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48 Schro Èdinger wave mechanics
1
hHi E ø (x)ø(x)dx E
ÿ1
where we have also applied equation (2.32). We have just shown that the
separation constant E is the expectation value of the Hamiltonian, or the total
energy for the stationary state, so that `E' is a desirable designation. Since the
energy is a real physical quantity, the assumption that E is real is justi®ed.
È
In the application of Schrodinger's equation (2.30) to speci®c physical
examples, the requirements that ø(x) be continuous, single-valued, and square-
integrable restrict the acceptable solutions to an in®nite set of speci®c functions
ø n (x), n 1, 2, 3, ... , each with a corresponding energy value E n . Thus, the
energy is quantized, being restricted to certain values. This feature is illustrated
in Section 2.5 with the example of a particle in a one-dimensional box.
Since the partial differential equation (2.6) is linear, any linear superposition
of solutions is also a solution. Therefore, the most general solution of equation
(2.6) for a time-independent potential energy V(x)is
X
Ø(x, t) c n ø n (x)e ÿiE n t=" (2:33)
n
where the coef®cients c n are arbitrary complex constants. The wave function
Ø(x, t) in equation (2.33) is not a stationary state, but rather a sum of
stationary states, each with a different energy E n .
2.5 Particle in a one-dimensional box
È
As an illustration of the application of the time-independent Schrodinger
equation to a system with a speci®c form for V(x), we consider a particle
con®ned to a box with in®nitely high sides. The potential energy for such a
particle is given by
V(x) 0, 0 < x < a
1, x , 0, x . a
and is illustrated in Figure 2.1.
È
Outside the potential well, the Schrodinger equation (2.30) is given by
2
" d ø
2
ÿ 1ø Eø
2m dx 2
for which the solution is simply ø(x) 0; the probability is zero for ®nding
the particle outside the box where the potential is in®nite. Inside the box, the
Schrodinger equation is
È
2
2
" d ø
ÿ Eø
2m dx 2
