QuantumPhysics.dvi

The solution to the harmonic oscillator problem will then provide an approximation to the

problem for general potentialV for reasonably low energies.

Of course, one may derive the spectrum of the harmonic oscillator by solving the Schr ̈odinger

equation. The harmonic oscillator actually provides the perfect example of a system that may

be solved much more directly using operator methods. The techniques we shall present also

will also illustrate more directly how the principles of quantum mechanics may be applied.

5.6.1 Lowering and Raising operators

We begin by reformulating the harmonic oscillator in terms oflowering and raising operators

aanda†, defined as follows,

a =

1

√

2 Mω ̄h

(

Mω X+iP

)

a† =

1

√

2 Mω ̄h

(

Mω X−iP

)

(5.70)

For self-adjoint operatorsXandP, clearlya†is the adjoint operator ofa, as the notation

indeed indicates. This change of variables has been chosen so that

H= ̄hω

(

a†a+

1

2

)

[a,a†] = 1 (5.71)

The terminology oflowering and raising operatorsforaanda†, derives from the fact that

they satisfy the following commutation relations with the HamiltonianH,

[H,a] = − ̄hω a

[H,a†] = + ̄hω a† (5.72)

For this simple system, there are no possible degeneracies and the Hamiltonian by itself may

be used to span the complete set of commuting observables. One may verify that no operator

built fromxandpcommutes withH, lest it be functions ofH. Thus, any state|n〉may be

uniquely labeled by its energyEn,

H|n〉=En|n〉 (5.73)

Applying the operatorsaanda†respectively lowers and raises the energy of a state by ̄hω,

H a|n〉 =

(

aH+ [H,a]

)

|n〉= (En− ̄hω)a|n〉

H a†|n〉 =

(

a†H+ [H,a†]

)

|n〉= (En+ ̄hω)a†|n〉 (5.74)