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 $\times$, $+$!
5,354 questions
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Gaussian integration with operator-valued sources and time ordering
I have a coherent-state path integral of complex scalar fields $\psi^+,\psi^-$ which I want to integrate out, but the linear terms are operators.
$$ \mathbb{P}(t) = \int D[\psi^+]D[\psi^-] \exp(S_0)T\...
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How do we define time-ordering operations in QFT?
Given two bosonic operators $A$, $B$ (in the Heisenberg picture) in a QFT, the time-ordered product of $A$ and $B$ is defined as
$$
T\{A(t_1)B(t_2)\}=\theta(t_1-t_2)A(t_1)B(t_2)+\theta(t_2-t_1)B(t_2)A(...
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In principle, do we have precise result of ensemble expectation value of interacting Hamiltonian? [closed]
We calculate Tr$(\rho H)$ using the eigenbasis $\sum \Pi a_n|0\rangle$(linear combination of all kinds of creation operator product acting a the vacuum state).
And $\langle0|\text{creation/...
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Is zero-point energy a mathematical artifact?
From Wikipedia, Pauli has said in his Nobel lecture that "It is clear that this zero-point energy has no physical reality". This feels natural - I've always been slightly puzzled by the ...
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Commuting operators and simultaneous eigenstates [duplicate]
If two operators commute, they share simultaneous eigenvectors. I understand the proof of this statement, however, I'm trying to gain physical insight as to what this means practically. When two ...
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Can any Foulis&Randall Test Space be equivalently formulated in terms of Yes/No Tests?
By test space I specifically mean this (RG) (see Section 5.1 for formal definition). A somewhat more thorough (and more specific) treatment is the article "Test Spaces" by Alexander Wilce, ...
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Do all operators which fulfil mathematical conditions in Quantum Mechanics correspond to an observable? [duplicate]
In my understanding of Quantum mechanics, which is demonstrably limited, Operators have corresponding observables. I imagine it like operators helping draw a graph, with probability on the y-axis and ...
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Path integral in quantum mechanics and expectation values [closed]
How to calculate any expectation value from path integral in quantum mechanics? In QM path integral the initial and final points are fixed and points between them are varied. But as far as i ...
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Why the Expectation Value of a unnormalised state $|\psi\rangle$ is $\langle A\rangle_\psi=\frac{\langle\psi|A|\psi\rangle}{\langle\psi|\psi\rangle}$? [closed]
The typical expectation value formula given most places is
$$\langle A \rangle_\psi = \langle\psi|A|\psi\rangle.$$
this assumes that the state is normalised.
For unnormalised states the formula is
$$\...
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Dyson series expression for the two-point Green function
On Chapter 7 of Fetter & Walecka, the authors prove Dyson formula for the (imaginary time) propagator $U(t,t_{0}) = e^{H_{0}t_{0}}e^{-H(t_{0}-t)}e^{-H_{0}t}$, where I am ommitting the $\hbar$'s. ...
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The physical meaning of the "coupling operator"
I am reading Vassili N. Kolokoltsov's paper arXiv:2505.14605, "On the Mathematical Theory of Quantum Stochastic Filtering Equations for Mixed States", and having trouble understanding the ...
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Time ordering vs symmetry transformation in Euclidean correlation functions
In a QFT with Euclidean signature, the correlation functions can only be well-defined in a time-ordered manner (This is Claim 1 on Page 2 of Simmons-Duffin's lecture note). For example, a scalar 2pt ...
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Does Wigner's theorem only imply left inverse?
From wikipedia
https://en.wikipedia.org/wiki/Wigner%27s_theorem
For unitary case
$$\langle U \Psi, U \Phi \rangle = \langle \Psi, \Phi \rangle .\tag{1} $$
If I apply the definition of adjoint
https://...
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Surjectivity of the d-vector mapping in topological insulators
An effective two-band Hamiltonian can be written in terms of the Pauli matrices as:
$$H=\mathbf{d}(\mathbf{k})\cdot \mathbf{\sigma}$$
The unit vector
$$\hat{\mathbf{d}}(\mathbf{k}) = \frac{\mathbf{d}(\...
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A paradox in using completeness relation $\sum |\rangle\langle|=1$ of quantum mechanics
Suppose we compute an expectation value of $r_{12} r_{13}^{-1}$ over a wave function $\phi_p (1) \otimes \phi_q(2) \otimes \phi_r (3)$, we denote it as $$\langle pqr | r_{12} r_{13}^{-1} |pqr \rangle. ...
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