Physical Chemistry Third Edition

658 15 The Principles of Quantum Mechanics. I. De Broglie Waves and the Schrödinger Equation

Nonrigorous Derivation of the Time-Independent

Schrödinger Equation

In the formal theory of quantum mechanics, the Schrödinger wave equation is taken

as a postulate (fundamental hypothesis). In order to demonstrate a relationship with

the classical wave equation, we obtain the time-independent Schrödinger equation

nonrigorously for the case of a particle that moves parallel to thexaxis. For a standing

wave along thexaxis, the classical coordinate wave equation of Eq. (14.3-10) is

d^2 ψ

dx^2

+

4 π^2

λ^2

ψ 0 (15.2-1)

where we have used Eq. (14.3-21) to replace the constantκin terms of the wavelength

λand where we use the letterψinstead ofφfor the coordinate factor. Use of the

de Broglie relation, Eq. (15.1-3), to replaceλgives

d^2 ψ

dx^2

+

4 π^2

h^2

m^2 v^2 ψ 0 (15.2-2)

We eliminate the speedvfrom our equation by using the relation

EK +V

1

2

mv^2 +V(x) (15.2-3a)

which is the same as

m^2 v^2  2 m[E−V(x)] (15.2-3b)

whereKis the kinetic energy,Vis the potential energy, andEis the total energy. Use

of Eq. (15.2-3b) in Eq. (15.2-2) gives thetime-independent Schrödinger equationfor

de Broglie waves moving parallel to thexaxis:

h^2

8 π^2 m

d^2 ψ

dx^2

+V(x)ψEψ (15.2-4)

We introduce the symbolh ̄(“h-bar”):

h ̄

h

2 π

(15.2-5)

and rewrite the equation:

−

h ̄^2

2 m

d^2 ψ

dx^2

+V(x)ψEψ (15.2-6)

This equation is made to apply to a specific case by specifying the appropriate potential

energy function.