bei48482_FM

166 Chapter Five

and the second partial derivative is

 y y

since i^2 1. The partial derivative of ywith respect to t(now holding xconstant) is

iy

and the second partial derivative is

i^2 ^2 y^2 y

Combining these results gives



which is Eq. (5.3). Hence Eq. (5.5) is a solution of the wave equation.

5.3 SCHRÖDINGER’S EQUATION: TIME-DEPENDENT FORM

A basic physical principle that cannot be derived from anything else

In quantum mechanics the wave function corresponds to the wave variable yof

wave motion in general. However, , unlike y, is not itself a measurable quantity and

may therefore be complex. For this reason we assume that for a particle moving

freely in the xdirection is specified by

Aei(tx) (5.7)

Replacing in the above formula by 2and by gives

Ae^2 i(tx) (5.8)

This is convenient since we already know what and are in terms of the total energy

Eand momentum pof the particle being described by . Because

Eh 2   and 

we have

Free particle Ae(i^ )(Etpx) (5.9)

Equation (5.9) describes the wave equivalent of an unrestricted particle of total

energy Eand momentum pmoving in the xdirection, just as Eq. (5.5) describes, for

example, a harmonic displacement wave moving freely along a stretched string.

The expression for the wave function given by Eq. (5.9) is correct only for freely

moving particles. However, we are most interested in situations where the motion of

a particle is subject to various restrictions. An important concern, for example, is an

electron bound to an atom by the electric field of its nucleus. What we must now do

is obtain the fundamental differential equation for , which we can then solve for 

in a specific situation. This equation, which is Schrödinger’s equation, can be arrived

at in various ways, but it cannotbe rigorously derived from existing physical principles:

2 

p

h

p

^2 y

t^2

1

^2

^2 y

x^2

^2 y

t^2

y

t

^2

^2

i^2 ^2

^2

^2 y

x^2

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