Question

(a) Cite a Hamiltonian from Chapter 2 (other than the harmonic oscillator) that has only a discrete spectrum.

(b) Cite a Hamiltonian from Chapter 2 (other than the free particle) that has only a continuous spectrum.

(c) Cite a Hamiltonian from Chapter 2 (other than the finite square well) that has both a discrete and a continuous part to its spectrum.

Short answer

a) Infinite square well.

b) Finite rectangular barrier.

c) Finite square well.

Step-by-step solution

Example of a Hamiltonian that has a discrete spectrum

a)

The infinite square well Hamiltonian is an example of a Hamiltonian with a discrete spectrum. Recall that all energies $E_n$corresponding to this Hamiltonian are discrete.

Example of a Hamiltonian that has a continuous spectrum

b)

The finite rectangular barrier Hamiltonian is an example of a Hamiltonian with a continuous spectrum. Recall that all energies corresponding to this Hamiltonian are continuous.

Example of a Hamiltonian that has both continuous and discrete spectrum

c)

The finite square well Hamiltonian is an example of a Hamiltonian with both a continuous and discrete spectrum. Recall that this Hamiltonian admits both bound (discrete spectrum) and scattering (continuous spectrum) status.