The physical quantum harmonic oscillator (h.o.) in one dimension is a linear combination of the operators and. Its properties are described by the wave function representing its state jxi in the position representation spanned by the eigenvectors. The energy eigenvalue problem of the h.o. in the position representation amounts to solving the second-order Schrödinger equation for x(x). The solution of that equation has been derived in Section 11.2 as a special case of the general solution of exactly solvable potentials.

Very often one encounters Hamiltonians which are linear combinations of operators and where and obey the same commutation relation as and but those operators do not act on fjxig. We call them h.o. like Hamiltonians. In such cases, one cannot define (x) and consequently the eigenvalue problem of such Hamiltonians cannot be reduced to the one of solving Schrödinger's differential equation.

The eigenvalue problem of h.o. like Hamiltonians therefore requires different approaches for its solution. This chapter discusses the method of solving that problem using only the commutation relation between, without reference to any representation. That approach proves useful even for the physical h.o. as it reduces its eigenvalue problem to the one of solving a simple first-order differential equation, a task arguably simpler than solving the second-order differential equation. The eigenvalue spectrum found by the two methods turns out to be the same.

In general, the method of solving the eigenvalue problem by using only the commutation relations between the operators constituting the Hamiltonian is indispensable while dealing with Hamiltonians, such as the ones involving spin, which have no classical counterpart.