2
votes
1answer
12 views

Unitary transformation between complete and orthonormal bases

I'm using the Dirac notation for vectors here, since this is a quantum mechanics question. Suppose the complete orthonormal bases $\{|\psi_n\rangle\}$ and $\{|\psi{'}_n\rangle\}$ are related by the ...
0
votes
0answers
15 views

Domain of the quantum free Hamiltonian in 1D

Consider the quantum free Hamiltonian $H_0 =-\frac{d^2}{dx^2}$ (the Laplacian on the real line). I want to show that it is (essentially) self-adjoint in its domain of definition. The usual approach ...
5
votes
1answer
71 views

Spectral theorem in Quantum Mechanics

In Quantum Mechanics one often looks at self-adjoint(unbounded and closed) linear operators $A,B$ that are defined dense on $L^2$. My question is: Is it true that if we have $[A,B]=0$, where $[,]$ is ...
0
votes
0answers
47 views

Question arising from quantum mechanics concerning groups and symmetries

I'm trying to understand a calculation my professor did in my quantum mechanics script. Here it is: Each rotation $R \in O(3)$ induces a unitary transformation in $L^2(R^3)$, i.e. the space of square ...
3
votes
1answer
90 views

If the expectation $\langle v,Mv \rangle$ of an operator is $0$ for all $v$ is the operator $0$?

I ran into this a while back and convinced my self that it was true for all finite dimensional vector spaces with complex coefficients. My question is to what extent could I trust this result in the ...
6
votes
1answer
115 views

How to find interesting operators for a quantum system?

How can we find "interesting" operators for a quantum mechanical system? I can think of the following method: Given some system with an associated Hilbert space $V$ and Hamiltonian $H:V\rightarrow ...
3
votes
1answer
87 views

A question about Moyal product

In Geometric quantization theory the Moyal product is one of main tools. We know Moyal product for the smooth functions $f$ and $g$ on $ℝ^{2n}$ takes the form $f\star g = fg + \sum_{n=1}^{\infty} ...
1
vote
1answer
111 views

How to prove $[x,p]=i$ $\implies$ $[x,p^n]=inp^{n-1}$?

How to prove $[x,p]=i$ $\implies$ $[x,p^n]=inp^{n-1}$? I can do this using $p=i\frac{d}{dx}$, but my book hasn't introduced this yet so is there another proof without using this ? These are just ...
0
votes
0answers
26 views

Relation $mod$ and Hermitian operators?

Modular arithmetic is likely a representation of some type of discrete oscillator - clocks. Hermitian operators on the other hand can also be applied to describe oscillators, such as quantum ...
0
votes
0answers
44 views

Proving that this is not a positive operator

Let $\rho$ be a density operator (i.e., it is an ortho projection with rank one, and also a positive operator). Say $X = X^*$ with a spectral decomposition $X = 1P_1 + 4P_4 + 16P_{16}$, and $Y = Y^*$ ...
1
vote
1answer
32 views

Expectation value of pure state in quantum mechanics

It's well known that in quantum mechanics, the expectation value of a self-adojint operator $A$ in pure state $|\psi\rangle$ is $\langle\psi |A|\psi\rangle = \operatorname{Tr}(A |\psi \rangle ...
2
votes
0answers
46 views

Doubt about the spectrum of an operator

I consider the Laplacian operator $$A=-\Delta$$ in the domain $$H^2(\mathbb{R}^3)$$ where it is selfadjoint. We know that its spectrum is $[0,+\infty)$. Now I want to consider the restriction of $A$ ...
6
votes
1answer
190 views

Kernel of adjoint operator

This problem is puzzling me, even though it should be really simple. Let $L=-\partial_x^2 + \frac 1 2 x^{-2}$ be an operator defined on $D(L)=C^\infty_c(0,+\infty)\subset L^2(0,+\infty)$. Its adjoint ...
0
votes
1answer
325 views

Show that $A^{\dagger^{\dagger}} = A $

How do we show that $A^{\dagger^{\dagger}} = A $ without assuming $A$ to be a explicit matrix. That is, given a linear operator $A$, let us define $A^\dagger$ to be a unique operator such that ...
1
vote
1answer
413 views

Commutator relationship proof $[A,B^2] = 2B[A,B]$

I'm trying to find the condition necessary for this commutator relationship equality: $$[A,B^2]=2B[A,B]$$ So far I've done this: \begin{align*} [A,B^2] & = B[A,B] + [A,B]B \\ ...
3
votes
1answer
199 views

What are the requirements for a “test” function to show operators commute?

To show that two operators $\hat{A}$ and $\hat{B}$ commute, we can check whether $\hat{A}\hat{B}f(x)$ = $\hat{B}\hat{A}f(x)$. My question is regarding the function $f(x)$. To check that $\hat{A}$ and ...
11
votes
1answer
361 views

Quantization of angular momentum: is Dirac's proof wrong?

I'm trying to understand the physicist's proof of the theorem on the spectral structure of angular momentum operators (I'm being told that this proof is due to Dirac). I will refer to Ballentine's ...
2
votes
2answers
161 views

Perturbation theorem of Weyl

Does anyone know where to find something about the perturbation theorem of Weyl, preferably on the internet. The theorem I'm talking about states: let $A$ be a self-adjoint operator on a Hilbert ...