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Electron in a Box
Michael Fowler, University of Virginia 9/1/08
Plane Wave Solutions
The best way to gain understanding of Schrödinger’s equation is to solve it for various
potentials. The simplest is a onedimensional ―particle in a box‖ problem. The
appropriate potential is
V
(
x
) = 0 for
x
between 0,
L
and
V
(
x
) = infinity otherwise—that is
to say, there are infinitely high walls at
x
= 0 and
x
=
L
, and the particle is trapped
between them. This turns out to be quite a good approximation for electrons in a long
molecule, and the threedimensional version is a reasonable picture for electrons in
metals.
Between
x
= 0 and
x
=
L
we have
V
= 0, so the wave equation is just
22
2
( , )
( , )
2
x t
x t
i
t
m
x
.
A possible plane wave solution is
( , )
(
)
x t
Ae
i
px
Et
.
On inserting this into the zeropotential Schrödinger equation above we find
E
=
p
2
/2
m
,
as we expect.
It is very important to notice that the complex conjugate, proportional to
()
i
px Et
e
, is
not
a solution to the Schrödinger equation! If we blindly put it into the equation we get
E
= –
p
2
/2
m
,
an unphysical result.
However, a wave function proportional to
i
px Et
e
gives
E
=
p
2
/2
m
, so this plane wave
is
a solution to the equation.
Therefore, the two allowed planewave solutions to the zeropotential Schrödinger
equation are proportional to
e
i
px
Et
(
)
and
i
px Et
e
respectively.
Note that these two solutions have the
same
time dependence
iEt
e
.
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To decide on the appropriate solution for our problem of an electron in a box, of course
we have to bring in the walls—what they mean is that
ψ
= 0 for
x
< 0 and for
x
>
L
because remember 
ψ

2
tells us the probability of finding the particle anywhere, and,
since it’s in the box, it’s trapped
between
the walls, so there’s zero probability of finding
it outside.
The condition
ψ
= 0 at
x
= 0 and
x
=
L
reminds us of the vibrating string with two fixed
ends—the solution of the string wave equation is standing waves of sine form. In fact,
taking the difference of the two permitted planewave forms above gives a solution of
this type:
( , )
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 Fall '07
 MichaelFowler
 wave function, plane wave solutions, IET

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