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Simple Harmonic Oscillator

The classical Hamiltonian of a simple harmonic oscillator is

 (C.106)

where is the so-called force constant of the oscillator. Assuming that the quantum-mechanical Hamiltonian has the same form as the classical Hamiltonian, the time-independent Schrödinger equation for a particle of mass and energy moving in a simple harmonic potential becomes

 (C.107)

Let , where is the oscillator's classical angular frequency of oscillation. Furthermore, let

 (C.108)

and

 (C.109)

Equation (C.107) reduces to

 (C.110)

We need to find solutions to the previous equation that are bounded at infinity. In other words, solutions that satisfy the boundary condition as .

Consider the behavior of the solution to Equation (C.110) in the limit . As is easily seen, in this limit, the equation simplifies somewhat to give

 (C.111)

The approximate solutions to the previous equation are

 (C.112)

where is a relatively slowly-varying function of . Clearly, if is to remain bounded as then we must chose the exponentially decaying solution. This suggests that we should write

 (C.113)

where we would expect to be an algebraic, rather than an exponential, function of .

Substituting Equation (C.113) into Equation (C.110), we obtain

 (C.114)

Let us attempt a power-law solution of the form

 (C.115)

Inserting this test solution into Equation (C.114), and equating the coefficients of , we obtain the recursion relation

 (C.116)

Consider the behavior of in the limit . The previous recursion relation simplifies to

 (C.117)

Hence, at large , when the higher powers of dominate, we have

 (C.118)

It follows that varies as as . This behavior is unacceptable, because it does not satisfy the boundary condition as . The only way in which we can prevent from blowing up as is to demand that the power series (C.115) terminate at some finite value of . This implies, from the recursion relation (C.116), that

 (C.119)

where is a non-negative integer. Note that the number of terms in the power series (C.115) is . Finally, using Equation (C.109), we obtain

 (C.120)

for .

Hence, we conclude that a particle moving in a harmonic potential has quantized energy levels that are equally spaced. The spacing between successive energy levels is , where is the classical oscillation frequency. Furthermore, the lowest energy state ( ) possesses the finite energy . This is sometimes called zero-point energy. It is easily demonstrated that the (normalized) wavefunction of the lowest-energy state takes the form

 (C.121)

where .

Next: Angular Momentum Up: Wave Mechanics Previous: Three-Dimensional Wave Mechanics
Richard Fitzpatrick 2016-01-25