BioPHYSICAL chemistry

Properties of the solutions

The quantum-mechanical problem was solved by substituting the potential
for the simple harmonic oscillator:

(11.10)

Vx()= kx

1

2

2

CHAPTER 11 VIBRATIONAL MOTION 225

This is called the Hermite equation. The general solution of this equation would require

special approaches involving the use of summations of polynomials. However, the critical

part can be determined by realizing that a solution can be written by looking at the limit

of y →∞then ε<<y^2 and ε−y^2 ≈−y^2. For this limiting case, the last equation reduces to:

(db11.8)

The appearance of this equation is similar to what we have seen before with the second

derivative of the wavefunction, yielding the wavefunction times another term, so we expect

that the solution case be expressed in terms of an exponential. In this case, the multiplying

term is not a constant but rather a variable, so the exponential term is somewhat more involved

than being a constant times the variable. The solutions for this are given by:

ψ(y) =e−y

(^2) /2
(db11.9)
so
(db11.10)
(db11.11)
The correctness of this solution is shown by substitution of the second derivative into eqn
db11.8:
[y^2 ψ(y)] −y^2 ψ(y) = 0 (db11.12)
This shows us that the solution to this problem will always have an exponential term. The form
used above actually represents the ground-state wavefunction. The higher-level wavefunctions
are given by the exponential times a polynomial term as described in the main text.
d
d
d
d
2
2
(^2222)
y
y
y
ψ()[()]=−=+eyeye−−−yyy//
(^22) // 2222
()−≈yyeyy−y =ψ()
d
dy
ψ()()yey=−−y^2 /^2
d
d
2
2
(^20)
y
ψψ()()yyy−=