Physical Chemistry , 1st ed.

derivative with respect to time, and so on). Further, if the potential energy V

is a function only of position, that is, the coordinates x,y, and z:

VV(x, y, z) (9.5)

then the Lagrangian function L(or simply “the Lagrangian”) of the particle is

defined as

L( ̇x, ̇y, ̇z,x, y, z) K( ̇x, ̇y, ̇z) V(x, y, z) (9.6)

Lhas units of joules, which is the SI unit of energy. (1 J 1 Nm 1 kgm^2 /s^2 )‡

Understanding that the coordinates x,y, and zare independent of each other,

one can now rewrite Newton’s second law in the form of Lagrange’s equations

of motion:



d

d

t







L

x ̇







L

x

 (9.7)



d

d

t







L

y ̇







L

y

 (9.8)



d

d

t







L

z ̇







L

z

 (9.9)

We are using partial derivatives here, because Ldepends on several variables.

One of the points to notice about the laws of motion equations (9.7–9.9) is

that the equations have exactly the same form regardless of the coordinate.

One can show that this holds true for any coordinate system, like the spheri-

cal polar coordinate system in terms ofr,, and that we will use later in our

discussion of atoms.

Lagrange’s equations, mathematically equivalent to Newton’s equations, rely

on being able to define the kinetic and potential energy of a system rather than

the forces acting on the system. Depending on the system, Lagrange’s differen-

tial equations of motion can be easier to solve and understand than Newton’s

differential equations of motion. (For example, systems involving rotation

about a center, like planets about a sun or charged particles about an oppo-

sitely charged particle, are more easily described by the Lagrangian function

because the equation that describes the potential energy is known.)

Irish mathematician Sir William Rowan Hamilton was born in 1805, eight

years before the death of Lagrange. Hamilton (Figure 9.3) also came up with a

different but mathematically equivalent way of expressing the behavior of mat-

ter in motion. His equations are based on the Lagrangian and they assume

that, for each particle in the system,Lis defined by three time-dependent co-

ordinates ̇qj,where j1, 2, or 3. (For example, they might be ̇x, ̇y,or ̇zfor a

particle having a certain mass.) Hamilton defined three conjugate momentafor

each particle,pj, such that

pj



L

q ̇j

, j1, 2, 3 (9.10)

The Hamiltonian function(“the Hamiltonian”) is defined as

H(p 1 ,p 2 ,p 3 ,q 1 ,q 2 ,q 3 ) 

3

j 1

pjq ̇jL (9.11)

244 CHAPTER 9 Pre-Quantum Mechanics

‡Equations 9.4 and 9.5 embody the definitions of kinetic and potential energies: kinetic

energy is energy of motion, and potential energy is energy of position.

Figure 9.3 Sir William Rowan Hamilton

(1805–1865). Hamilton reformulated the law of

motion of Newton and Lagrange into a form that

ultimately provided a mathematical basis for

modern quantum mechanics. He also invented

matrix algebra.

© CORBIS-Bettmann