130_notes.dvi

9.2 The 1D Harmonic Oscillator

Theharmonic oscillator is an extremely important physics problem. Many potentials look
like a harmonic oscillator near their minimum. This is the first non-constant potential for which we
will solve the Schr ̈odinger Equation.

The harmonic oscillator Hamiltonian is given by

H=

p^2

2 m

+

1

2

kx^2

which makes theSchr ̈odinger Equation for energy eigenstates

− ̄h^2

2 m

d^2 u

dx^2

+

1

2

kx^2 u=Eu.

Note that this potential also has a Parity symmetry. The potentialis unphysical because it does
not go to zero at infinity, however, it is often a very good approximation, and this potential can be
solved exactly.

It is standard to remove the spring constantkfrom the Hamiltonian, replacing it with theclassical
oscillator frequency.

ω=

√

k
m
TheHarmonic Oscillator Hamiltonianbecomes.

H=

p^2

2 m

+

1

2

mω^2 x^2

Thedifferential equation to be solved is

− ̄h^2

2 m

d^2 u

dx^2

+

1

2

mω^2 x^2 u=Eu.

To solve the Harmonic Oscillator equation (See section 9.7.4), we will first change to dimensionless
variables, then find the form of the solution forx→±∞, then multiply that solution by a polynomial,
derive a recursion relation between the coefficients of the polynomial, show that the polynomial series
must terminate if the solutions are to be normalizable, derive the energy eigenvalues, then finally
derive the functions that are solutions.

Theenergy eigenvaluesare

En=

(

n+

1

2

)

̄hω

forn= 0, 1 , 2 ,.... There are a countably infinite number of solutions withequal energy spacing.
We have been forced to have quantized energies by the requirement that the wave functions be
normalizable.

Theground state wave functionis.