238
QUANTUM WELLS, WIRES, AND DOTS
c o s ( T )
4nx n = 4
3nx
cos (T) n = 3
c o s ( a )
2nx n = 2
cos(+)
n = l
odd
even
odd
even
Figure 9.11. Sketch of wavefunctions for the four lowest energy levels ( n = 1-4) of the
one-dimensional infinite square well. For each level the form of the wavefunction is given. on
the left, and its parity (even or odd) is indicated on the right (From C. P. Poole, Jr., Handbook of
Physics, Wiley, New York, 1998, p. 289.)
The electrons confined to the potential well move back and forth along the
direction x, and the probability of finding an electron at a particular value of x is
given by the square of the wavefunction I$,(X))~ for the particular level n where the
electron is located. There are even and odd wavefunctions $,(x) that alternate for the
levels in the one-dimensional square well, and for the infinite square well we have
the unnormalized expressions
$, = cos(nnx/a)
n = 1 , 3 , 5 , . . .
even parity
(9.7)
$, = sin(nnx/a)
n = 2; 4 , 6 , . . .
odd parity
(9.8)
These wavehnctions are sketched in Fig. 9.1 1 for the infinite well. The property
called parity is defined as even when $,(-x) = $,(x), and it is odd when
= -$n(x).