The hydrogen atom
Compute the wave functions of the hydrogen atom for \(n=1\), \(j=1/2\). Plot them. Compare them with the solution of the Schrödinger equation for \(n=1\).
Compute and plot the emission spectrum of the hydrogen atom in Dirac’s theory.
The energy levels of a hydrogen atom change in the gravitational field of a star. For the photons emitted by the atoms this leads to the gravitational redshift \[z= \frac{\Delta\nu}{\nu}=\frac{GM}{Rc^2},\] where \(G\) is the gravitational constant and \(M\) and \(R\) are the mass and the radius of the star. Compute \(\Delta\nu/\nu\) for different types of stars. Which of them allow one to observe the gravitational redshift? Compare with experiment.