3
Bound States
The Schr¨
odinger equation is a second order differential equation because it contains
the second derivative of the wavefunction. For a second order equation, the solutions
have two degrees of freedom.
In the case of a free particle, there were no boundary
conditions, so that the wavefunction had both degrees of freedom. One is the energy (or
wavelength), which could take any positive value. The second is the phase of the wave.
If a single wall is introduced (for example at
x
= 0), the value of the wavefunction is
constrained at the wall, and the phase is restricted, leaving only one degree of freedom
 the energy. We can see this by looking at the general solution (in terms of sin and
cos)
ψ
(
x
) =
A
sin(
kx
) +
B
cos(
kx
)
(3.1)
and imposing the constraint that
ψ
(0) = 0 at the wall (at
x
= 0).
To satisfy this
condition, there can be no cos component, so that
B
= 0 and the solution for
x >
0 is
ψ
(
x
) =
A
sin(
kx
)
.
(3.2)
In this section, we will look at the simplest example of a potential which imposes two
constraints on the wavefunction. With two constraints, the second order Schr¨
odinger
equation fully defines the allowed wavefunctions.
With no degrees of freedom, the
wavefunction and its energy are restricted to precise,
quantized
values.
3.1
Particle in a box
A potential energy defined by,
V
(
x
) =
0
for
0
≤
x
≤
L
∞
for
x <
0
and
x > L
(3.3)
is a box that constrains a particle between
x
= 0 and
x
=
L
.
Within this box,
the potential is constant, and the particle is free.
At the boundaries of the box the
potential becomes infinite, so that the wavefunction must become zero.
This can be
seen by looking at the Schr¨
odinger equation
−
2
2
m
∂ψ
(
x
)
∂x
+
V
(
x
)
ψ
(
x
) =
Eψ
(
x
)
.
(3.4)
To have a finite energy value, the second term,
V
(
x
)
ψ
(
x
), must be well behaved.
If
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 Fall '06
 henkelman
 Physical chemistry, pH, Energy, Sin, Schr¨dinger equation, ny y Ly

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