# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 $[A,B] = 0$, ...