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, as in classical mechanics, a wide variety of physical situations can be reduced to it either exactly or approximately. In particular, a system near an equilibrium configuration can often be described in terms of one or more harmonic oscillators. Furthermore, it is one of the few quantum mechanical systems for which a simple exact solution is known.

The following discussion of the quantum harmonic oscillator relies on the article mathematical formulation of quantum mechanics.

One-dimensional harmonic oscillator

Hamiltonian and energy eigenstates

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 (p = −iℏ ∂ /∂x). 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 H_n(θ) are the Hermite polynomials:

They should not be confused with the Hamiltonian, which is unfortunately also denoted by H. The corresponding energy levels are

This energy spectrum is noteworthy for two reasons. Firstly, the energies are "quantized", and may only take the discrete values of ℏω 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 ℏω/2, which is called the "ground state energy" or zero-point energy. 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.

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.

Ladder operator method

The power series solution, though straightforward, is rather tedious. The "ladder operator" method, due to Paul Dirac, allows us to extract the energy eigenvalues without directly solving the differential equation. Furthermore, it is readily generalizable to more complicated problems, notably in quantum field theory. Following this approach, we define the operators a and its adjoint a^†

The operator a is not Hermitian since it and its adjoint a^† are not equal.

In deriving the form of a^†, we have used the fact that the operators x and p, which represent observables, are Hermitian.

The x and p operators obey the following identity, known as the canonical commutation relation:

The square brackets in this equation are a commonly-used notational device, known as the commutator, defined as

Using the above, we can prove the identities

Now, let |ψ_E⟩ denote an energy eigenstate with energy E. The inner product of any ket with itself must be non-negative, so

Expressing a^†a in terms of the Hamiltonian:

so that E ≥ (ℏω / 2). Note that when (a|ψ_E⟩) is the zero ket (i.e. a ket with length zero), the inequality is saturated, so that E = (ℏω/2). It is straightforward to check that there exists a state satisfying this condition; it is the ground (n = 0) state given in the preceding section.

Using the above identities, we can now show that the commutation relations of a and a^† with H are:

Thus, provided (a|ψ_E⟩) is not the zero ket,

Similarly, we can show that

In other words, a acts on an eigenstate of energy E to produce, up to a multiplicative constant, another eigenstate of energy (E − ℏω), and a^† acts on an eigenstate of energy E to produce an eigenstate of energy (E + ℏω.) For this reason, a is called a "lowering operator", and a^† a "raising operator". The two operators together are called "ladder operators". In quantum field theory, a and a^† are alternatively called "annihilation" and "creation" operators because they destroy and create particles, which correspond to our quanta of energy.

Given any energy eigenstate, we can act on it with the lowering operator, a, to produce another eigenstate with ℏω less energy. By repeated application of the lowering operator, it seems that we can produce energy eigenstates down to E = −∞. However, this would contradict our earlier requirement that E ≥ (ℏω / 2). Therefore, there must be a ground-state energy eigenstate, which we label |0⟩ (not to be confused with the zero ket), such that

In this case, subsequent applications of the lowering operator will just produce zero kets, instead of additional energy eigenstate. Furthermore, we have shown above that

Finally, by acting on |0⟩ with the raising operator and multiplying by suitable normalization factors, we can produce an infinite set of energy eigenstates |0⟩,|1⟩,|2⟩, ..., |n⟩, ...}, such that

which matches the energy spectrum which we gave in the preceding section.

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 we measure energy in units of ℏω and distance in units of (ℏ/(mω))^1/2, then the Schrödinger equation becomes:

and the energy eigenfunctions and eigenvalues become

To avoid confusion, we will not adopt these natural units in this article. However, they frequently come in handy when performing calculations.

N-dimensional harmonic oscillator

The one-dimensional harmonic oscillator is readily generalizable to N dimensions, where N = 1, 2, 3, ... . In one dimension, the position of the particle was specified by a single coordinate, x. In N dimensions, this is replaced by N position coordinates, which we label x₁, ..., x_N. Corresponding to each position coordinate is a momentum; we label these p₁, ..., p_N. The canonical commutation relations between these operators are

The Hamiltonian for this system is

As the form of this Hamiltonian makes clear, the N-dimensional harmonic oscillator is exactly analogous to N independent one-dimensional harmonic oscillators with the same mass and spring constant. In this case, the quantities x₁, ..., x_N would refer to the positions of each of the N particles. This is a happy property of the r² potential, which allows the potential energy to be separated into terms depending on one coordinate each.

This observation makes the solution straightforward. For a particular set of quantum numbers {n} the energy eigenfunctions for the N-dimensional oscillator are expressed in terms of the 1-dimensional eigenfunctions as:

In the ladder operator method, we define N sets of ladder operators,

By a procedure analogous to the one-dimensional case, we can then show that each of the a_i and a^†_i operators lower and raise the energy by ℏω respectively. The energy levels of the system are

.
:

As in the one-dimensional case, the energy is quantized. The ground state energy is N times the one-dimensional energy, as we would expect using the analogy to N independent one-dimensional oscillators. There is one further difference: in the one-dimensional case, each energy level corresponds to a unique quantum state. In N-dimensions, except for the ground state, the energy levels are degenerate, meaning there are several states with the same energy.

The degeneracy can be calculated relatively easily, as an example, consider the 3-dimensional case: Define n = n₁ + n₂ + n₃. All states with the same n will have the same energy. For a given n, we choose a particular n₁. Then n₂ + n₃ = n − n₁. There are n − n₁ + 1 possible groups {n₂, n₃}. n₂ can take on the values 0 to n − 1, and for each n₂ the value of n₃ is fixed. The degree of degeneracy therefore is: