3.II.20A

The radial wavefunction for the hydrogen atom satisfies the equation

$\frac{-\hbar^{2}}{2 m} \frac{1}{r^{2}} \frac{d}{d r}\left(r^{2} \frac{d}{d r} R(r)\right)+\frac{\hbar^{2}}{2 m r^{2}} \ell(\ell+1) R(r)-\frac{e^{2}}{4 \pi \epsilon_{0} r} R(r)=E R(r) .$

Explain the origin of each term in this equation.

The wavefunctions for the ground state and first radially excited state, both with $\ell=0$, can be written as

$\begin{aligned} &R_{1}(r)=N_{1} \exp (-\alpha r) \\ &R_{2}(r)=N_{2}(r+b) \exp (-\beta r) \end{aligned}$

respectively, where $N_{1}$ and $N_{2}$ are normalization constants. Determine $\alpha, \beta, b$ and the corresponding energy eigenvalues $E_{1}$ and $E_{2}$.

A hydrogen atom is in the first radially excited state. It makes the transition to the ground state, emitting a photon. What is the frequency of the emitted photon?

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