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15 Waves in A Well

1) A particle of mass m is trapped in an infinite well of length L, where the potential is zero inside and infinity outside. 2) Treating the particle as a wave, the stable wave patterns are standing waves with fixed ends and a wavelength of λn = L/n, where n = 0,1,2,... 3) Combining de Broglie's momentum relation with the wavelength expression gives the momentum and energy En for the nth standing wave as En = n2ħ2/2mL2.

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40 views3 pages

15 Waves in A Well

1) A particle of mass m is trapped in an infinite well of length L, where the potential is zero inside and infinity outside. 2) Treating the particle as a wave, the stable wave patterns are standing waves with fixed ends and a wavelength of λn = L/n, where n = 0,1,2,... 3) Combining de Broglie's momentum relation with the wavelength expression gives the momentum and energy En for the nth standing wave as En = n2ħ2/2mL2.

Uploaded by

Andrew
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Quantum Physics 1

names:

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
doesnt 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 Planks solution to the black body radiation problem, a


particles energy is equal to hf where h is Planks constant and f the
frequency of the wave. De Broglie added the relation between
wavelength and momentum: p = h/(2).

Combine de Broglies 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 2 in one wavelength. So the wave function is

sin( 2 x/)
into which your expression for 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 n . 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.
Then we need to determine the amplitude for the wave equation. This
is done by requiring the integral of the square of the wave function over
the entire well to equal 1. This is because the square of the wave
function is the probability of finding the particle and the probability
that the particle is in the well is 1 = 100%.

So,

An2

n x
sin 2
dx =1
L

determine An :

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