Find the first-order corrections to the energies of the hydrogen atom that result from the relativistic mass increase of the electron. Hint. The energy is related to the momentum by \(E=\left(p^{2} c^{2}+m^{2} c^{4}\right)^{1 / 2}+V .\) When \(p^{2} c^{2} \ll m^{2} c^{4}\), \(E \approx \mu c^{2}+p^{2} / 2 \mu+V-p^{4} / 8 \mu^{3} c^{2},\) where the reduced mass \(\mu\) has replaced \(m\). Ignore the rest energy \(\mu c^{2},\) which simply fixes the zero. The term \(-p^{4} / 8 \mu^{3} c^{2}\) is a perturbation; hence calculate \(\left\langle n l m_{l}\left|H^{(1)}\right| n l m_{l}\right\rangle=-\left(1 / 2 \mu c^{2}\right)\left\langle n l m_{l}\left|\left(p^{2} / 2 \mu\right)^{2}\right| n l m_{l}\right\rangle\) \(=-\left(1 / 2 \mu c^{2}\right)\left\langle n l m_{l}\left|\left(E_{n l m_{l}}-V\right)^{2}\right| n l m_{l}\right\rangle .\) We know \(E_{n l m} ;\) therefore calculate the matrix elements of \(V=-e^{2} / 4 \pi \varepsilon_{0} r\) and \(V^{2}\).

Step by step solution

01

Define the given values and equations

Initially, we recognize that the problem has provided several important pieces of information needed to solve this problem: \(E=\sqrt{(p^{2} c^{2}+m^{2} c^{4})}+V\) which can be approximated to \(E \approx \mu c^{2}+p^{2} / 2 \mu+V-p^{4} / 8 \mu^{3} c^{2}\) when \(p^{2} c^{2} << m^{2} c^{4}\). The perturbative term is \(-p^{4} / 8 \mu^{3} c^{2}\) and hence \(\left\langle n l m_{l}\left|H^{(1)}\right| n l m_{l}\right\rangle=-\left(1 / 2 \mu c^{2}\right)\left\langle n l m_{l}\left|\left(p^{2} / 2 \mu\right)^{2}\right| n l m_{l}\right\rangle\); using \(E_{n l m_{l}}-V\) for \(\left(p^{2} / 2\mu\right)^{2}\).

02

Calculate the matrix element of the potential \(V\)

Calculate the matrix element of the potential \(V=-e^{2} / 4 \pi \varepsilon_{0} r\). The final part of the exercise demands the calculation of \(\left\langle n l m_{l}\left|V\right| n l m_{l}\right\rangle\), hence it needs to be calculated.

03

Calculate the matrix element of the potential \(V^{2}\)

Calculate the matrix element of the potential \(V^{2}\) as \(\left\langle n l m_{l}\left|V^{2}\right| n l m_{l}\right\rangle \) and compute the expression of \(V^{2}\).

04

Finalize the calculation

Combine the matrix elements from Step 2 and Step 3 to find the first order corrections to the energies of the hydrogen atom

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