The operators tag has no wiki summary.
4
votes
1answer
104 views
Operator Ordering Ambiguities
I have been told that $$[\hat x^2,\hat p^2]=2i\hbar (\hat x\hat p+\hat p\hat x)$$ illustrates operator ordering ambiguity.
What does that mean?
I tried googling but to no avail.
4
votes
1answer
79 views
Constructing the space of quantum states
I want to learn how to construct spaces of quantum states of systems.
As an exercize, I tried to build the space of states and to find hamiltonian spectrum of the quantum system whose Hamiltonian is ...
0
votes
1answer
58 views
Adjoint of momentum operator
In position basis, we have,
$$\langle x \mid \hat p \mid \Psi(t) \rangle = -\imath \hbar \frac{\partial{\langle x \mid \Psi(t) \rangle}}{\partial{x}} $$
Now i know $\hat{p}$ is a hermitian operator ...
1
vote
2answers
57 views
Notation for differential operators and wave function math
I know that $[\frac {d^2}{dx^2}]\psi$ is $\frac {d^2\psi}{dx^2}$ but what about this one $[\frac {d^2\psi}{dx^2}]\psi^*$? Is it this like $\frac {d^2\psi\psi^*}{dx^2}$ or this like $\frac ...
2
votes
6answers
232 views
Is H=H* sloppy notation or really just incorrect, for Hermitian operators?
I saw it in this pdf, where they state that
$P=P^\dagger$ and thus $P$ is hermitian.
I find this notation confusing, because an operator A is Hermitian if
$\langle \Psi | A \Psi \rangle=\langle A ...
3
votes
4answers
203 views
Is the momentum operator well-defined in the basis of standing waves?
Suppose I want to describe an arbitrary state of a quantum particle in a box of side $L$. The relevant eigenmodes are those of standing waves, namely
$$ \left<x|n\right>=\sqrt{\frac{2}{L}}\cdot ...
3
votes
1answer
115 views
Evolution operator for time-dependent Hamiltonian
When i studyed QM I'm only working with non time-dependent Hamiltonians. In this case unitary evolution operator has the form $$\hat{U}=e^{-\frac{i}{\hbar}Ht}$$ that follows from this equation
$$
...
4
votes
1answer
89 views
Existence of adjoint of an antilinear operator, time reversal
The time reversal operator $T$ is an antiunitary operator, and I saw $T^\dagger$ in many places (for example when some guy is doing a "time reversal" $THT^\dagger$), but I wonder if there is a ...
1
vote
3answers
79 views
Hamiltonian in position basis
Let $ H = \frac{-h^2}{2m}\frac{\partial^2 }{\partial x^2}$. I want to find the matrix elements of $H$ in position basis. It is written like this:
$\langle x \mid H \mid x' \rangle = ...
2
votes
1answer
177 views
The cleverest way to calculate $\left[\hat{a}^{M},\hat{a}^{\dagger N}\right]$ with $\left[\hat{a},\hat{a}^{\dagger}\right]=1$
Who can provide me some elegant solution for
$$\left[\hat{a}^{M},\hat{a}^{\dagger N}\right]\qquad\text{with} \qquad\left[\hat{a},\hat{a}^{\dagger}\right]~=~1$$
other than brute force calculation?
...
1
vote
2answers
186 views
Dirac notation question
I don't understand this equality
$$\int \!d^3p~\langle\textbf{x}|e^{-i(\hat{\textbf{p}}^2/2m)t}|\textbf{p}\rangle\langle\textbf{p} | \textbf{x}_0 \rangle ~=~\int\! ...
5
votes
3answers
130 views
What does it mean to apply an operator to a state?
Let's say I have an operator $\hat{A}$ and a state $|\psi\rangle$. What exactly is the state $\hat{A}|\psi\rangle$? Is it just another different state that I am describing using my $\hat{A}$ and ...
2
votes
1answer
69 views
What is the operator for the edge current of a fracional quantum Hall state?
The edge of a fractional quantum Hall state is a chiral conformal field theory. In the Laughlin case it corresponds to the chiral boson,
$$ S = \frac{1}{4\pi} \int dt dx ...
6
votes
2answers
321 views
Difficulties with bra-ket notation
I have started to study quantum mechanics. I know linear algebra,functional analysis, calculus, and so on, but at this moment I have a problem in Dirac bra-ket formalism. Namely, I have problem with ...
5
votes
2answers
195 views
What does the Canonical Commutation Relation (CCR) tell me about the overlap between Position and Momentum bases?
I'm curious whether I can find the overlap $\langle q | p \rangle$ knowing only the following:
$|q\rangle$ is an eigenvector of an operator $Q$ with eigenvalue $q$.
$|p\rangle$ is an eigenvector of ...
