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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1answer
89 views

When to use a Kronecker sum vs a tensor product of Hamiltonians?

Let $H_1$ and $H_2$ be Hamiltonians on Hilbert spaces $\mathcal{H}_1$ and $\mathcal{H}_2$. My question in general concerns how one would form a Hamiltonian on the tensor product Hilbert space $$\...
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25 views

Do time-invariant self-adjoint operators have locally orthogonal eigenfunctions?

Cross-posting from the Math stackexchange, because I haven't gotten any replies there. Let $T$ be some self-adjoint, time-invariant (in that it commutes with any shift) operator on $L^2(\mathbb{R})$. ...
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2answers
161 views

Taylor expansion of time evolution operator

Given the definition of an exponential of an operator $$e^{\hat{O}}=\sum_{k=0}^{\infty}\frac{\hat{O}^{k}}{k!}$$ the expansion of the time evolution operator $\hat{U}=e^{-\frac{i\hat{H}t}{\hbar}}$ is $\...
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82 views

What is the interpretation of $\pi(x) |0\rangle $?

In the second chapter of Peskin and Schroeder An Introduction to Quantum Field Theory after second quantization of the Klein-Gordon field we arrive at $$|p\rangle =\sqrt{2 E_p} a^{\dagger}_{p}\ |0\...
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953 views

Inverse Laplacian

I have seen the following operator somewhere in a paper on cosmology $$ \frac{\partial_i \partial_j}{\nabla^2} - \frac{1}{3} \delta_{ij}. $$ What is the definition of the inverse Laplacian? What is ...
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42 views

Using commuting operators to express states

I read that it is natural to use commuting operators to express physical state-vectors in QM. For example, if the energy momentum four vector operators all commute, it is natural to express state-...
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1answer
108 views

Conflict between Bra-Ket notation and Integration

Suppose, I have a wavefunction given by $\psi(x,t)$. This wavefunction, over time, becomes $\psi(\alpha x,t)$. I've been asked to compute the final kinetic energy of this new wavefunction, in terms of ...
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30 views

Acting with a quantum operator

I have a very basic question that might sound silly. But I have noticed that in some cases when we act with a quantum operator, say $\hat{A}$ on some state, say $\rho$, we sometimes just write: \begin{...
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1answer
84 views

Is spectrum of Hamiltonian all you need?

This should be well-known, but I don't seem to know it... Quantum mechanics is defined by a Hamiltonian, and a Hamiltonian (as any Hermitian operator) is determined by its spectrum. Hence, it seems as ...
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45 views

Proof of Time-Energy-Uncertainty $\Delta H\frac{\Delta A}{|\frac{d}{dt}\langle A(t)\rangle|}\geq\frac{\hbar}{2}$

I am interested in mathematically proving the time-energy uncertainty relation by Mandelstamm-Tamm in a system seen in the Schrödinger-Picture (time-dependent states but time-independent operators). ...
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1answer
57 views

Matrix Representation of an operator in Non-Orthogonal bases [closed]

Given that a set of orthonormal basis $\psi$ is constructed by a set of non-orthogonal basis $\chi$ by the following relation: $$\psi_{\mu}=\sum^{\mu}_{i=1}t_{i\mu}\chi_{i} \tag{1}$$ The matrix ...
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66 views

Confused with the velocity operator

The velocity operator is defined as the commutator of the position operator and the Hamiltonian $$ \mathbf{v} = -\frac{i}{\hbar}[\mathbf{r},H] $$ Say $H$ is a crystal Hamiltonian with eigenstates of ...
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How to ascertain entanglement in Heisenberg picture?

It appears to me that the definition of entanglement explicitly refers to the state of the system in the Schrodinger picture, i.e., if a system $\psi\in\mathcal{\otimes_i\mathcal{H}_i}$ is such that $\...
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39 views

Continuum bosonic commutation relation with an additional coefficients

In some papers in quantum optics involving the structure of continuum, their commutation relations of the creation and annihilation operators are a bit weird. In the paper https://journals.aps.org/pra/...
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1answer
82 views

Mathematically, why aren't the creation and annihilation operators Hermitian? [closed]

Take for instance the one-dimensional creation operator $\hat a$: $$\begin{equation} \hat a = \sqrt{\frac{m\omega}{2\hslash}}\left(\hat x + i \frac{\hat{p}_x}{m\omega}\right) \tag{1}\label{1} \end{...
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What are the physical dimensions of the photonic plane wave operators in the multimode regime?

