller operator $\Omega^{(+)}$ maps in-states that belong to the continuum spectrum of the free ...
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Why does the Pauli objection not disqualify the existence of the position operator?
According to the Pauli objection (see for example here or the answer to this question) there can be no time operator $\hat{T}$ canonically conjugate to the Hamiltonian $\hat{H}$ of a physical system ...
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Equivalent definitions of Wick ordering
Let $\phi$ denote a field consisting of creation and annihilation operators. In physics, the Wick ordering of $\phi$, denoted $:\phi:$, is defined so that all creation are to the left of all ...
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Expectation Value of Momentum Question [closed]
I recently started learning QM and am currently learning about how to use operators to find expectation values.
I recently received a question concerning this and have been struggling to solve it. I ...
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How to prove if ${\hat A}^{\dagger}{\hat A}\le {\hat 1}$ then ${\hat A}{\hat A}^{\dagger}\le {\hat 1}$ for any square matrix ${\hat A}$? [migrated]
My note says that for any square matrix ${\hat A}$, if ${\hat A}^{\dagger}{\hat A}\le {\hat 1}$, then ${\hat A}{\hat A}^{\dagger}\le {\hat 1}$, but I don't know how to prove this. Can anyone help me ...
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Is there any difference between Wick time order and Dyson time order?
Reading A Guide to Feynman Diagrams in the Many-Body Problem by R. Mattuck, I am getting the feeling that I missed something subtle related to time order. When deriving the Dyson series for the ...
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What are single-, double- and multi-trace operators in AdS/CFT?
Can someone explain what are single-, double- and multi-trace operators are in AdS/CFT? I am a senior undergrad and only recently started studying AdS/CFT from TASI lectures and could not make much ...
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Picture Number in String Vertex Operator
How can I know what is the Picture of a particular vertex operator?
For example in 8.3.15 in Polchinski's book Vol.1, the Vertex Operators for the Enhanced Gauge symmetry are given by
\begin{equation}...
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Pauli matrix exponentials [closed]
Just a short query to confirm my understanding.
Given the Pauli-X operator $\hat{X}$ and it's eigenstates $|+\rangle:=\frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$ and $|-\rangle:=\frac{1}{\sqrt{2}}(|0\...
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Is this a valid way to write Schwinger's Action Principle? [closed]
δS = ±iħδ
I added the + sign for situations where the Action has maxima or the mass is negative. I am specifically wondering if this form without the brakets is valid so that a Calculus of Variations ...
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Integration by parts to switch sign on anti-Hermitian Louivillian
I am self studying non-equilibrium stat mech and a common theme is that I am unable to reproduce the "this follow from integration by parts" parts of derivations. Here is the most recent:
In ...
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Is there a problem with the commuting axiom in QFT? [closed]
All existing quantum field theories include the axiom that fields that are space like separated commute with each other. This also means that a field at one point commutes with the future light cone ...
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Choice of spacetime foliation while quantising a conformal field theory
I was reading Rychkov's EPFL lectures on $D\geq 3$ CFT (along with these set of TASI lectures) and in chapter 3, he starts discussing radial quantisation and OPE (operator product expansion). I ...
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Exercise on self-adjointness of Hamiltonian [closed]
I am struggling with some exercise I have to solve for my quantum mechanics class.
PROBLEM:
Suppose $|\psi\rangle, |\phi\rangle$ are normalised and linearly independent (but not necessarily ...
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How do we know at the operator-level that the tadpole $\langle\Omega|\phi(x)|\Omega\rangle=0$ vanishes in scalar $\phi^4$ theory?
I'm a mathematician slowly trying to teach myself quantum field theory. To test my understanding, I'm trying to tell myself the whole story from a Lagrangian to scattering amplitudes for scalar $\phi^...
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Extracting the dimension of an operator from algebra
I may misinterpret the question. In the lecture note of conformal field theory, arXiv:2207.09474, it says the following
where for $P^\mu=i\partial_\mu$ and $D=ix^\mu \partial_\mu$. I am confused ...
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Dictionary between interpretations of field operators
For now, let $\hat{\phi}(x)$ be a quantization of a classical, real scalar field $\phi(x)$.
