<P> For a particle moving on a cone under the influence of 1 / r and r potentials, centred at the tip of the cone, the conserved quantities corresponding to accidental symmetry will be two components of an equivalent of the Runge - Lenz vector, in addition to one component of the angular momentum vector . These quantities generate SU (2) symmetry for both potentials . </P> <P> A particle moving under the influence of a constant magnetic field, undergoing cyclotron motion on a circular orbit is another important example of an accidental symmetry . The symmetry multiplets in this case are the Landau levels which are infinitely degenerate . </P> <P> In atomic physics, the bound states of an electron in a hydrogen atom show us useful examples of degeneracy . In this case, the Hamiltonian commutes with the total orbital angular momentum L 2 ^ (\ displaystyle (\ hat (L ^ (2)))), its component along the z - direction, L z ^ (\ displaystyle (\ hat (L_ (z)))), total spin angular momentum S 2 ^ (\ displaystyle (\ hat (S ^ (2)))) and its z - component S z ^ (\ displaystyle (\ hat (S_ (z)))). The quantum numbers corresponding to these operators are l (\ displaystyle l), m l (\ displaystyle m_ (l)), s (\ displaystyle s) (always 1 / 2 for an electron) and m s (\ displaystyle m_ (s)) respectively . </P> <P> The energy levels in the hydrogen atom depend only on the principal quantum number n . For a given n, all the states corresponding to l = 0 (\ displaystyle l = 0) → n − 1 (\ displaystyle n - 1) have the same energy and are degenerate . Similarly for given values of n and l, the (2 l + 1) (\ displaystyle (2l + 1)), states with m l = − l (\ displaystyle m_ (l) = - l) → l (\ displaystyle l) are degenerate . The degree of degeneracy of the energy level E is therefore: ∑ l = ⁡ 0 n − 1 (2 l + 1) = n 2 (\ displaystyle \ sum _ (l \ mathop (=) 0) ^ (n - 1) (2l + 1) = n ^ (2)), which is doubled if the spin degeneracy is included . </P>

Calculate the degeneracy of each of the first three energy levels for a particle in a cubic box