Operator Self-Adjointness Condition
A second-order differential operator, denoted as $ L = P_2(x)\frac{d^2}{dx^2} + P_1(x)\frac{d}{dx} + P_0(x) $, is formally self-adjoint if its coefficients satisfy the relationship $ P_1(x) = \frac{dP_2}{dx}(x) $. This condition ensures the operator equals its formal adjoint.
We will examine each operator provided to determine if it meets this criterion.
Analysis of Operators
| Operator ID | Operator Expression | Coefficient $P_2(x)$ | Coefficient $P_1(x)$ | Derivative $P_2'(x)$ | Self-Adjoint Condition Met? |
| 1 | $x^2 \frac{d^2}{dx^2} + 3x \frac{d}{dx} + x^2$ |
$x^2$ | $3x$ | $2x$ | No (since $3x \neq 2x$) |
| 2 (B) | $(1 - x^2)\frac{d^2}{dx^2} - 2x\frac{d}{dx} + 3x$ |
$1 - x^2$ | $-2x$ | $-2x$ | Yes (since $-2x = -2x$) |
| 3 (C) | $(3x - 4x^3)\frac{d^2}{dx^2} + (3 - 12x^2)\frac{d}{dx} + 12$ |
$3x - 4x^3$ | $3 - 12x^2$ | $3 - 12x^2$ | Yes (since $3 - 12x^2 = 3 - 12x^2$) |
| 4 | $x\frac{d^2}{dx^2} + x^2\frac{d}{dx} + \frac{5x}{3}$ |
$x$ | $x^2$ | $1$ | No (since $x^2 \neq 1$) |
Conclusion on Self-Adjoint Operators
The analysis shows that operators 2 (B) and 3 (C) satisfy the self-adjoint condition $ P_1(x) = P_2'(x) $. Therefore, these are the self-adjoint operators among the choices.
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.
- 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 $|\phi(p)|^2$ ?- If $H$ is the Hamiltonian for a free particle with mass $m$, the commutator $[x, [x, H]]$ is
