Physical Chemistry Third Edition

666 15 The Principles of Quantum Mechanics. I. De Broglie Waves and the Schrödinger Equation

0.0

Energy or wave function value

Energy

20

18

16

14

12

10

8

6

4

0

2

20

18

16

14

12

10

8

6

4

0

2

0.2 0.4

x/a

0.6 0.8

(a) (b)

Figure 15.4 The Solutions to the Schrödinger Equation for a Particle in a

One-Dimensional Box.(a) The energy eigenvalues. (b) The energy eigenfunctions.

EXAMPLE15.2

Find the zero-point energy of an electron in a box of length 1.000 nm, roughly equal to the

length of a 1,3-butadiene molecule.

Solution

E

(6. 6261 × 10 −^34 Js)^2 (1)^2

(8)(14. 109 × 10 −^31 kg)(1. 000 × 10 −^9 m)^2

 6. 025 × 10 −^20 J

Exercise 15.3

How does the energy for a given value ofnchange if the length of the box is doubled? How

does it change if the mass of the particle is doubled?

The Schrödinger Equation and De Broglie Waves

The particle in a box model provides an illustration of the fact that the energy eigen-

function represents de Broglie waves. In the case of zero potential energy

E

1

2

mv^2 

p^2

2 m

(15.3-12)

where we use the definition of the momentum,pmv. From the de Broglie wavelength

formula in Eq. (15.1-3) and Eq. (15.3-12)

λ

h

p



h

√

2 mE

(15.3-13)

which is the same as

Eh^2 / 2 mλ^2 (15.3-14)