Note: The graphs are not normalized, and the signs of some of the functions differ from those given in the text. The first term in the Commutation relations in quantum mechanics pdf represents the kinetic energy of the particle, and the second term represents its potential energy. It turns out that there is a family of solutions. This energy spectrum is noteworthy for three reasons.
The ground state probability density is concentrated at the origin, which means the particle spends most of its time at the bottom of the potential well, as one would expect for a state with little energy. As the energy increases, the probability density becomes peaked at the classical “turning points”, where the state’s energy coincides with the potential energy. See the discussion below of the highly excited states. For this reason, they are sometimes referred to as “creation” and “annihilation” operators. In this case, subsequent applications of the lowering operator will just produce zero kets, instead of additional energy eigenstates.
The preceding analysis is algebraic, using only the commutation relations between the raising and lowering operators. Once the algebraic analysis is complete, one should turn to analytical questions. Once the ground states is computed, one can show inductively that the excited states are Hermite polynomials times the Gaussian ground state, using the explicit form of the raising operator in the position representation. The quantum harmonic oscillator possesses natural scales for length and energy, which can be used to simplify the problem. To avoid confusion, we will mostly not adopt these “natural units” in this article. However, they frequently come in handy when performing calculations, by bypassing clutter.
The eigenstates are peaked near the turning points, that is the points at the ends of the classically allowed region where the classical particle changes direction. This observation makes the solution straightforward. As in the one-dimensional case, the energy is quantized. There is one further difference: in the one-dimensional case, each energy level corresponds to a unique quantum state. The degeneracy can be calculated relatively easily.