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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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What is a singular continuous spectrum?

I read some answers about this and the wikipedia page that basically always say that a spectrum can be decomposed into: $$\mu = \mu_{ac} + \mu_{sc} + \mu_{pp}, $$ where $\mu_{ac}$ is absolutely ...
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3answers
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Why doesn't the anticommutator $\{x,p_x\}$ have an unique value?

The commutator of position and momentum, $[x,p_x]$, has a unique value given by $i\hbar$. Why doesn't the anticommutator $\{x,p_x\}$ also have a definite value?
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How to prove $e^{it\hat X}=e^{it\hat Y}+\int_0^te^{i(t-\tau)\hat X}i(\hat X-\hat Y)e^{i\tau\hat Y}d\tau$ where $\hat X$ and $\hat Y$ are operators?

When deriving the generalized Langevin equation with Mori-Zwanzig formalism, I was taught that one identity should be used, that is, $$e^{it\hat X}=e^{it\hat Y}+\int_0^te^{i(t-\tau)\hat X}i(\hat X-\...
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Total angular momentum operator on $|m\rangle$ and $|m-1\rangle$ results in different eigenvalue [duplicate]

In the lectures by Prof. Leonard Susskind, he mentioned that the total angular momentum squared operator can be represented by $$ L^2 = L_z^2 + L_z + L_- L_+ $$ ($L_+, L_-$ being ladder operators). ...
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2answers
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Can the Hamiltonian operator act on a bra, if it was once acting on a ket?

I was watching a MIT Quantum Physics III class when I got a doubt about a specific bra-ket manipulation. My doubt is about the step from the expression $(3.7)$ to the expression $(3.8)$ of the lecture ...
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0answers
59 views

How to understand the quantum operators [on hold]

I am reading a research article based on quantum image watermarking, Sang, J., Wang, S. & Li, Q. Quantum Inf Process (2016) 15: 4441. https://doi.org/10.1007/s11128-016-1411-z The authors ...
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2answers
92 views

Mathematical formulation of quantum mechanics

I am reading a book on quantum mechanics, but it is difficult to understand. Quantum mechanics is roughly formulated as follows: Physicsl state is a normalized ray $\{e^{i\theta}\psi|\theta \in \...
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1answer
46 views

Wigner map of the product of two operators

Does anyone know how to prove that for the product of two operators $\hat{A}\hat{B}$ the Weyl-Wigner correspondence reads $$ (AB)(x,p) = A\left (x-\frac{\hbar}{2i}\frac{\partial}{\partial p}, p+\frac{\...
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1answer
37 views

Difference on the invariance of operators and their transformations under unitary operators

I am confused about, what I believe, refers to passive and active transformations in QM. What I have understood so far is that the matrix elements $\langle \psi| \hat{H}|\phi\rangle$ should remain ...
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Ladder operators in quantum mechanics [closed]

How do ladder operators work? And how exactly do they interact with the Hamiltonian operator?
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Find the function $G(x,y,t,t_0)$ such that $\psi(x,t)=\int_{-\infty}^{\infty}dy G(x,y,t,t_0) \psi(y,t_0)$ is the time evolution equation [closed]

The problem I know that if the Hamiltonian $\hat{H}$ is independent of time, then the time evolution operator $\hat{T}(t,t_0)$ has the form $$\hat{T}(t,t_0)=e^{-\frac{i}{\hbar}(t-t_0)\hat{H}} \, .$$ ...
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Why is the definition of sin and cos in terms of exponentials is similar to the definition of $L_x$ & $L_y$ in terms of raising & lowering operators? [closed]

The angular momentum operators in $x$ and $y$ direction can be written \begin{align} L_x &= \frac{1}{2}(L_++L_-) \\ L_y &= \frac{1}{2i}(L_+-L_-) \, . \end{align} Similarly, $\cos(x)$ and $...
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How can I prove this relationship between the S-matrix and the Gamma matrices?

For an infinitesimal Lorentz transformation: $$ S(\Lambda)=1+i\epsilon_{\rho\sigma}s^{\rho\sigma} $$ $$ S(\Lambda)^{-1}=1-i\epsilon_{\rho\sigma}s^{\rho\sigma} $$ and apparently if we just plug that ...
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1answer
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Non-commutative Fourier transform of an operator

Wigner-Weyl transform relates an operator to its distribution function in phase space through an operator Fourier transform which is said to be non-commutative. $$ \hat{\rho} \xrightarrow[non-comm]{...
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2answers
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About superposition of states

In quantum computing, we can always create an arbitrary superposition of states by rotation of $|0\rangle$ state for one qubit. This raises a question: for arbitrary superposition of states, is there ...
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0answers
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Bivector Gravity and d'Alembertian [on hold]

Just wanting to check with the community if they can spot anything wrong with the following. First of all, let's quickly rehash the main equation, I arrived at independently for a construction of ...
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1answer
102 views

Normal ordered products of operators and inverses

I have been reading this paper ("Operator ordering in quantum optics theory and the development of Dirac’s symbolic method" by Hong-yi Fan), and on page 3 (right-hand column) the author writes that $:...
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1answer
40 views

Variance of a hermitian operator

Take an hermitian operator $O$ such that $O|\psi\rangle = x|\psi\rangle$. The variance of an operator $O$ is defined as $$ (\Delta O)^2 = \langle{O^2}\rangle - \langle{O}\rangle^2.$$ Let's consider ...
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In the quantum hamiltonian, why does kinetic energy turn into an operator while potential doesn't?

