Physical Chemistry Third Edition

654 15 The Principles of Quantum Mechanics. I. De Broglie Waves and the Schrödinger Equation

15.1 De Broglie Waves

Niels Bohr received the 1922 Nobel Prize in physics for his hydrogen atom theory,

based on an assumption of quantization of angular momentum. In 1923 a graduate

student at the University of Paris named Prince Louis de Broglie was trying to find a

physical justification for Bohr’s hypothesis of quantization. In classical physics, one

thing that is quantized is the wavelength of a standing wave. De Broglie sought a way

to relate this to Bohr’s theory and came up with the idea that a moving particle such as

an electron is accompanied by a “fictitious wave.”^1

Prince Louis Victor de Broglie,
1892–1977, won the Nobel Prize in
physics in 1929 for this work.
According to Einstein’s theory of relativity, a particle of energyEhas a massm
such that

Emc^2 (15.1-1)

wherecis the speed of light. If we apply this to a photon and use the Planck–Einstein

relation, Eq. (14.4-8), for the energy and if we replacemcby the momentump,

Eq. (15.1-1) becomes

hc

λ

pc or λ

h

p

(15.1-2)

whereλis the wavelength of the photon andhis Planck’s constant. De Broglie deduced

that the velocity of the wave accompanying a particle was the same as the velocity of

the particle if Eq. (15.1-2) is applied to a particle of massm.

λ

h

p



h

mv

(15.1-3)

We omit de Broglie’s argument, which is more complicated than simply saying that

Eq. (15.1-3) is analogous to Eq. (15.1-2).

The quantization assumption of Bohr’s theory arises naturally from Eq. (15.1-3)

if one assumes that the circumference of a circular electron orbit in a hydrogen atom

is equal to an integral number of wavelengths. This assumption means that the wave

repeats itself with the same phase (with crests in the same positions) on each trip around

the orbit, as depicted in Figure 15.1a. The situation depicted in Figure 15.1b is assumed

not to occur. For a circular orbit

2 πrnλ

nh

mv

(15.1-4)

This equation is the same as Eq. (14.4-14), the hypothesis of Bohr:

mvrnh/ 2 π (15.1-5)

De Broglie had shown that it was not necessary to assume quantization in the hydro-

gen atom as a hypothesis if one assumes the matter-wave relation of Eq. (15.1-3) for

the motion of the electron. This proposal ofmatter waveswas revolutionary. When

(^1) M. Jammer,The Conceptual Development of Quantum Mechanics, McGraw-Hill, New York, 1966,
p. 243ff.