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Quantum Harmonic Oscillator
The quantum harmonic oscillator is the quantum mechanical analogue of the classical
harmonic oscillator. It is one of the most important model systems in quantum mechanics
because an arbitrary potential can be approximated as a harmonic potential at the vicinity
of a stable equilibrium point. Furthermore, it is one of the few quantum mechanical
systems for which a simple exact solution is known.
In the one-dimensional harmonic oscillator problem, a particle of mass m is subject to a
potential V(x) = (1/2)mω2 x2. The Hamiltonian of the particle is:
where x is the position operator, and p is the momentum operator
.
The first term represents the kinetic energy of the particle, and the second term represents
the potential energy in which it resides. In order to find the energy levels and the
corresponding energy eigenstates, we must solve the time-independent Schrödinger
equation,
.
We can solve the differential equation in the coordinate basis, using a power series
method. It turns out that there is a family of solutions,
The first six solutions (n = 0 to 5) are shown on the right. The functions Hn are the
Hermite polynomials:
They should not be confused with the Hamiltonian, which is also denoted by H. The
corresponding energy levels are
.
This energy spectrum is noteworthy for three reasons. Firstly, the energies are
"quantized", and may only take the discrete values of
times 1/2, 3/2, 5/2, and so forth.
This is a feature of many quantum mechanical systems. In the following section on ladder
operators, we will engage in a more detailed examination of this phenomenon. Secondly,
the lowest achievable energy is not zero, but
, which is called the "ground state
energy" or zero-point energy. In the ground state, according to quantum mechanics, an
oscillator performs null oscillations and its average kinetic energy is positive. It is not
obvious that this is significant, because normally the zero of energy is not a physically
meaningful quantity, only differences in energies. Nevertheless, the ground state energy
has many implications, particularly in quantum gravity. The final reason is that the
energy levels are equally spaced, unlike the Bohr model or the particle in a box.
Note that the ground state probability density is concentrated at the origin. This means the
particle spends most of its time at the bottom of the potential well, as we would expect
for a state with little energy. As the energy increases, the probability density becomes
concentrated at the "classical turning points", where the state's energy coincides with the
potential energy. This is consistent with the classical harmonic oscillator, in which the
particle spends most of its time (and is therefore most likely to be found) at the turning
points, where it is the slowest. The correspondence principle is thus satisfied.