When we go from the classical many-body hamiltonian $$H = \sum_i \frac{\vec{p}_i^2}{2m_e} - \sum_{i,I} \frac{Z_I e^2 }{|\vec{r}_i - \vec{R}_I|} + \frac{1}{2}\sum_{i,j} \frac{ e^2 }{|\vec{r}_i - \vec{...
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1answer
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Implication of the Jacobian map for the structure of the Euclidean space-time

I'm listening to Alain Connes "On the Fine-Structure of Space-Time" around minute 23 saying that it was disappoing that the solution Y to the equation $$ <Y[D,Y]^{2m} >= \gamma $$ with $D$ a ...
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2answers
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Initial values of creation/annihilation operators

I have a question about creation/annihilation operators. For example, if I have an evolution equation for annihilation operator of photon $$ \frac{da_k}{dt} = -i \omega_k a_k$$ I obviously obtain $$...
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2answers
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Relation between giving the form of an operator in a given representation, and bra ket notation [on hold]

So I understand that kets are abstract objects that are the elemnets of a Hilberts space. Say $|\psi \rangle$. We can write this ket in a position representation $\langle r|\psi \rangle = \psi(r)$, ...
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Doubt regarding the angular momentum in quantum mechanics

What does $L^2|l,m\rangle$ indicate? Can anyone specify the 2 states in a ket vector of angular momentum $|l,m\rangle$? $$L^2|l,m\rangle=\hbar^2(l+1)l|l,m\rangle$$ and how to differentiate between $\...
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1answer
50 views

Quantum expressions for the Virasoro constraints

I am trying to derive the quantum form of the Virasoro constraints. $$ L_{m} = \frac{1}{2} \sum_{n} :\alpha_{m-n}.\alpha_{n}: $$ :...: meaans normal ordering. Using the common commutator between ...
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1answer
59 views

How is expectation of $x^2$ at time $t$ calculated?

Ehrenfest theorem for position operator states $$\frac{d\left<x \right>}{dt} = \left<[H,x]\right> + \left<\frac{\partial x}{\partial t}\right>$$ where $H = \frac{p^2}{2m}$ and $\...
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1answer
46 views

Two spin-1 system and the projector onto total spin 2 subspace [on hold]

I am having trouble grasping the projection operators in the context of composite spins system, e.g. with two spin-1. First off, a projector $P$ is said to be an operator that squares to itself, $P^2=...
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1answer
39 views

Hausdorff expansion

Could someone explain me, what the Hausdorff expansion is? In my quantum mechanics homework I should use something called the Hausdorff expansion which looks like the following: $$e^ABe^{-A}=B+[A,B]+\...
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Transformations of contravariant and covariant tensor operators

I've been able to convince myself that a set of contravariant tensor operators $\hat{O}^{x}$ for $x=1,2,...,n$ respond to a small transformation $\hat{A}$ as, \begin{equation} [\hat{A},\hat{O}^{x}]=-(\...
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1answer
55 views

Expressing the vacuum projection operator in terms of number operator

I've been reading this book, in which the author expresses the vacuum projection operator $\vert 0\rangle\langle 0\vert$ in terms of the number operator $\hat{N}=\hat{a}^{\dagger}\hat{a}$, where $\hat{...
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1answer
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Definition of QFT in Vertex Operator Algebra by Kac

QFT is composed of the following data with some axioms(I omitted them here). (1) Hilbert space $H$. (2) Vacuum belongs to $H$. (3) There is unitary representation of Poincare group. (4) A ...
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1answer
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What is the meaning of the notation $\langle a_1, \ldots, a_n \mid X_i(u) \mid a_1', \ldots, a_n' \rangle$? [closed]

I am from the math department and reading Belavin & Gebner's On the Algebraic Approach to Solvable Lattice Models. I am trying to understand the left-hand side of Equation (2.2) on page 4. What ...
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Wick's theorem and bilinear Hamiltonian assumption

I read books (Mahan) talking about Wick's theorem is valid when the Hamiltonian is bilinear in creation and annillation operators. From the proof, for example see this post, we require that the ...
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1answer
95 views

Representation of operators and wavefunctions as matrices and vectors

I remember reading somewhere that in quantum mechanics you can always set up your Hilbert space to be finite or countably infinite. However, I don't see how it's possible to to that we want to ...
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1answer
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The charge given by a commutator

I saw in the text that $[Q,X]=cX$ and says the operator $X$ has charge $c$ under the generator $Q$. I tried to understand why the coefficient $c$ means the charge. So I used this relation to get the ...
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1answer
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Hermitian operator

