Quantum Operator Commutation Relations and Implications
This solution analyzes the implications of given commutation relations between quantum operators $\hat{A}$, $\hat{B}$, and $\hat{C}$, specifically focusing on simultaneous diagonalization and the uncertainty principle.
Simultaneous Diagonalization from Commutation
Two operators can be simultaneously diagonalized if and only if they commute. This means they share a common set of eigenvectors.
- The relation $[$\hat{A}, \hat{B}$] = 0$ indicates that operators $\hat{A}$ and $\hat{B}$ commute. Therefore, they can be simultaneously diagonalized. This validates option 3.
- Because $[$\hat{A}, \hat{C}$] $$\neq$$ 0$, operators $\hat{A}$ and $\hat{C}$ do not commute. Consequently, they cannot be simultaneously diagonalized. This rules out option 4, which requires simultaneous diagonalization of all three operators ($\hat{A}, \hat{B}, \hat{C}$), an condition not met due to $\hat{A}$ and $\hat{C}$ not commuting.
Uncertainty Principle and Non-Commutation
The Heisenberg uncertainty principle provides a lower bound for the product of uncertainties of two observables represented by operators $\hat{X}$ and $\hat{Y}$ if they do not commute:
$ \Delta X \Delta Y \ge \frac{1}{2} | \langle [\hat{X}, \hat{Y}] \rangle | $
If the commutator $[$\hat{X}, \hat{Y}$]$ is non-zero, the product of uncertainties $\Delta X \Delta Y$ must be strictly positive.
- For operators $\hat{A}$ and $\hat{C}$, the commutator is given as $[$\hat{A}, \hat{C}$] $$\neq$$ 0$. Applying the uncertainty principle, we get $\Delta A \Delta C \ge \frac{1}{2} | \langle [\hat{A}, \hat{C}] \rangle | $. Since the expectation value of a non-zero operator is generally non-zero, this implies $\Delta A \Delta C > 0$. This validates option 2.
- For operators $\hat{A}$ and $\hat{B}$, $[$\hat{A}, \hat{B}$] = 0$. The uncertainty principle does not impose a condition that $\Delta A \Delta B > 0$; it is possible for $\Delta A \Delta B = 0$. Therefore, option 1 is not necessarily implied by the commutation relations.
Summary of Validated Options
The commutation relations directly imply:
- $\Delta A \Delta C > 0$ (Option 2)
- $\hat{A}, \hat{B}$ can be simultaneously diagonalized (Option 3)
Important Questions from Operators Commutators Heisenberg Picture
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 $|\phi(p)|^2$ ?- From the pairs of operators given below, identify the ones which commute. Here $l$ and $j$ correspond to the orbital angular momentum and the total angular momentum, respectively.
- An electromagnetic pulse has a pulse width of $10^{-3}$ s. The uncertainty in the momentum of the corresponding photon is of the order of $10^{-N}$ kg m $s^{-1}$, where $N$ is an integer. The value of $N$ is ________ (speed of light = $3 \times 10^8$ m $s^{-1}$, h = $6.6 \times 10^{-34}$ J s)
- In cylindrical coordinates $(s, \varphi, z)$, which of the following is a Hermitian operator?
- 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
