The Hydrogen Atom and Degeneracy

We have begun to explore the Hydrogen atom. One of the important features of this system is the degeneracy associated with different states of the atom. For this problem, your group will engage in the inverse problem of consrtucting normalized superposition states that have particular features. To remind you, for the Hydrogen atom we know the energy eigenstates $\ket{nlm}$ are simultaneous eigenstates of $H$, $\mathbf{L}^2$, and $L_z$ with eigenvalues given by:

\[H\ket{nlm} = -\dfrac{c}{n^2}\ket{nlm}\;\mathrm{where}\; c = 13.6\mathrm{eV}\] \[\mathbf{L^2}\ket{nlm} = l(l+1)\hbar^2\ket{nlm}\] \[L_z\ket{nlm} = m\hbar\ket{nlm}\]

For the following acitvity, you will construct normalized superpositon states that satisfy the criteria below. There are likely multiple solutions to each question below.

Construct a normalized superposition state that:

  1. is only degenerate in $E$
  2. is only degenerate in $L^2$
  3. is only degenerate in $L_z$
  4. has $\langle H \rangle = -\dfrac{c}{9}$, $\langle L_z \rangle = 0$, and possible measures of $\mathbf{L}^2 = 6\hbar^2$ or $2\hbar^2$.
  5. has energy measurements of $-\dfrac{c}{9}$ and $-\dfrac{c}{4}$, $\langle \mathbf{L}^2 \rangle =2\hbar^2$, and possible measures of $L_z = 1\hbar$ or $0\hbar$.