From this question, it appears that the creation/annihilation operators have units "$1$", i.e. they are of unit dimensionality. However, in the multimode regime, the commutator of two ...
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38 views

Sum of integrals for time-ordered products

Say I have a sum of $n$ time-ordered products in the following way: \begin{eqnarray} \hat{T}\left[A(t_{1})O(t_{2})...O(t_{k})...O(t_{n})\right] +...+\hat{T}\left[O(t_{1})O(t_{2})...A(t_{k})...O(t_{n})\...
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1answer
51 views

Deriving Uncertainty Relation for Canonical Commuting Observables from Schrödinger Uncertainty Relation

In case the names are not standard: \begin{equation} \sigma_{\hat{A}}^{2}\sigma_{\hat{B}}^{2} \geq \left\vert \frac{1}{2i} \left\langle \left[\hat{A},\hat{B}\right] \right\rangle \right\...
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50 views

Function of noncommutative operators: how should the powers in its Taylor expansion be arranged, and how to take partial derivatives?

Let $F:\mathbb R ^n\to\mathbb R$ be a function that has a Taylor expansion, then it can be written (expanded at $a$) as $$ F(x)=\sum_{\alpha} \frac{(x_1 - a_1)^{\alpha_1}\dots(x_n - a_n)^{\alpha_n}}{\...
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2answers
83 views

How to understand the time reversal symmetry of position operator?

How to understand the fact that position operator is symmetric under time reversal? I can visualize the momentum and magnetic field being odd under time reversal. Got the same doubt for Electric field ...
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Contradictory statements in Dirac's "Principles of Quantum Mechanics"?

At the end of Section 9 in Dirac's Principles of Quantum Mechanics (p. 34), there is a sentence that is very confusing to me. I am hoping that someone can explain whatever it is that I am missing. The ...
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How to work with the denominator of the Feshbach Projection

I have been reading this paper and am quite confused about the Feshbach projection method they introduce on page 3. They calculate an effective Hamiltonian on a subspace of the system using projection ...
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861 views

Physical interpretation of Uncertainty

Uncertainty of an operator $\hat{A}$ when observing a state $|\psi\rangle$ is defined as \begin{equation} \Delta A_{\psi} = |(\hat{A} - \langle\hat{A}\rangle)\psi| \end{equation} Now assume that there ...
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1answer
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Heat kernel expression for index calculation/Nakahara Exercise 12.5

A quick question. I was reading this paper, on the last page, the authors wrote $$ \lim_{\mu^2\rightarrow\infty} \text{Tr} \int_0^\infty dT\ e^{-T} \Big(e^{-(T/\mu^2)LL^\dagger}-e^{-(T/\mu^2)L^\dagger ...
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Why are vertex operators integrated over the worldsheet?

In chapter $3$ of Polchinski after discussing why vertex operators are used for preparing states in S-matrix. We are given the vertex operator for closed string tachyon is $$V_0=2g_c\int d^2\sigma\...
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1answer
119 views

What does this bra-ket equation mean? [closed]

I was reading Griffiths Introduction to Quantum Mechanics and I came across this equation: $$\langle \psi_a |z|\psi_b\rangle$$ Where z is direction. I don't understand why there are those subscripts a ...
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Doubt about property of hermitian operator

For any hermitian operator M, prove that \begin{equation} \langle Ma|b \rangle = \langle a|Mb \rangle \end{equation} My attempt: Let \begin{eqnarray} \langle a| = \sum_i a_i^*\langle i|\\ |b\rangle = \...
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1answer
33 views

Doubt in a commutator relation

There is a proof in J.J Sakurai Mordern Quantum Mechanics page 30 which says Suppose that A and B are compatible observables, and the eigenvalues of A are nondegenerate. Then the matrix elements $\...
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1answer
83 views

What is the proper translation of a field operator?

I am trying to write the correct expression for a translated quantum field operator. There appear to be conflicting expressions given in this PSE post, and this one. In the former linked PSE post, the ...
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139 views

Noncommutativity and Quantum Fluctuations

I have been reading a book called Quantum Phase Transitions in Transverse Field Spin Models: From Statistical Physics to Quantum Information. Part I on this book gives an introduction to Quantum Phase ...
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1answer
43 views

Eigenvectors of commuting hermitian operators?

Didn't know if this belonged here or on the maths StackExchange, let me know if I should switch over. Currently going through a quantum mechanics class and I'm reading the following theorem (...
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How to prove $(A|\Psi\rangle)^{\dagger}=\langle\Psi|A$ $ $?

In the context of quantum mechanics we postulate that every observable operator $A$ acting on the corresponding Hilbert space $\mathcal{H}$ is self-adjoint (Hermitian), i.e. $$\forall \Psi,\varphi\in\...
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3answers
156 views

Field operators and their Fourier transform

In "QFT for the gifted amateur" page 37, the author defines the field operators: $$ \hat{\psi}^\dagger(x)=\frac1 {\sqrt V} \sum_p{\hat a ^\dagger _p e^{-ipx}} \tag{4.7}$$ $$ \hat{\psi}(x)=\...
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Is the Hamiltonian the only quantum observable with a mixed spectrum?

