Question

The aim of this problem is to understand how eigenstates of L₂, L², H can form a complete set of simultaneous eigenstates for a particle in a spherical potential in 3D. We take as our starting point the commutation relations for position and momentum [x, px] = ih, [x, y] = 0 etc. The angular momentum operators are simply found from the classical vector cross product L = r x p: Lz = xpy -ýpx, etc. In the following, make sure you use the ABC rule for commutators [A,BC] = [A, B]C + B[Â, Ĉ] - rather that the spatial form of the operators. I neglect "hats" on operators hereafter.

Show that the Hamiltonian H = p²/2m+V commutes with each component of L = [Lx, Ly, Lz] provided V is a spherical potential, and thus only depends on r.

Answer 1

The student's physics question involves understanding the quantization of angular momentum in three dimensions within a spherical potential and its compatibility with energy eigenstates of the Hamiltonian in quantum mechanics.

Step-by-step explanation:

The question is concerned with the quantization of angular momentum in a spherical potential and its implications in quantum mechanics and spectral analysis. Angular momentum operators, L, are generated from the cross-product of position and momentum vectors in classical mechanics.

In a spherical potential, the Hamiltonian H, which includes the kinetic energy (p²/2m) and a spherical symmetric potential energy (V(r)), commutes with the angular momentum operators. This commutation implies that the energy eigenstates can also be eigenstates of angular momentum. To illustrate this, consider Lz which has quantized eigenvalues given by hbar ml/2π where ml is the magnetic quantum number and can range from -l to l.

The ground state of hydrogen emphasizes this quantization, as for n=1, the angular momentum is zero, which aligns with Heisenberg's uncertainty principle that prohibits definite knowledge of all components of angular momentum simultaneously.