2.3.4_slides solving for the particle-in-a-box
TRANSCRIPT
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2.3 Particle in a box
Slides: Video 2.3.4 Solving for the
particle in a boxText reference: Quantum Mechanics
for Scientists and EngineersSection 2.6 (first part)
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The particle in a box
Solving for the particle in a bo
Quantum mechanics for scientists and engineers David M
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Particle in a box
We consider a particle of mass m
with a spatially-varying potential V ( z ) in th z direction
so we have a Schrödinger equation
where E is the energy of the particleand ( z ) is the wavefunction
22
22
d zV z z E z
m dz
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Particle in a box
Suppose the potential energy is asimple “rectangular” potential well
thickness L z
Potential energy is constant inside
we choose thererising to infinity at the walls
i.e., at andWe will sometimes call thisan infinite or infinitely deep
(potential) well
E n e r g y0V
0 z z z L
0 z
0V
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Particle in a box
Because these potentials at andat are infinitely highbut the particle’s energy E is
presumably finite
we presume there is no possibilityof finding the particle outside
i.e., for orso the wavefunction is 0 thereso should be 0 at the walls
E n e r g y
0 z z z L
0 z
0 z
z z L
0V
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Particle in a box
With these choicesinside the well
the Schrödinger equation
becomes
with the boundary conditions
and
E n e r g y
0 z
0V
22
22
d zV z z E z
m dz
22
22
d z E z
m dz
0 0 0 z L
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Particle in a box
The general solution to the equation
is of the form
where A and B are constants
andThe boundary condition
means because
E n e r g y
0 z
0V
22
22
d z E zm dz
sin cos z A kz B kz
22 /k mE
0 0
0 B cos 0 1
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Particle in a box
With nowand the condition
kz must be a multiple of , i.e.,
where n is an integerSince, therefore,
the solutions are
with
E n e r g y
sin z A kz 0 z L
22 / / zk mE n L
2 2
2
k E
m
sinn n z
n z
z A L
22
2n z
n
E m L
1n
2n
3n
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Particle in a box
We restrict n to positive integersfor the following reasonsSince for any real
number a
the wavefunctions with negative nare the same as those withpositive n
within an arbitrary factor, here -1the wavefunction for is trivial
the wavefunction is 0 everywhere
E n e r g y
1n
2n
3n
1, 2,n
sin sina a
0n
n z A
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Particle in a box
We can normalize the wavefunctions
To have this integral equal 1choose
Note A n can be complex
All such solutions are arbitrarywithin a unit complex factor
Conventionally, we choose A n
real for simplicity in writing
E n e r g y
1n
2n
3n 2 22
0sin 2
z L
zn n
z
Ln z A dz A L
2 /n z L
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Particle in a box
E n e r g y
1n
2n
3n
2sinn
z z
n z z
L L
22
2n
z
n E m L
1, 2,n 0
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