Physical Chemistry Third Edition

664 15 The Principles of Quantum Mechanics. I. De Broglie Waves and the Schrödinger Equation

(a)

(b)

0

0

t

vx

x

Figure 15.3 Mechanical Variables of a Particle in a One-Dimensional Box.(a) The

position according to classical mechanics. (b) The velocity according to classical mechanics.

According to classical mechanics, the particle would move back and forth at constant

speed inside the box, reversing direction as it elastically collides with the ends of the

box. Figure 15.3a shows the position of the particle as a function of time according to

classical mechanics, and Figure 15.3b shows the velocity of the particle as a function

of time. Quantum mechanics predicts a very different behavior.

The time-independent Schrödinger equation is

− ̄

h^2

2 m

d^2 ψ/d x^2 +V(x)ψ(x)Eψ(x) (15.3-1)

whereV(x) is the potential energy function. We divide thexaxis into three regions

and solve separately in each region:

Region I: x< 0

Region II: 0 ≤x≤a

Region III: a<x

We require thatψis finite and that it is continuous everywhere, including the boundaries

between the regions.

In regions I and III the potential energy approaches an infinite value, so the

Schrödinger equation is

d^2 ψ

dx^2

− lim

V→∞

2 mV

h ̄^2

ψ−

2 mE

h ̄^2

ψ (15.3-2)

We require thatψis finite, and we assume thatEis finite, so the right-hand side of

this equation is finite. SinceVapproaches an infinite value, the left-hand side would

be infinite unlessψvanishes, so the solution in regions I and III must be

ψ(I)(x)ψ(III)(x) 0 (15.3-3)

For region II (inside the box) the Schrödinger equation is

d^2 ψ(II)

dx^2

−κ^2 ψ(II) (15.3-4)