Mathematically, the relation of degeneracy with symmetry can be clarified as follows. Consider a symmetry operation associated with a unitary operator S. Under such an operation, the new Hamiltonian is related to the original Hamiltonian by a similarity transformation generated by the operator S, such that H ′ = S H S − − 1 = S H S † † , since S is unitary. If the Hamiltonian remains unchanged under the transformation operation S, we have S H S † † = H S H S − − 1 = H S H = H S [ S , H ] = 0 Now, if | α α ⟩ ⟩ is an energy eigenstate, H | α α ⟩ ⟩ = E | α α ⟩ ⟩ where E is the corresponding energy eigenvalue. H S | α α ⟩ ⟩ = S H | α α ⟩ ⟩ = S E | α α ⟩ ⟩ = E S | α α ⟩ ⟩ which means that S | α α ⟩ ⟩ is also an energy eigenstate with the same eigenvalue E. If the two states | α α ⟩ ⟩ and S | α α ⟩ ⟩ are linearly independent (i.e. physically distinct), they are therefore degenerate.