Page 112 - Bruno Linder Elementary Physical Chemistry

August 18, 2010 11:36 9in x 6in b985-ch09 Elementary Physical Chemistry

Chapter 9

Applications of Quantum Theory

9.1. Translational Motion. Particle-in-a-Box

Consider a particle of mass m in a one-dimensional box of length L.
Assuming that the potential energy is V = 0 inside box and V = ∞ outside
o
the box, the solution of the Schr¨dinger equation yields the energy (see
Fig. 9.1):
2
2
E n = n h /8 mL 2 n =1, 2,... (9.1)
and wave-function
ψ n =(2/L) 1/2 sin(nπx/L) n =1, 2,... (9.2)

Note that the spacing between energy levels increases with increasing n and
decreases with increasing L.When L becomes very large (of macroscopic
dimensions), the energy distribution becomes practically continuous.
Example 9.1. A conjugated polyene molecule is sometimes simulated by
a one-dimensional-box. If L =2.0 nm and an electron in the box is excited
from state 5 to state 6, what is the transition energy?

Solution
The energy difference between level 5 and level 6 is
2
2
∆E 6←5 =11 h /(8 mL )
) J s
11 × (6.6260 × 10 −34 2 2 2
=
) m
8 × 9.10939 × 10 −31 kg × (2.0 × 10 −9 2 2
=1.6554 × 10 −19 J ≈ 1.0 eV (9.3)
2 −2
[Note: I J =1 Nm=1kg ms −2 m= 1kgm s ;1 eV = 1.607 × 10 −19 J]
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