Physical Chemistry , 1st ed.

10.8 An Analytic Solution: The Particle-in-a-Box

Very few systems have analytic solutions (that is, solutions that have a specific

mathematical form, either a number or an expression) to the Schrödinger

equation. Most of the systems having analytic solutions are defined ideally,

much as an ideal gas is defined. This should not be a cause for despair. The few

ideal systems whose exact solutions can be determined have applications in the

real world, so they are not wasted on ideality! Several of these systems were rec-

ognized by Schrödinger himself as he developed his equation.

The first system for which there is an analytic solution is a particle of mat-

ter stuck in a one-dimensional “prison” whose walls are infinitely high barri-

ers. This system is called the particle-in-a-box.The infinitely high barriers cor-

respond to potential energies of infinity; the potential energy inside the box

itself is defined as zero. Figure 10.5 illustrates the system. Arbitrarily, we are

setting one side of the box at x0 and the other at some length a. Inside this

box the potential energy is 0. Outside, the potential energy is infinity.

The analysis of this system using quantum mechanics is similar to the analy-

sis that we will apply to every system. First, consider the two regions where the

potential energy is infinity. According to the Schrödinger equation

^2



m

2





x

2

    2 E

must hold true for x0 and xa. The infinity presents a problem, and

in this case the way to eliminate infinity is to multiply it by zero. Thus,

must be identically zero in the regions x0 and xa. It does not matter

288 CHAPTER 10 Introduction to Quantum Mechanics

Table 10.1 Operators for various observables and their classical counterpartsa

Observable Operator Classical counterpart

Position ˆxx x

And so forth for coordinates

other than x

Momentum (linear) ˆpxi (^) x pxmvx
And so forth for coordinates
other than x
Momentum (angular) ˆLxiˆy (^) z ˆz (^) y (^)  Lxypzzpy
Kinetic energy, 1-Db Kˆ 2 m
2
(^) ddx
2
2 K
1
2 mvx
(^2) 
2
p
m
x^2
Kinetic energy, 3-Db Kˆ 2 m
2
(^)  (^) x
2
2  (^) 

y
2
2  (^) 

z
2
(^2)  K
1
2 m(vx
(^2) vy (^2) vz (^2) )

 px

(^2) 
2
p
m

y^2 pz^2

Potential energy:

Harmonic oscillator Vˆ     12   kx^2 V    12   kx^2

Coulombic Vˆ q 41



 

0

q

r

    2

 V q

4

1

 0

q

r

    2

Total energy Hˆ 2 m

2

(^)  (^) x
2
2  (^) 

y
2
2  (^) 

z
2
(^2) Vˆ H (^2)
p
m
2
V
aOperators expressed in x,y, and/or zare Cartesian operators; operators expressed in r,, and/or are spherical po-
lar operators.b
The kinetic energy operator is also symbolized by Tˆ.