Quantum harmonic oscillator

The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Furthermore, it is one of the few quantum-mechanical systems for which an exact, analytical solution is known.

One-dimensional harmonic oscillator

Hamiltonian and energy eigenstates

The Hamiltonian of the particle is:
where is the particle's mass, is the force constant, is the angular frequency of the oscillator, is the position operator, and is the momentum operator. The first term in the Hamiltonian represents the kinetic energy of the particle, and the second term represents its potential energy, as in Hooke's law.
The time-independent Schrödinger equation is,
where denotes a real number that will specify a time-independent energy level, or eigenvalue, and the solution denotes that level's energy eigenstate.
Then solve the differential equation representing this eigenvalue problem in the coordinate basis, for the wave function, using a spectral method. It turns out that there is a family of solutions. In this basis, they amount to Hermite functions,
The functions are the physicists' Hermite polynomials,
The corresponding energy levels are
The expectation values of position and momentum combined with variance of each variable can be derived from the wavefunction to understand the behavior of the energy eigenkets. They are shown to be and owing to the symmetry of the problem, whereas:
The variance in both position and momentum are observed to increase for higher energy levels. The lowest energy level has value of which is its minimum value due to uncertainty relation and also corresponds to a gaussian wavefunction.
This energy spectrum is noteworthy for four reasons. First, the energies are quantized, meaning that only discrete energy values are possible; this is a general feature of quantum-mechanical systems when a particle is confined. Second, these discrete energy levels are equally spaced, unlike in the Bohr model of the atom, or the particle in a box. Third, the lowest achievable energy is not equal to the minimum of the potential well, but above it; this is called zero-point energy. Because of the zero-point energy, the position and momentum of the oscillator in the ground state are not fixed, but have a small range of variance, in accordance with the Heisenberg uncertainty principle. Fourth, the energy levels are nondegenerate implying that every eigenvalue is associated with only one solution.
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 peaks 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 more of its time near the turning points, where it is moving the slowest. The correspondence principle is thus satisfied. Moreover, special nondispersive wave packets, with minimum uncertainty, called coherent states oscillate very much like classical objects, as illustrated in the figure; they are not eigenstates of the Hamiltonian.

Ladder operator method

The "ladder operator" method, developed by Paul Dirac, allows extraction of the energy eigenvalues without directly solving the differential equation. It is generalizable to more complicated problems, notably in quantum field theory. Following this approach, we define the operators and its adjoint,
Note these operators classically are exactly the generators of normalized rotation in the phase space of and, i.e they describe the forwards and backwards evolution in time of a classical harmonic oscillator.
These operators lead to the following representation of and ,
The operator is not Hermitian, since itself and its adjoint are not equal. The energy eigenstates, when operated on by these ladder operators, give
From the relations above, we can also define a number operator, which has the following property:
The following commutators can be easily obtained by substituting the canonical commutation relation,
and the Hamilton operator can be expressed as
so the eigenstates of are also the eigenstates of energy.
To see that, we can apply to a number state :
Using the property of the number operator :
we get:
Thus, since solves the TISE for the Hamiltonian operator, is also one of its eigenstates with the corresponding eigenvalue:
QED.
The commutation property yields
and similarly,
This means that acts on to produce, up to a multiplicative constant,, and acts on to produce. For this reason, is called an annihilation operator, and a creation operator. The two operators together are called ladder operators.
Given any energy eigenstate, we can act on it with the lowering operator,, to produce another eigenstate with less energy. By repeated application of the lowering operator, it seems that we can produce energy eigenstates down to. However, since
the smallest eigenvalue of the number operator is 0, and
In this case, subsequent applications of the lowering operator will just produce zero, instead of additional energy eigenstates. Furthermore, we have shown above that
Finally, by acting on with the raising operator and multiplying by suitable normalization factors, we can produce an infinite set of energy eigenstates
such that
which matches the energy spectrum given in the preceding section.
Arbitrary eigenstates can be expressed in terms of,

Analytical questions

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. First, one should find the ground state, that is, the solution of the equation. In the position representation, this is the first-order differential equation
whose solution is easily found to be the Gaussian
Conceptually, it is important that there is only one solution of this equation; if there were, say, two linearly independent ground states, we would get two independent chains of eigenvectors for the harmonic oscillator. Once the ground state 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. One can also prove that, as expected from the uniqueness of the ground state, the Hermite functions energy eigenstates constructed by the ladder method form a complete orthonormal set of functions.
Given that Hermite functions are either even or odd, it can be shown that the average displacement and average momentum is 0 for all states in QHO.
Explicitly connecting with the previous section, the ground state |0⟩ in the position representation is determined by,
hence
so that, and so on.

Natural length and energy scales

The quantum harmonic oscillator possesses natural scales for length and energy, which can be used to simplify the problem. These can be found by nondimensionalization.
The result is that, if energy is measured in units of and distance in units of, then the Hamiltonian simplifies to
while the energy eigenfunctions and eigenvalues simplify to Hermite functions and integers offset by a half,
where are the Hermite polynomials.
To avoid confusion, these "natural units" will mostly not be adopted in this article. However, they frequently come in handy when performing calculations, by bypassing clutter.
For example, the fundamental solution of, the time-dependent Schrödinger operator for this oscillator, simply boils down to the Mehler kernel,
where. The most general solution for a given initial configuration then is simply

Coherent states

The coherent states of the harmonic oscillator are special nondispersive wave packets, with minimum uncertainty, whose observables' expectation values evolve like a classical system. They are eigenvectors of the annihilation operator, not the Hamiltonian, and form an overcomplete basis which consequentially lacks orthogonality.
The coherent states are indexed by and expressed in the basis as
Since coherent states are not energy eigenstates, their time evolution is not a simple shift in wavefunction phase. The time-evolved states are, however, also coherent states but with phase-shifting parameter instead:.
Because and via the Kermack-McCrae identity, the last form is equivalent to a unitary displacement operator acting on the ground state:. Calculating the expectation values:
where is the phase contributed by complex. These equations confirm the oscillating behavior of the particle.
The uncertainties calculated using the numeric method are:
which gives. Since the only wavefunction that can have lowest position–momentum uncertainty,, is a Gaussian wavefunction, and since the coherent state wavefunction has minimum position–momentum uncertainty, we note that the general Gaussian wavefunction in quantum mechanics has the form:Substituting the expectation values as a function of time, gives the required time varying wavefunction.

The probability of each energy eigenstates can be calculated to find the energy distribution of the wavefunction:
which corresponds to a Poisson distribution.

Highly excited states

When is large, the eigenstates are localized into the classical allowed region, that is, the region in which a classical particle with energy can move. The eigenstates are peaked near the turning points: the points at the ends of the classically allowed region where the classical particle changes direction. This phenomenon can be verified through asymptotics of the Hermite polynomials, and also through the WKB approximation.
The frequency of oscillation at is proportional to the momentum of a classical particle of energy and position. Furthermore, the square of the amplitude is inversely proportional to , reflecting the length of time the classical particle spends near. The system behavior in a small neighborhood of the turning point does not have a simple classical explanation, but can be modeled using an Airy function. Using properties of the Airy function, one may estimate the probability of finding the particle outside the classically allowed region, to be approximately
This is also given, asymptotically, by the integral