Physical Chemistry , 1st ed.

Example 10.9

Consider an electron confined to some finite system. The state of the electron

is described by the wavefunction  2 sin k x,where kis some constant.

Assume that the potential energy is zero, or V(x) 0. What is the energy of

the electron?

Solution

Since the potential energy is zero, the electron has only kinetic energy. The

Schrödinger equation reduces to

^2



m

2





x

2

(^2) E
We rewrite it as

2



m

2





2

x



2 E

We need to evaluate the second derivative of, multiply it by the appropri-

ate set of constants, and regenerate the original wavefunction and find out

what constant Eis multiplying . That Eis the energy of the electron.

Evaluating the second derivative:





x

2

    2 ( 2 sin k x) k

(^2
2) ( 2 sin k x) k (^2
2) 
Therefore, we can substitute k^2
2 into the left side of the Schrödinger
equation:

2



m

2

(k^2

2 ) 

^2

2

k

m

2

2



From this expression, we should see that the energy eigenvalue has the ex-

pression

E

k^2

2



m

2

2

The kinetic energy part of the Hamiltonian has a similar form for all sys-
tems (although it may be described using different coordinate systems, as we
will see in rotational motion). However, the potential energy operator Vˆde-
pends on the system of interest. In the examples of systems using the
Schrödinger equation, different expressions for the potential energy will be
used. What we will find is that the exact form of the potential energy deter-
mines if the second-order differential equation is exactly solvable. If it is, we
say that we have an analyticsolution. In many cases, it is notsolvable analyti-
cally and must be approximated. The approximations can be very good, good
enough for their predictions to agree with experimental determinations.
However, exact solutions to the Schrödinger equation, along with specific pre-
dictions of various observables like energy, are necessary to illustrate the true
usefulness of quantum mechanics.
Among the quantum mechanical operators presented so far, the Hamiltonian
is probably the most important one. As a summary, a short list of quantum-
mechanical operators and their classical counterparts is provided in Table 10.1.

10.7 The Schrödinger Equation 287