Quantum Mechanics | Part IB, 2003

The radial wavefunction for the hydrogen atom satisfies the equation

22m1r2ddr(r2ddrR(r))+22mr2(+1)R(r)e24πϵ0rR(r)=ER(r).\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 =0\ell=0, can be written as

R1(r)=N1exp(αr)R2(r)=N2(r+b)exp(βr)\begin{aligned} &R_{1}(r)=N_{1} \exp (-\alpha r) \\ &R_{2}(r)=N_{2}(r+b) \exp (-\beta r) \end{aligned}

respectively, where N1N_{1} and N2N_{2} are normalization constants. Determine α,β,b\alpha, \beta, b and the corresponding energy eigenvalues E1E_{1} and E2E_{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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