Biomedical Engineering Reference
In-Depth Information
We notice also that the values of
±
n
give wavefunctions that differ only by multiplication
by
−
1, since
sin
sin
n
π
x
L
n
π
x
L
−
=−
According to the Born interpretation, both give the same probability density and so we
only need one set. It is conventional to take the quantum numbers as positive and the value
of
A
as the positive square root so the results can be collected together as
n
2
h
2
8
mL
2
ε
n
=
U
0
+
2
L
sin
n
π
x
(11.11)
ψ
n
=
L
n
=
1, 2, 3, ...
Plots of the squares of the wavefunctions for
n
=
1 and
n
=
3 are shown in Figure 11.2.
For a particle with
n
1 there is zero probability of finding the particle at the end points
and a maximum probability of finding the particle at the centre point. For
n
=
=
3 there are
three equal maxima of probability.
2
1.5
1
0.5
0
0.2
0.4
0.6
0.8
1
x
/L
Figure 11.2
Squares of wavefunction for n
=
1
(full curve) and n
=
3
(dotted curve)
11.4 Correspondence Principle
In 1923, Neils Bohr formulated a principle that he had already used in his treatment of the
hydrogen atom, called the
correspondence principle
; this states that quantum and classical
predictions have to agree with each other in the limit, as the quantum number(s) get large.
In other words, the results of classical physics may be considered to be the limiting cases of
