All Questions
6
questions
1
vote
1
answer
42
views
Find self-adjoint of $P=|a \rangle \langle b |$
Let $P$ be an operator s.a. $P=|a \rangle \langle b |$ and $P|f \rangle = \langle b | f \rangle | a \rangle $.
Find the self-adjoint and the $P^2$ operator.
My attempt:
We know to find the self ...
- 463
0
votes
1
answer
437
views
Show that the adjoint of two operators is the sum of the adjoints
Problem
Show that for any two operators $\hat{A}$ and $\hat{B}$, the adjoint $(\hat{A} + \hat{B})^\dagger = \hat{A}^\dagger + \hat{B}^\dagger$. Do so using the integral form of the definition of ...
- 1,381
0
votes
0
answers
27
views
Question regarding conjugate operators and the harmonic operator.
Let us consider the operator $\hat{n}=\hat{a}^\dagger\hat{a}$ as the number operator of the harmonic oscillator. Let $|n\rangle$ be the eigenstates. Then we can say : $$\hat{n}|n\rangle=n|n\rangle$$
I'...
3
votes
2
answers
4k
views
Definition of Adjoint Operator for Quantum Mechanics
While learning about adjoint operators for quantum mechanics, I encountered two definitions.
The first definition is given by Shankar in The Principle of Quantum Mechanics:
Given a ket
$$ A\lvert ...
- 771
0
votes
0
answers
17
views
Necessary condition for Hermician lin operators
Let $H$ be a Hilbert space and define $T: H \to H$ as a linear unbounded operator.
Say that the spectrum of $T$ is real and the eigenvectors are orthogonal and span the space.
Does that imply that $...
- 1,622
5
votes
1
answer
423
views
Schrodinger with real potential is self-adjoint?
Suppose I define the operator
$$
-\frac{d^2}{dx^2}+V(x)
$$
on the space of Schwartz class functions $\mathcal{S}(\mathbb{R})$ and take its closure to form the operator $H$ acting on a domain in $L^2(\...
- 1,035