My understanding is that, for fixed $x$, there are three ways to regard the operator $\hat{\phi}(x)$:
The ...
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Total momentum operator of the Klein-Gordon field (before limit to the continuum)
I'm following K. Huang's QFT: From Operators to Path Integrals book. In the second chapter, he introduces the Klein-Gordon equation (KGE), and its scalar field $\phi(x)$, which satisfies this equation....
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Ladder operators and creation & annihilation operators - different between $a$, $b$ and $c$ [closed]
Usually, the ladder operator denoted by $a$ and $a^\dagger$. In some case, people talk about the creation operator and denote it by $c$ and $c^\dagger$. Recently I see another notation, $b$ and $b^\...
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Displacement operator and phase space
I read a paper on open quantum system, it's about non-Markovian process with memory effects.
They describe a generic model of two qubits interacting with correlated multimode field.
They describe the ...
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Where does the "arbitrary constant" in the $L_{0}$ Virasoro operator come from?
In the 2007 "String Theory and M-Theory" textbook by Becker, Becker, Schwartz there is the following claim about the canonical first quantization of a bosonic string: the quantization of the ...
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How to find the expectation value of momentum operator? [closed]
I have struggled with the following steps in finding the expectation value of the momentum operator $\hat{p}$.
$$\left \langle \hat{p} \right \rangle=\int_{-\infty}^{\infty}\psi^{*}(x)\hat{p}\psi{x}dx=...
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Generator of two-qubit quantum gate
I would like to know how to derive the explicit form of the GENERATOR of a general two-qubit gate (also here), e.g., controlled-rotation Y.
From the definition: $$\exp(-i\theta G) \ ,$$
I see it is: $$...
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Wigner's formula for the kinetic energy density in QM
In the Schroedinger equation the kinetic energy is represented by the operator $T = -\frac {\hbar^2} {2m} \Delta$ which acts on a wavefunction $\Psi$. If we multiply this by the complex conjugate of ...
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Why do QM books point out that $S^2$ commutes with $S_x$, $S_y$, and $S_z$?
The spin angular momentum magnitude squared operator:
$$S^2=S_x^2+S_y^2+S_z^2=\frac{3\hbar^2}{4}
\begin{pmatrix}1&0\\0&1\end{pmatrix}$$
Obviously $S^2$ commutes with everything, so why do QM ...
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What can we say about the eigendecomposition of quantum channels?
It is known that quantum channels, being CPTP maps, map density operators to density operators. And thus, they can be seen as superoperators. Similar to operators, where eigenstates and eigenvalues ...
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Spin 1 operator and photon
$\newcommand{\ket}[1]{| #1 \rangle}$
The general spin formulation is the same as the cinetic momentum :
\begin{equation}
[S_{x},S_{y}]=i\hbar S_{z},\quad [S_{y},S_{z}]=i\hbar S_{x},\quad
[S_{z}, S_{x}]...
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Generator of time shift when the Hamiltonian is time dependent
Let's consider the unitary group $\hat{S_{\tau}^†}$ such that :$$\hat{S^†_{\tau}}|\psi(t)\rangle=|\psi(t-\tau)\rangle$$
Since we know that:
$$\hat{U}(t,t_0)|\psi(t_0)\rangle=|\psi(t)\rangle$$
Where ${...
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Introduction of symmetries in quantum mechanics
The (Italian) book that I am currently reading introduces the topic of symmetries in quantum mechanics in the following way:
Let O and O' be two distinct observers and let $A$ and $B$ be two ...
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Confused about square of time-reversal operator $T$
I am reading An Introduction to Quantum Field Theory by Peskin & Schroeder, and I am confused about what is the square $T^2$ of time reversal operator $T$.
My guess is that for $P^2$, $C^2$ and $T^...
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Understanding the basics of second quantization [duplicate]
I am new to quantum field theory and I am trying to understand how to work with quantum field operators and the notations that are used here.
Context:
Assume a hamiltonian with operator: $\hat{W} = t\...
