Page 45 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

2

È
Schrodinger wave mechanics

2.1 The Schro Èdinger equation
In the previous chapter we introduced the wave function to represent the
motion of a particle moving in the absence of an external force. In this chapter
we extend the concept of a wave function to make it apply to a particle acted
upon by a non-vanishing force, i.e., a particle moving under the in¯uence of a
potential which depends on position. The force F acting on the particle is
related to the potential or potential energy V(x)by
dV
F ˆÿ (2:1)
dx
As in Chapter 1, we initially consider only motion in the x-direction. In Section
2.7, however, we extend the formalism to include three-dimensional motion.
In Chapter 1 we associated the wave packet

1 1
Ø(x, t) ˆ p   A( p)e i( pxÿEt)=" dp (2:2)
2ð" ÿ1
with the motion in the x-direction of a free particle, where the weighting factor
A( p)isgiven by

1 1
A(p) ˆ p   Ø(x, t)e ÿi( pxÿEt)=" dx (2:3)
2ð" ÿ1
This wave packet satis®es a partial differential equation, which will be used as
the basis for the further development of a quantum theory. To ®nd this
differential equation, we ®rst differentiate equation (2.2) twice with respect to
the distance variable x to obtain
@ Ø ÿ1 1 2 i( pxÿEt)="
2
ˆ p  p A(p)e dp (2:4)
@x 2 2ð" 5 ÿ1
Differentiation of (2.2) with respect to the time t gives

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