Hamiltonian (quantum Mechanics) - Energy Eigenket Degeneracy, Symmetry, and Conservation Laws

Energy Eigenket Degeneracy, Symmetry, and Conservation Laws

In many systems, two or more energy eigenstates have the same energy. A simple example of this is a free particle, whose energy eigenstates have wavefunctions that are propagating plane waves. The energy of each of these plane waves is inversely proportional to the square of its wavelength. A wave propagating in the x direction is a different state from one propagating in the y direction, but if they have the same wavelength, then their energies will be the same. When this happens, the states are said to be degenerate.

It turns out that degeneracy occurs whenever a nontrivial unitary operator U commutes with the Hamiltonian. To see this, suppose that is an energy eigenket. Then is an energy eigenket with the same eigenvalue, since

Since U is nontrivial, at least one pair of and must represent distinct states. Therefore, H has at least one pair of degenerate energy eigenkets. In the case of the free particle, the unitary operator which produces the symmetry is the rotation operator, which rotates the wavefunctions by some angle while otherwise preserving their shape.

The existence of a symmetry operator implies the existence of a conserved observable. Let G be the Hermitian generator of U:

It is straightforward to show that if U commutes with H, then so does G:

Therefore,

In obtaining this result, we have used the Schrödinger equation, as well as its dual,

Thus, the expected value of the observable G is conserved for any state of the system. In the case of the free particle, the conserved quantity is the angular momentum.