Creation and Annihilation Operators - Derivation For Quantum Harmonic Oscillator

Derivation For Quantum Harmonic Oscillator

In the context of the quantum harmonic oscillator, we reinterpret the ladder operators as creation and annihilation operators, adding or subtracting fixed quanta of energy to the oscillator system. Creation/annihilation operators are different for bosons (integer spin) and fermions (half-integer spin). This is because their wavefunctions have different symmetry properties.

First consider the simpler bosonic case of the phonons of the quantum harmonic oscillator.

Start with the Schrödinger equation for the one dimensional time independent quantum harmonic oscillator

Make a coordinate substitution to nondimensionalize the differential equation

.

and the Schrödinger equation for the oscillator becomes

.

Note that the quantity is the same energy as that found for light quanta and that the parenthesis in the Hamiltonian can be written as

The last two terms can be simplified by considering their effect on an arbitrary differentiable function f(q),

which implies,

Therefore

and the Schrödinger equation for the oscillator becomes, with substitution of the above and rearrangement of the factor of 1/2,

.

If we define

as the "creation operator" or the "raising operator" and

as the "annihilation operator" or the "lowering operator"

then the Schrödinger equation for the oscillator becomes

This is significantly simpler than the original form. Further simplifications of this equation enables one to derive all the properties listed above thus far.

Letting, where "p" is the nondimensionalized momentum operator then we have

and

.

Note that these imply that

in contrast to the so-called "normal operators" of mathematics, which have a similar representation (e.g. with self-adjoint But in the case of normal operators one would be dealing with commuting i.e. with so that the 1 at the extreme r.h.s. of the previous equation would be replaced by 0, which would have the consequence of one-and-the-same set of eigenfunctions (and/or eigendistributions) for both and, whereas here common eigenfunctions or eigendistributions of the operators p and q don't exist.

Thus, although in the present case one is explicitly dealing with non-normal operators, by the commutation relation given above, the Hamiltonian operator can be expressed as

And and operators give the following commutation relations with the Hamiltonian

These relations can be used to find the energy eigenstates of the quantum harmonic oscillator. Assuming that is an eigenstate of the Hamiltonian . Using these commutation relations it can be shown that

This shows that and are also eigenstates of the Hamiltonian with eigenvalues and . This identifies the operators and as lowering and rising operators between the eigenstates. Energy difference between two eigenstates is .

The ground state can be found by assuming that the lowering operator will collapse it, . And then using the Hamiltonian in terms of rising and lowering operators,

the wave-function on the right is non-zero, thus term in brackets must be. This gives the ground state energy . This allows to identify the energy eigenvalue of any eigenstate as

Furthermore it can be shown that the first-mentioned operator, the number operator plays a most-important role in applications, while the second one, can simply be replaced by So one simply gets

.