When we say that an operator is Hermitian in QM, does it depend on the Hilbert space under consideration, or not? Are there operators that are Hermitian in one Hilbert space but not in another?
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Real symmetric matrix in Wigner's theorem

A consequence of Wigner's theorem is that if a Hamiltonian matrix obeys time reversal symmetry then it is real-symmetric. It seems to me that for this to make sense then "real symmetric" should be a ...
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1answer
94 views

Summation of an exponential operator on quantum amplitude

For a quantum Dirac field interacting with a classical EM field, one can (through the Quantum Dynamical Principle) write the vacuum transition amplitude as $$\langle0_+|0_-\rangle=\exp\left[ie_0\int ...
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1answer
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What is the commutator of a reasonably-behaved function of an operator and the derivative of that function? What is $[f(\hat x) ,f'(\hat x) ]$?

I tried to do the usual procedure and expand the commutator, but couldn't proceed after I Taylor-expanded $f(\hat x)$. $$\Big[f(\hat x),\frac{d}{dx}f(\hat x)\Big]=$$ $$f(\hat x)f'(\hat x)-f'(\hat x)...
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1answer
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Why do we think that the relation $\vec{\mu}_L=\frac{e}{2m_e}\vec{L}$ will be valid in quantum mechanics?

Assuming the electrons to revolve round the nucleus in circular orbits and using the fact from classical electromagnetism that a current loop behaves like a magnetic dipole of dipole moment $\vec{\mu}...
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1answer
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How to interpret $\left[f(A), B\right] = \left[A, B \right]\frac{\partial f}{\partial A}$ - how to differentiate with respect to an operator?

From here and here I know the commutation relation for two operators are: $$\left[f(A), B\right] = \left[A, B \right]\frac{\partial f}{\partial A}$$ if $\left[A, \left[A, B \right]\right] =0$ and $f$ ...
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1answer
366 views

General derivative of the exponential operator w.r.t. a parameter

I am interested in the calculation of the general $N$th derivative w.r.t. a parameter $\lambda$ of a quantum mechanical exponential operator with the following structure: \begin{equation*} \frac{\...
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0answers
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Hamiltonian ordering ambiguity in quantum cosmology/gravity

I am trying to study several quantum cosmology models. The standard procedure for quantization consists typically in several steps: People write the theory as an action, or Hamiltonian $H(p^i,q^j)$ ...
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1answer
79 views

What's the intuitive interpretation of quantum uncertainty $\Delta \hat{A}=\sqrt{\langle\hat{A}^2\rangle-\langle\hat{A}\rangle^2}$?

As per this video, if $\hat{A}$ is a quantum operator, the uncertainty is given by $$\Delta \hat{A}=\sqrt{\langle\hat{A}^2\rangle-\langle\hat{A}\rangle^2}$$ I understand what this expression means ...
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2answers
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Definition of the fully depolarizing quantum channel

The fully depolarizing quantum channel in a $d$ dimensional Hilbert space is defined by $$ \mathcal N^D (\rho) = \text{Tr}[\rho]\frac I d $$ I've seen that definition in several places but I don't ...
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1answer
37 views

What's the minimum condition for time evolution operator to be written as $U(t,t_0)=e^{-\frac{i}{\hbar} \int_{t_0}^t H(t') dt'}$?

Is $\frac{d }{dt} e^{H(t)}=H(t)' e^{H(t)}$ the minimum condition for time evolution operator to be written as $U(t,t_0)=e^{-\frac{i}{\hbar} \int_{t_0}^t H(t') dt'}$? Further, what's the minimum ...
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2answers
94 views

How does a Hamiltonian 'generate' a unitary?

I know that the unitary (propagator) is given by $$U=e^{iHt}\tag{1}.$$ But I actually never saw how a Hamiltonian translates into a unitary. For example when I consider a two-level rotation in a ...
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2answers
66 views

Computation of $e^{i \hbar \omega a^{\dagger} a} a e^{-i \hbar \omega a^{\dagger} a}$

I need to compute terms like : $$e^{i \omega t a^{\dagger} a} a e^{-i \omega t a^{\dagger} a}$$ Where $[a,a^{\dagger}]=1$ (they are the bosonic annihilation/creation operators). I wonder if there ...
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1answer
54 views

Potential must be real for Hamiltonian to be Hermitian?

I have seen a few proofs specify for finite wells, step functions, and harmonic oscillators, that $V$ must be real for $H$ to be Hermitian. Why is that? If we're solving the Schrodinger equation, we ...
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1answer
88 views

What is energy in quantum mechanics?

Is it wrong to say energy is the expectation value of Hamiltonian? Or should I say energy is the eigenvalue of Hamiltonian?
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2answers
77 views

Simple question on Angular Momentum

Need to know why $L^2$ and ONLY ONE of $L_x$, $L_y$, $L_z$ are constants of motion. Main problem arrives when $V = f(r, \theta, \phi)$ causing none of the $L_x$, $L_y$, $L_z$ to commute with ...