Free Particle Hamiltonian Commutator Calculation
The Hamiltonian ($H$) for a free particle of mass ($m$) is purely kinetic:
$H = \frac{p^2}{2m}$
where $p$ is the momentum operator.
Inner Commutator Calculation
First, compute the commutator $[x, H]$:
$[x, H] = [x, \frac{p^2}{2m}]$
Using the linearity property of commutators and the fundamental commutation relation $[x, p] = i\hbar$:
$[x, H] = \frac{1}{2m} [x, p^2]$
$= \frac{1}{2m} ([x, p]p + p[x, p])$
$= \frac{1}{2m} (i\hbar p + p(i\hbar))$
$= \frac{1}{2m} (2i\hbar p)$
$= \frac{i\hbar p}{m}$
Outer Commutator Calculation
Next, compute the double commutator $[x, [x, H]]$ using the result from the previous step:
$[x, [x, H]] = [x, \frac{i\hbar p}{m}]$
Factor out the constants and apply the fundamental commutation relation $[x, p] = i\hbar$ again:
$[x, [x, H]] = \frac{i\hbar}{m} [x, p]$
$= \frac{i\hbar}{m} (i\hbar)$
$= \frac{i^2 \hbar^2}{m}$
Since $i^2 = -1$:
$= -\frac{\hbar^2}{m}$
Result
The commutator $[x, [x, H]]$ for the free particle Hamiltonian is $-\$\hbar^2$/m.
Important Questions from Operators Commutators Heisenberg Picture
- Consider an operator $\hat{A}$ which is not Hermitian. Find the possible values of $c$ and $d$ such that the operator $(c\hat{A} - d\hat{A}^\dagger)$ is Hermitian.
- Which of the following operators is/are self-adjoint?
- Consider operators $\hat{A}$, $\hat{B}$, and $\hat{C}$ for three observables of a quantum system satisfying $[\hat{A}, \hat{B}] = 0$, $[\hat{B}, \hat{C}] = 0$, and $[\hat{A}, \hat{C}] \neq 0$, with uncertainties $\Delta A, \Delta B, \Delta C$, respectively. From the options given below, which is/are implied by the commutation relations among $\hat{A}, \hat{B}$, and $\hat{C}$?
- Let $|m\rangle$ and $|n\rangle$ denote the energy eigenstates of a one-dimensional simple harmonic oscillator. The position and momentum operators are $\hat{X}$ and $\hat{P}$, respectively. The matrix element $\langle m|\hat{P}\hat{X}|n\rangle$ is non-zero when
The wavefunction of a particle in one dimension is given by
$\psi(x) = \begin{cases} M, & -a < x < a \\ 0, & \text{otherwise.} \end{cases}$
Here $M$ and $a$ are positive constants. If $\phi(p)$ is the corresponding momentum space wavefunction, which one of the following plots best represents $|\phi(p)|^2$ ?
