Wave goodbye to familiar territory, you are about to
enter the quantum zone.
Your first quantum mechanics problem.
With the addition of a few modern ideas, you will be
able to find the solution to one of the five quantum
problems which can be solved exactly.
The problem is that of a particle of mass m trapped
in a well of length L having a potential of zero inside
and infinity outside.
Idea number one: Based on the observation that
electrons reflected off crystal faces produce
diffraction patterns, particles can be treated as
waves.
Using this idea for the particle in an infinite well of
length L, stable wave patterns are the standing waves
with fixed ends. The particle doesn’t have an infinite
amount of energy and so its wave must be zero at the
walls.
Write out the expression for the wavelength of the
nth standing wave in terms of L, where n = 0,1,2,.. .
Idea 2: From Plank’s solution to the black body
radiation problem, a particle’s energy is equal to hf
where h is Plank’s constant and f the frequency of the
wave. De Broglie added the relation between
wavelength and momentum: p = h/(2πλ).
Combine de Broglie’s momentum relation with your
wavelength expression above to get an expression for
the momentum for the nth standing wave.
Now we can apply some straight-forward mechanics.
Using the idea that the kinetic energy is given by
E = p2/2m ,
Obtain an expression for the energy of the nth
standing wave.
This is the set of possible energies for the stable
states in the infinite well of length L.
We can even get the wave functions for the states
based on the wavelength. To do so, recall that the
argument of the sine function goes through 2π in one
wavelength. So the wave function is
sin( 2π x/λ)
into which your expression for λ can be substituted to
get the wave function for the nth stable state.
nth wave function:
So that particle of mass m bouncing around in the well
can have energies En and wave functions ψn . States
with other energies are not stable and particles
confined to such a well are not found with energies
other than the En in the set you have found.