The Chemistry Maths Book, Second Edition

352 Chapter 12Second-order differential equations. Constant coefficients

equilibrium position, potential energy is converted into kinetic energy and, atx 1 = 10 ,

V 1 = 10 and the kinetic energy is a maximum,T 1 = 1 E.

0 Exercises 22, 23

12.6 The particle in a one-dimensional box

The ‘particle in a one-dimensional box’ is the name given to the system consisting of

a body allowed to move freely along a line of finite length. In quantum mechanics, this

is one of the simplest systems that demonstrate the quantization of energy.

The Schrödinger equation for a particle of mass mmoving in the x-direction is

(12.42)

whereV(x) is the potential energy of the particle at position x, Eis the (constant)

total energy, and ψis the wave function. For the present system the potential-energy

function is (Figure 12.5)

(12.43)

The constant value ofVinside the box ensures that no force acts on the particle

in this region; settingV 1 = 10 means that the energyEis the (positive) kinetic energy

of the particle. The infinite value ofVat the ‘walls’ and outside the box ensures that

the particle cannot leave the box; in quantum mechanics this means that the wave

function is zero at the walls and outside the box.

For the particle within the box, we therefore have the boundary value problem

(12.44)

with boundary conditions

ψ(0) 1 = 1 ψ(l) 1 = 10 (12.45)

Equation (12.44) is identical in form to equation (12.20), or to equation (12.35) for the

harmonic oscillator. Thus, setting

ω (12.46)

2

2

2

=

mE



−=



22

2

2 m

dx

dx

Ex

ψ

ψ

()

()

Vx

xl

x

()=

≤

0

0

for 0

for a

<<

∞nnd xl≥











−+=



22

2

2 m

dx

dx

Vx x E x

ψ

ψψ

()

()() ()

.............................................................................................................................................................................................................................................................................................................................

V=∞ V= 0 V=∞

0 l

x

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Figure 12.5