bei48482_FM

experiments performed on hydrogen atoms always show that each one contains a whole

electron, not 27 percent of an electron in a certain region and 73 percent elsewhere.

The probability is one of findingthe electron, and although this probability is smeared

out in space, the electron itself is not.

Operators and Eigenvalues

The condition that a certain dynamical variable Gbe restricted to the discrete values

Gn—in other words, that Gbe quantized—is that the wave functions (^) nof the system
be such that
Eigenvalue equation Gˆ (^) nGn (^) n (5.34)
whereGˆ is the operator that corresponds to Gand each Gnis a real number. When
Eq. (5.34) holds for the wave functions of a system, it is a fundamental postulate of
quantum mechanics that any measurement of Gcan only yield one of the values Gn.
If measurements of Gare made on a number of identical systems all in states described
by the particular eigenfunction k, each measurement will yield the single value Gk.
Example 5.3
An eigenfunction of the operator d^2 dx^2 is e^2 x. Find the corresponding eigenvalue.
Solution
HereGˆd^2 dx^2 , so
Gˆ  (e^2 x)
(e
2 x)
 (2e
2 x) 4 e 2 x
But e^2 x , so
Gˆ  4

From Eq. (5.34) we see that the eigenvalue Ghere is just G4.
In view of Eqs. (5.25) and (5.26) the total-energy operatorEˆof Eq. (5.24) can also
be written as
HˆU (5.35)
and is called the Hamiltonian operatorbecause it is reminiscent of the Hamiltonian
function in advanced classical mechanics, which is an expression for the total energy
of a system in terms of coordinates and momenta only. Evidently the steady-state
Schrödinger equation can be written simply as
Schrödinger’s Hˆ (^) nEn (^) n (5.36)
equation
^2
x^2
2
2 m
Hamiltonian
operator
d
dx
d
dx^2
d
dx
d^2
dx^2
176 Chapter Five
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