# 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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### Laplace transform: How to evaluate partial derivative in the denominator of a fraction?

I am solving a differential equation using the Laplace transform. However, to evaluate it I need to evaluate some strange terms. Specifically, I have a partial derivative in the denominator of the ...
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### Algebra of observables in Quantum Mechanics

When reading books about Quantum Mechanics, it is generally stated (in a kind of axiomatic way) that in Quantum Mechanics, the state of the system is represented by a vector in some Hilbert space $H$, ...
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1 vote
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### Are partition functions invariant under Bogoliubov transformations?

Consider a Hamiltonian $H(a_i, a^{\dagger}_i)$ as a function of some ladder operators $a_i, a^{\dagger}_i$. Now, consider a partition function $H(a'_i, a'^{\dagger}_i)$ where $a', a'^{\dagger}$ are ...
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### Bra-Ket Notation vs Wavefunction Notation [duplicate]

We know that the rule for creating excited states for a Quantum Harmonic Oscillator is $|n\rangle=\frac{(a^\dagger)^n(|0\rangle)}{\sqrt{n!}}$. I wanted to derive from this the familiar rule in terms ...
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### Expectation value of vacuum state [closed]

I'm confused about the expectation value of the vacuum state b. Here's my understating: the a' and b' are defined by operator a and b which are similar to the annihilation operator. So when we act ...
1 vote
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### Wick rotation and Exponential mapping of an "imaginary" differential operator acting on a real-valued wavefunction

The position shift operator $T^{a}$ (where $a \in \mathbb{R}$ ) takes a real valued wavefunction $\psi$ on $\mathbb{R}$ to its translation $\psi_{a}$, $T^{a} \psi(x)=\psi_{a}(x)=\psi(x+a)$. A ...
1k views

### Why are expectation values of an observable important in QM?

I've been reading that expectation values of an observable is all what we can get and are the key quantities of the theory, but performing the same experiment many times would generate a distribution ...
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### On the Born-Jordan quantization being an equally weighted average of all operator orderings

On my way studying quantization schemes, I came across the expression saying that the Born-Jordan quantization rule is the equally weighted average of all the operator orderings and that the Weyl's ...
1 vote
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### Questions of lower boundness of Hamiltonians in quantum theories

In general spectral analysis, we have examples of unbounded from below hamiltonians with discrete spectrum. Is it okay to say that they have no sense in physical context, because for me it looks like ...
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### On the symmetry (and anti-symmetry) of operators in QM

In quantum mechanics, the terms symmetric and antisymmetric typically refer to states; specifically, a state is said to be '(anti)symmetric' if it is an eigenstate of the exchange operator with an ...
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### What is the difference between $UXU^{-1}$ and $UX$?

Suppose we have a state $X$ living in some vector space $S$ and a linear operator $U$ that acts on $S$. Now, my understanding is that if $X$ and $U$ are expressed in the same representation, say both ...
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### Spectrum of Klein-Gordon operator in AdS Black Hole

I'm working on obtaining the spectrum of the Klein-Gordon operator in $AdS_2$ for black hole coordinates. To accomplish that, I first consider the problem in hyperbolic space $H_2$ and then Wick ...
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### The meaning of the equivalence of the Schroedinger and Heisenberg pictures

On my way to explore the equivalence between the Heisenberg and the Schroedinger pictures, I cannot see why textbooks quickly deduce this equivalence from the following: If $\psi$ is the state of the ...
1 vote
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### Projection operator onto support of distinct observables

Suppose $P_i$ is the projection operator onto the support of the observable $O_i$ defined on some (say, finite dimensional) Hilbert space. I'm curious as to whether we can define the projection ...
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### Uncertainty on the sum of two non-commuting operators

Suppose that I have an observable $$\hat{E} = \sin(\alpha) \hat{Q} + \cos({\alpha}) \hat{P}$$ with $\hat{Q}, \hat{P}$ being non-commuting operators satisfying $$[\hat{Q}, \hat{P}] = i \hbar$$ It ...
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### How do quantum Fields know about changes in normal ordering?

According to Transformation of the energy-momentum tensor under conformal transformations The schwartzian term in the transformation properties arises due to the stress tensor being defined as the ...