The spin-orbit energy is proportional to $\mathbf{S} \cdot \mathbf{L}$. To evaluate $\mathbf{S} \cdot \mathbf{L}$, note that since $\mathbf{J}=\mathbf{S}+\mathbf{L}$,

$$J^2=(\mathbf{S}+\mathbf{L}) \cdot(\mathbf{S}+\mathbf{L})=S^2+L^2+2 \mathbf{S} \cdot \mathbf{L}$$

You can solve this for $\mathbf{S} \cdot \mathbf{L}$ and then put in the values for $J^2, S^2$, and $L^2$. [For instance, $J^2=j(j+1) \hbar^2$ where $j=l \pm \frac{1}{2}$.] Show that if $j=l+\frac{1}{2}$ then $\mathbf{S} \cdot \mathbf{L}=$ $l^2 / 2$, and if $j=l-\frac{1}{2}$, then $\mathbf{S} \cdot \mathbf{L}=-(l+1) \hbar^2 / 2$. Use these answers to prove that the spin-orbit splitting increases with increasing $l$, as indicated in the above figure.

(a) Calculate to first-order perturbation theory the contribution due to the spin-orbit interaction for the $n$th excited state for a positronium atom.
(b) Use the result of part (a) to obtain numerical values for the spin-orbit correction terms for the $2 \mathrm{p}$ level and compare them to the energy of $n=2$.

Find the values of the radius where the n = 2, l = 0 radial probability density has its maximum values.

a) Evaluate, in electron volts, the energies of the three levels of the hydrogen atom in the states for n = 1, 2, 3. b) Then calculate the frequencies in hertz, and the wavelengths in angstroms, of all the photons that can be emitted by the atom in transitions between these levels. c) In what range of the electromagnetic spectrum are these photons?

Show that the energy eigenstates of the infinite square well are orthogonal.

What is the difference between the probability density and the radial probability density?

What is the difference between the probability density and the radial probability density?

Calculate the radial probability density P(r) for the hydrogen atom in its ground state at (a) r = 0, (b) r = a, and (c) r = 2a, where a is the Bohr radius.

For the hydrogen atom in its ground state, calculate (a) the probability density ψ²(r) and (b) the radial probability density P(r) for r = a, where a is the Bohr radius.

Consider a system of two electrons, each with $\ell=1$ and $s=\frac{1}{2}$. (a) What are the possible values of the quantum number for the total orbital angular momentum $\mathrm{L}=\mathrm{L}_1+\mathrm{L}_2$ ? (b) What are the possible values of the quantum number $S$ for the total spin $\mathrm{S}=\mathrm{S}_1+\mathrm{S}_2$ ? (c) Using the results of parts (a) and $(b)$, find the possible quantum numbers $j$ for the combination $\mathbf{J}=\mathbf{L}+\mathbf{S}$. (d) What are the possible quantum numbers $j_1$ and $j_2$ for the total angular momentum of each particle? $(e)$ Use the results of part $(d)$ to calculate the possible values of $j$ from the combinations of $j_1$ and $j_2$. Are these the same as in part $(c)$ ?

What are the possible values for all four quantum numbers?

Enumerate the quantum numbers $(n, \ell, \text { and } m)$ for all the independent states of a hydrogen atom with definite $E,|\vec{L}|^{2},$ and $L_{z},$ up to n = 3. Check that the number of independent states for level n is equal to $n^{2}.$

For each choice of $n$ and $l$ (with $l \neq 0$ ), a nucleon has two different possible energy levels with $j=l \pm \frac{1}{2}$. Prove that the sum of the degeneracies of these two levels is equal to the total degeneracy that the level $n l$ would have had in the absence of any $\mathbf{S} \cdot \mathbf{L}$ splitting.

Solve. The reciprocal of 3, plus the reciprocal of 6, is the reciprocal of what number?