3.4: Symmetry and Degeneracy

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In real systems degeneracy almost always related to symmetry. In general if the probability density has lower symmetry than the Hamiltonian, the wavefunction will be degenerate.

There is a clear physical reason behind this. Consider the \(2p_x\) orbital in hydrogen: it has a lobe along the \(x\)-axis. However, there is no measurable quantity which defines an \(x\)-axis - the coordinate system is just introduced by physicists to help solve the equations. The lobe could just as well point in the \(y\) or \(z\) or (27, 43.2, −12) direction. Thus the \(p_x\) orbital has lower symmetry than the Hamiltonian (spherically symmetric potential), and is degenerate with \(p_y\) and \(p_z\). Likewise the spin: we talk about ‘spin up’, but there is no way to define ‘up’ from the Hamiltonian. Thus there is degeneracy between spin states ‘up’ and ‘down’.

If we reduce the symmetry of the Hamiltonian, we now ‘lift’ the degeneracy. (i.e. the levels no longer have the same energy). For example, an applied magnetic field defines an axis and lowers the symmetry of the Hamiltonian. If the field is weak, we can use perturbation theory and assume we still have \(p\) orbitals (Zeeman effect). Now, the orbitals must be eigenstates not only of \(\hat{H}_0\), but also of \(\mu.{\bf B}\) where \(\mu\) is the magnetic dipole moment. The degenerate energy level splits into several different energy levels, depending on the relative orientation of the moment and the field: The degeneracy is lifted by the reduction in symmetry.