Linear potential
February 8, 2008
Linear Potential  Gravity
The Schrodinger equation for the wave function of a bouncing ball is

¯
h
2
2
m
d
2
ψ
dx
2
+
mgxψ
=
Eψ
(1)
where we assume a perfectly elastic collision of the ball with the floor. So
V
(
x
) =
∞
for
x <
0, and
V
(
x
) =
mgx
for
V >
0. There is zero probability
to find the ball at
x <
0 so
ψ
(0) = 0. If we define a characteristic length,
l
0
=
¯
h
2
2
m
2
g
!
1
3
and energy
E
0
=
¯
h
2
mg
2
2
!
1
3
then
x
=
yl
0
and
E
=
E
0
where
y
and
are dimensionless and substitution
into Equation
1 gives

d
2
ψ
dy
2
+
yψ
=
ψ
(2)
Finally with the substitution of
z
=
y

, Equation
2 becomes

d
2
ψ
dz
2
+
zψ
= 0
(3)
Equation
3 is Airy’s equation and the solutions to it are Airy functions.
(Some of the properties of Airy functions are summarized in Chapter 8 of
Griffiths. For more details see http://dlmf.nist.gov/contents/AI/index.html)
The general solution to Airy’s equation is
ψ
(
z
) =
aA
i
(
z
) +
bB
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 RUBIN, D
 mechanics, Gravity, Schrodinger Equation, ORDINARY DIFFERENTIAL EQUATIONS, Fundamental physics concepts, Airy function

Click to edit the document details