Physics 401 Homework 9---Due November 11
The first three problems are based on the time-independent Schrödinger equation for an attractive
potential in terms of dimensionless variables:
)()()(
2
2
2ξψεξψξρ
ξ−=
+
∂
∂f
where
)()( 0
0x
x
fVxV −=
,
0
x
x
≡ξ
,
2
2
00
22
xVm
≡ρ
,
2
2
0
2
xEm
≡ε
.
1. As
ρ
becomes large the well becomes deep and supports many bound states. The
lowest lying states are well-localized at the bottom of the well where the well looks
harmonic.
a. From this you should be able to show that in the limit of large
ρ
the ground
state dimensionless energy,
0
ε
is given approximately by
2
)(
)( max
max
2
0
ξ
ρξρε f
f′′
−
+−≈
where
max
ξ
is the value of
ξ
which
maximizes f (i.e. miminimizes the potential which is attractive).
b. Show that the excitation energy of the first excited state in dimensionless
units is given approximately by
)(2 max01 ξρεε f′′
−≈−
.
The approximations in a. and b. should become more an accurate as ρ increases
and become exact in the
∞→ρ
limit.
c. Consider the case where
)cosh(
1
)( ξ
ξ =f
. Show that the approximate
ground state energy and excitation energy of the first excited state in
dimensionless units is given by
ρρε 2
1
2
0+−≈
ρεε 2
01 ≈−
2. Numerically solve for the ground state for the for the potential with
)cosh(
1
)( ξ
ξ =f
. Determine the bound state energies by adjusting the energy until
the wave function has a very small exponentially growing part. Plot the approximate
wave function. You may find the mathematica notebook on the course website
useful.
a. Do this analysis for the case where
.10=ρ
b. Do this analysis for the case where
.100=ρ
3. Verify that the energies obtained in 2a. and 2b. are close to those given by the
approximations in 1c. Which case works better in terms of number of accurately
predicted digits? Why?
Griffths: 2.30, 2.52