Let $\mathscr{H}$ be a complex separable Hilbert space of a quantum system. Assume that the Groenewold-van Hove no-go theorem did not necessarily apply and we are free to map all possible polynomial ...
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1answer
59 views

What measurable quantity is associated with parity?

In quantum mechanics, we learn that for any Hamiltonian with a symmetry, there exists a unitary operator associated with that symmetry. Consider the parity operator which is defined by its operation ...
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89 views

Matrix elements from commutation relations

Suppose we are given commutators of the spin operators: $[S_X, S_Y], [S_Y, S_Z]$ and $[S_Z, S_X]$. Then can we completely determine the matrix representation of the operators? Can we do it in any ...
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1answer
33 views

Lifting degeneracy when operators commute

What is meant by the term "lifting of Degeneracy"? I have been told in my class that if suppose and operator $A$ has some degeneracy, and we have another operator $B$ such that $[A,B] = 0$, ...
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How does the electromagnetic dipole operator appear in the decay $b \rightarrow s \gamma$?

I was analyzing the effective theory of the process $b \rightarrow s \gamma$ and and I was in doubt about the emergence of the effective operator of photon dipole $$O_7 = e m_b \bar{s}_L \sigma_{\mu\...
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98 views

Commutator of spin operators

Suppose we are given $\left[S_X, S_Y\right]$, $\left[S_Y, S_Z\right]$ and $\left[S_Z, S_X\right]$, that is the spin operator commutation relations, can we then determine the matrix representation of ...
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1answer
66 views

Why does $a^\dagger a a^\dagger \left|n\right\rangle = a^\dagger \left|n\right\rangle + a^\dagger a^\dagger a \left|n\right\rangle$?

$\def\textdagger{\dagger}\def\ket#1{\left|#1\right\rangle}$I'm reading through the Schwartz text on QFT & the Standard Model. I've found myself bogged down already in section 1.3! In a derivation ...
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What happens when you replace an identity matrix with a matrix full of ones?

In physics, we often use resolutions of identity $$\sum_n |n\rangle\langle n|=\mathbb{I}$$ to simplify expressions. Sometimes, the "full matrix" (for lack of a better term) $$\sum_{m,n}|m\...
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1answer
132 views

Sum of commutator and Anticommutator

Suppose $A$ and $B$ are Hermitian Operators. Then what will be the nature (purely real/purely imaginary/complex number of form $a + ib$, $a,b \in \mathbb{R}, a, b \neq 0$) of eigenvalues of $[A, B] + ...
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2answers
309 views

Eigenvalues of Product of 2 hermitian operators [closed]

Let $A$ and $B$ be two Hermitian operators. Let $C$ be another operator such that $C = AB$. What can we say about Eigenvalues of $C$? Will they be real/imaginary/complex? What I did was to search for ...
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1answer
94 views

If $\big[[\hat{A},\,\hat{B}],\, \hat{A}\big] = 0$ does that mean $\hat{A}$ and $\hat{B}$ commute?

Let's say we have the following identity: $$\Big[\big[\hat{A},\,\hat{B}\big],\, \hat{A}\Big] = 0.$$ Expand the LHS: \begin{align} \Big[\big[\hat{A},\,\hat{B}\big],\, \hat{A}\Big] &= 2\hat{A}\hat{B}...
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Is the evolution operator $U(t)$ unitary in quantum mechanics? [duplicate]

The Schrodinger equation: $$ i \frac{d}{d t}\left|\psi_{t}\right\rangle=H\left|\psi_{t}\right\rangle $$ and if $H$ does not depend of time the solutions are given by $$\left|\psi_{t}\right\rangle=...
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1answer
86 views

Operators in Heisenberg and Schrodinger pictures [duplicate]

I don't understand the difference between the Schrödinger picture and the Heisenberg picture in quantum mechanics. Here's some of my doubts: If in the Heisenberg picture state vectors are constant in ...
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86 views

Why does metaplectic correction fix the vacuum energy?

In geometric quantization we want to go from a symplectic manifold $\left( M, \omega \right)$ to a Hilbert space $H$. If $M$ is prequantizable, we find a prequantum bundle $L \rightarrow M$ with ...
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1answer
66 views

Is occupation number of a certain quantum state an observable? [closed]

I know that the number operators are projective. Can I use the number operator for measurement if it is projective? Can I use it if I want to measure the number of particles (fermions) in a certain ...
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6answers
534 views

What is meant by the "components" of operator?

So far, I have understood that vectors are represented in a basis, and operators are linear maps which map one vector to another in the same space or to a different space. What I don't understand is ...
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1answer
51 views

Polchinski OPE of spacetime translation current

I am trying to derive $$ j^\mu(z):e^{ik\cdot X(0,0)}: \;\sim \frac{k^\mu}{2z}:e^{ik\cdot X(0,0)} \tag{2.3.14a} $$ from Polchinski's String Theory vol.1 equation (2.3.14a). using $j^{\mu}=\frac{i}{\...

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