Lecture 16 Feb 27#

Today we discussed:

  • the symmetry of the hydrogen Hamiltonian under parity (\([\hat{H},\hat{P}]=0\)) and how that implies that the stationary states do not have a dipole moment, \(\braket{n, \ell, m|\hat{z}|n,\ell, m}=0\).

  • no intrinsic dipole moment means that we should see a quadratic Stark effect. The dipole moment is induced by the applied electric field, \(\mathbf{p} = \alpha \mathbf{E}\) where \(\alpha\) is the polarizability. The energy is then \(-\int \alpha E \ dE = -(1/2)\alpha |\mathbf{E}|^2\).

  • nonetheless, we see a linear Stark effect for the \(n=2\) states. This is because we have a degenerate subspace. Any linear combination of the \(n=2\) states is a stationary state, and we can choose linear combinations that have a dipole moment. This gives a linear energy shift in the applied electric field, \(-p_0 E\), where \(p_0\) is the dipole moment associated with the new eigenstates.

  • This is an example of symmetry breaking: the Hamiltonian is symmetric under parity, but in a degenerate subspace we can construct stationary states that break the symmetry, i.e. do not have a definite parity.

  • The double potential well as an example of symmetry breaking.

  • The ammonia molecule as a similar example, and what happens in more complicated molecules.

  • The idea that perfect degeneracy is not always required to apply degenerate perturbation theory.

Further reading#