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What does the identity operator mean in Quantum Mechanics? [duplicate]
I'm new to quantum mechanics, and I am beginning to study Dirac notation, but I do not understand the significance or meaning of the following equation:
$$\sum_n\left|e_n\right\rangle\left\langle e_n\...
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Azimuthal coordinate operator: Hermition or not? Self-adjoint or not?
I am told that the azimuthal coordinate operator $\hat{\phi}$ is not self-adjoint. I am told this by people who I am sure know much more about this stuff than I do. To my unsophisticated mind, "...
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Maximum value of angular momentum component in quantum mechanics [closed]
Since $[L^2, L_z]=0$, we can say that they share a common eigenbasis, call it $f$, and $L^2f=\lambda f$, $L_z f=\mu f$
The ladder operators for the $z$ component of angular momentum are
$$L_\pm=L_x\pm ...
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Photon and qubit representation
The photon spin-1 has two states, $\pm\hbar$, just like the spin qubit ($\pm\frac{\hbar}{2}$).
From a quantum information point of view, they can encode the same amount of data.
However, I am confused ...
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Property of the Hamiltonian's discrete spectrum
I have found a statement online saying that there must be an eigenvalue of the Hamiltonian inside the range $(E-\Delta H,E+\Delta H)$. Where the mean value and variance are defined for a random (...
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An exercise to find the expectation of quantum states [closed]
Given $\vert\psi(0)\rangle=\vert 0 \rangle$,
$ H =\hbar\omega(\vert 0\rangle \langle 1 \vert+\vert 1\rangle \langle 0 \vert) $,
$p=\vert 0\rangle \langle 0\vert$,
ask for $\langle \psi(t)\vert p \vert ...
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Some questions about derivation of uncertainty principle
In Introduction to Quantum Mechanics by Griffiths and Schroeter, they derive the Uncertainty principle in the following way:
First, they define
$$f=\left(\hat A-\langle A\rangle\right)|\Psi\rangle$$
$$...
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What does the Jacobi identity *mean* statistically?
Given that the commutator of a pair of operators shows up explicitly in the lower bound of the Robertson-Schrodinger inequality, I am wondering what, if any, statistical meaning/significance one can ...
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Conmutators and Jacobi's Identity
I've come across an exercise asking me to calculate:
$$[[A,B],[C,D]]$$
knowing $[A,C]=[B,D]=0$ and $[A,D]=[B,C]=1$
I've already solved it by "brute force", separating the commutator as ...
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Time Evolution of Eigenkets in the Heisenberg picture
I'm reading Modern Quantum Mechanics by Jun John Sakurai and in section 2.2 he talks about Base Kets and Transition Amplitudes. He goes to show, that $|a',t\rangle=\mathcal{U}^\dagger|a'\rangle$, (...
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Where is my mistake in using a measurement operator instead of Born’s rule to calculate the probability of detecting photons at an arbitrary angle?
As I asked in this question: https://quantumcomputing.stackexchange.com/questions/36998/how-can-i-calculate-the-measuring-probabilities-of-a-two-qubit-state-along-a-cer/37000#37000
From here I know ...
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Momentum probability density and its normalization
Let the (normalized) wave function $\Psi(x,y)$ represent a free particle in the XY plane. I know $|\Psi|^2$ gives me the probability density function of the particle's position, which I can then ...
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1
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On properties of open quantum dynamics and Lindbladians
It is well known that the open quantum dynamics is governed by the Lindblad master equation
$$\partial_t{\rho}=\mathbb{L}(\rho)=-\frac{i}{\hbar}[H, \rho]+\sum_i \gamma_i\left(L_i \rho L_i^{\dagger}-\...
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Measure of connectivity in Graph States
Given $G = (V,E)$, with the set of vertices $V$ and the set of edges $E$, the corresponding graph state is defined as
$$|G\rangle = \prod_{(a,b)\in E} U^{\{a,b\}} |+\rangle ^{\otimes V}$$
where the ...
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Derivation of the quantization of the EM field in a dielectrics
I'm currently studying the quantization of the EM field in a dielectric medium and trying to understand the quantization scheme of Huttner and Barnett (1992, see Phys. Rev. A 46, 4306). The system ...
