Angular Momentum Operator Commutation Fundamentals
Two operators, say $Â$ and $B̂$, commute if their commutator, defined as $[Â, B̂] = ÂB̂ - B̂Â$, equals zero.
Mathematically, this condition is $[Â, B̂] = 0$.
Analyzing Commuting Operator Pairs
We investigate the provided pairs involving orbital angular momentum (l) and total angular momentum (j).
Operator Pair: l2 and j2
- Total angular momentum j is defined as j = l + s, where s is spin angular momentum.
- Squaring this gives j2 = (l + s)2 = l2 + s2 + 2l·s.
- In standard quantum mechanics treatments, l2 and j2 are typically chosen to commute. This is because they can be simultaneously diagonalized in a common basis, reflecting different aspects of angular momentum magnitude.
- Therefore, the commutator $[l2, j2] = 0$, indicating this pair commutes.
Operator Pair: j2 and jz
- The square of an angular momentum operator (j2) inherently commutes with its projection onto any axis (jz).
- This is a fundamental property of angular momentum algebra in quantum mechanics.
- Consequently, $[j2, jz] = 0$, confirming this pair commutes.
Operator Pair: j2 and lz
- Using j2 = l2 + s2 + 2l·s, the commutator is $[j2, lz] = [l2, lz] + [s2, lz] + [2l·s, lz]$.
- The commutators $[l2, lz]$ and $[s2, lz]$ are both zero, as spin and orbital operators act on independent spaces and have commuting properties with each other's components.
- However, the term $[l·s, lz]$ is generally non-zero. It can be shown to equal $iħ(lysx - lxsy)$, which is not zero.
- Thus, $[j2, lz] ≠ 0$, meaning this pair does not commute.
Operator Pair: lz and jz
- The z-component of total angular momentum is jz = lz + sz.
- The commutator is $[lz, jz] = [lz, lz + sz] = [lz, lz] + [lz, sz]$.
- The commutator of an operator with itself is zero: $[lz, lz] = 0$.
- Components of orbital and spin angular momentum commute: $[lz, sz] = 0$.
- Therefore, $[lz, jz] = 0 + 0 = 0$, and this pair commutes.
Commuting Operator Identification Summary
The pairs of operators that commute, based on the analysis, are:
- l2, j2
- j2, jz
- lz, jz
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 $|\phi(p)|^2$ ?- 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
Let $|\psi_1\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$, $|\psi_2\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$ represent two possible states of a two-level quantum system. The state obtained by the incoherent superposition of $|\psi_1\rangle$ and $|\psi_2\rangle$ is given by a density matrix that is defined as $\rho ≡ c_1|\psi_1\rangle\langle\psi_1| + c_2|\psi_2\rangle\langle\psi_2|$. If $c_1 = 0.4$ and $c_2 = 0.6$, the matrix element $\rho_{22}$ (rounded off to one decimal place) is ________
