Quantum harmonic oscillator

The Quantum Harmonic Oscillator is a quantum mechanical analogue of the classical harmonic oscillator. It is one of the most important problems in quantum mechanics, because (i) a simple exact solution exists, and (ii) a wide variety of physical situations can be reduced to this. In particular, a system near an equilibrium configuration can often be described in terms of one or more harmonic oscillators.

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

One-dimensional Harmonic Oscillator

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:

H = p²/2m + (1/2) mω² x²

where x is the position operator, and p is the momentum operator (p = - iℏ ∂ /∂x). We wish to solve Schrödinger's wave equation,

H |ψ_E> = E |ψ_E>

to find energy eigenkets |ψ_E> and the corresponding energy levels E. We can solve the differential equation in the coordinate basis, using a power series method; the solution is

<x|ψ_n> = (2ⁿn!)^-1/2 (mω/πℏ)^1/4 exp(- mωx²/2ℏ) H_n((mω/ℏ)^1/2 x)

where n is a non-negative integer; the corresponding energy levels are

E = ℏ&omega ( n + 1/2).

The functions H_n(θ) are the Hermite polynomials. The first two Hermite polynomials are H₀(θ) = 1 and H₁(θ) = 2θ.

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. Following this approach, we define the operator

a = (mω/2ℏ)^1/2 ( x + (i /mω) p )

which, it should be noted, is not Hermitian. Its Hermitian conjugate is

a^† = (mω/2ℏ)^1/2 ( x - (i /mω) p )

We have used the fact that the operators x and p, which represent observables, are Hermitian. Using the canonical commutation relations, it is possible to show that

[a , a^†] ≡ a a^† - a^†a = 1

and that

H = ℏω (a^†a + 1/2)

Let |ψ_E> denote an energy eigenket with energy E. The inner product of any ket with itself must be non-negative, so

(a |ψ_E>, a |ψ_E>)

= <ψ_E | a^†a |ψ_E>

≥ 0

Expressing a^†a in terms of the Hamiltonian:

<ψ_E | (H/ℏω) - 1/2 |ψ_E>

= (E/ℏω) - 1/2

≥ 0.

That is:

E ≥ (ℏω / 2)

We have not yet shown that any energy eigenvalue has the value (ℏω / 2), but they can be no smaller than this. Furthermore, note that when (a|ψ_E>) is the zero ket (i.e. a ket with length zero), the equality E = (ℏω / 2) holds.

The commutation relations of a and a^&dagger with H are:

[H,a] = ℏω [a^†a + 1/2, a] = ℏω (a^†[a, a] + [a^†, a]a + [1/2, a]) = - ℏω a

[H,a^†] = ℏω [a^†a + 1/2, a^†] = ℏω (a^†[a, a^†] + [a^†, a^†]a + [1/2, a^†]) = ℏω a^†

Using these, consider the action of H on the ket (a|ψ_E>). Provided (a|ψ_E>) is not the zero ket,

H (a|ψ_E>)

= ([H,a] + a H) |ψ_E>

= (- ℏω a + a E) |ψ_E>

= (E - ℏω) (a|ψ_E>)

This informs us that (a|ψ_E>) is also an energy eigenket up to a normalization factor, with eigenvalue (E - ℏω). Similarly,

H (a^†|ψ_E>) = (E + ℏω) (a^†|ψ_E>)

so a^†|ψ_E> is also an energy eigenket up to a normalization factor, with eigenvalue (E + ℏω). As a result, a is referred to as the "annihilation" or "lowering" operator, and a^† is referred to as the "creation" or "raising" operator. Together, a and a^† are often called the "ladder operators".

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

a |0> = 0    (zero ket)

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

H |0> = (ℏω/2) |0>

Finally, by acting on |0> with the raising operator and multiplying by suitable normalization factors, we can produce an infinite set of energy eigenkets

{ |0>, |1>, |2>, ..., |n>, ... }

such that

H |n> = ℏω (n + 1/2) |n>

which is the desired energy spectrum.