References: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 6.36.
The Zeeman effect occurs when an atom is placed in an external magnetic field, resulting in the interaction between field and the magnetic dipole moments of the atom causing splitting of the energy levels. The electrical analogue of the Zeeman effect, when an atom is placed in an external electric field, is called the Stark effect. We can use perturbation theory to analyze the effect on the energy levels of the electron.
The perturbation hamiltonian is, assuming the electric field points in the direction:
To use perturbation theory, we’ll need the wave functions for unperturbed hydrogen, which are given in Griffiths as equation 4.89. For the ground state , we have
Since the ground state is non-degenerate, we can use non-degenerate perturbation theory:
Rather than writing out the integral, we observe that contains the integral of over which is zero, so .
To analyze , we need the four wave functions:
Since all four of these states have the same unperturbed energy, we need to use degenerate perturbation theory, so we’ll need to find the matrix with elements
where and represent one of the four states above.
First, we’ll look at the integrals. All matrix elements involve integrals of the form (remember that always contributes a and the spherical volume element always contributes a ):
For the possible values of and in this problem, the only non-zero integrals of this form are
arises in (and its transpose) and arises in and (and their transposes). Thus these are the only possible non-zero entries in . However, and involve integrating over which gives zero. Thus the only non-zero matrix elements are (and its transpose). This gives
(The integral can be done with software, or by hand using integration by parts.) The matrix is therefore
The eigenvalues are 0, 0, so the state splits into 3 states, one with energy (degeneracy 2) and two with energies (each with degeneracy 1). The eigenvectors are and for eigenvalue 0, for and for . Thus the ‘good’ states are
The electric dipole moment of hydrogen is (treating the proton and electron as point charges):
We can work out the expectation value of in each of the ‘good’ states by straightforward integration: where stands for one of the ‘good’ states. Note that if or , then has only a component that is non-zero, since the complex exponentials in cancel out and the integral of or in the or components is zero. Similarly, if or , the and components are again zero, since these wave functions are independent of so the integral of or in the or components gives zero again. Therefore, is always in the direction, and can be calculated from
Doing the integrals results in