All Questions
24
questions
-1
votes
2
answers
99
views
Are Hermitian operators Hermitian in any basis? [closed]
Given a Hilbert space and a Hermitian operator defined on it, will the operator exhibit Hermiticity in any basis used to span the space? My thought on this is that this must be the case, after all, if ...
- 4,399
1
vote
1
answer
132
views
A theorem about functions of self-adjoint operators
It is very common (see e.g. page 18 of Ballentine's Quantum Mechanics: A Modern Development) for the following development to take place. We couch the discussion in Dirac's bra-ket notation noting ...
- 1,095
1
vote
2
answers
378
views
How does one write Adjoint, Self-adjoint and Hermitian operators in Dirac notation?
The following portion is paraphrased from Mathematical Methods for Physics and Engineering by Riley, Hobson, and Bence.
The adjoint of a linear operator $\hat{A}$, denoted by $A^\dagger$, is an ...
- 11.8k
0
votes
1
answer
126
views
How can eigenstates of a hermitian operator be orthogonal without explicitly defining the inner product?
It's a well known fact that for any hermitian operator, say $H$ (assuming there is no degeneracy), $${\left< a_i \right.\left| a_j \right> \over \sqrt{\left< a_i \right.\left| a_i \right>...
- 317
3
votes
1
answer
290
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Mathematical definition of annihilation and creation operators
I am self-studying quantum field theory and have gotten to creation and annihilation operators, respectively denoted $A^\dagger$ and $A$. Conceptually I understand what these objects are, at least on ...
- 3,350
0
votes
2
answers
131
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Question on Dirac notation with operator [closed]
What does $\langle\psi|A|\phi\rangle$ mean if $A$ is some operator like how does $A$ acts on these two vectors $\phi$ and $\psi$ and what is it equal to and also does $A$ act on both vectors or just ...
- 33
0
votes
1
answer
364
views
Radial position operator
While trying to find the expectation value of the radial distance $r$ of an electron in hydrogen atom in ground state the expression is:
$$\begin{aligned}\langle r\rangle &=\langle n \ell m|r| n \...
- 1,270
4
votes
3
answers
529
views
$\left\langle{\hat{O}}^\dagger\varphi\middle|\psi\right\rangle$ How do I act the operator in bra?
$$\left\langle\varphi\middle|\hat{O}\middle|\psi\right\rangle=\left\langle{\hat{O}}^\dagger\varphi\middle|\psi\right\rangle.$$
In above formula, I have confused what does mean $\left\langle{\hat{O}}^\...
- 75
1
vote
1
answer
106
views
Action of permutation operator on other operators
I'm watching MIT 8.06 Quantum physics, lecture $23.2$ See for example [1] Particularly See $5:41$. It is shown that
$$P_{21}B(1)P^\dagger_{21}|u_i\rangle_1\otimes |u_j\rangle_2=|u_i\rangle_1\otimes |...
- 11.9k
0
votes
1
answer
58
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Doubt about property of hermitian operator
For any hermitian operator M, prove that
\begin{equation}
\langle Ma|b \rangle = \langle a|Mb \rangle
\end{equation}
My attempt:
Let
\begin{eqnarray}
\langle a| = \sum_i a_i^*\langle i|\\
|b\rangle = \...
2
votes
2
answers
900
views
Domain of an adjoint operator
I'm studying a bit of functional analysis for quantum mechanics and I'm stuck on a definition our professor gave us.
Given an operator and its domain $(A,\mathcal{D}(A))$ densely defined in $\mathcal ...
- 77
7
votes
1
answer
681
views
Why do self-adjoint operators have to be densely defined?
I have been watching the Schiller lectures on QM and have been going through ‘quantum mechanics and quantum field theory’ by Dimock.
Both seem to ensure operators are densely defined, especially if ...
- 1,933
2
votes
3
answers
723
views
Confused about definition of three dimensional position operator in QM
My QM text defines the position operator as follows:
The position operator $X= (X_1,X_2,X_3)$ is such that for $j=1,2,3: \ X_j \psi(x,y,z)= x_j \psi(x,y,z)$.
To me this can mean two things.
1) $...
- 151
2
votes
2
answers
1k
views
Physical meaning of Transpose of an Operator in Quantum Mechanics?
What's the physical meaning of transpose of a matrix in Quantum Mechanics?
Although for Unitary or Orthogonal operators, I know that transpose of that operator would reverse the action and that's ...
- 175
8
votes
3
answers
385
views
Equivalent definitions of total angular momentum
Consider the equality
\begin{equation}\exp\left(-\frac{i}{\hbar}\boldsymbol{\phi J}\right)\left|x\right>=\left|R(\phi)x\right>,\end{equation}
where $\left|x\right>$ denotes a position ...
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