Forces, Potentials, and the Shell Model
Recall the Infinite Square Well (1D)
Solve Shroedinger’s equation:
E
H
E
V
dx
d
2
2
Result:
Consideration of boundary conditions (the behavior of the wavefunction at the walls)
results in quantization.
Both wavefunctions and eigenstates (energy levels)
2
2
2
8
mL
h
n
E
n
Notice the dependence of the energy levels on the size of the box, and on the principal
quantum number.
Harmonic oscillator (1D)
Hooke’s law :
)
(
0
x
x
k
F
If
0
x
x
, the system is at equilibrium because there is no force. However if
x
is
different from
0
x
there is a force which acts to restore the position to the equilibrium
value (Notice the negative sign.)
dx
dV
F
Integrating we get,
2
0
)
(
2
1
x
x
k
V
Now solve Schrodinger’s equation using this potential.
Solution: Wavefunctions and eigenvalues
Eigenvalues:
)
2
1
(
n
E
n
where
m
k

This
** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*
Notice the energy spacing for the harmonic oscillator. What is the minimum energy of
the harmonic oscillator?