Concise Physical Chemistry

c16 JWBS043-Rogers September 13, 2010 11:28 Printer Name: Yet to Come

QUANTUM MECHANICAL SYSTEMS 251

group of constants times the kinetic energyT,∇^2 is called akinetic energy operator.

The total energy of a classical system is the sum of the kinetic energy and the potential

energy,E=T+V, so the kinetic energy is the differenceT=E−Vand

∇^2 =−

8 π^2 m

h^2

(E−V)

which is one form of the famous Schrodinger equation. ̈

Other notational conveniences include the definition of “h bar,” ̄h≡h/ 2 π, which

gives

∇^2 =−

2 m

̄h^2

(E−V)

and

−

h ̄^2

2 m

∇^2 +V=E

The kinetic energy operator plus the potential energy operator (− ̄h

2

2 m∇

(^2) +V)
is defined as theHamiltonian operatorHˆ by analogy to the classical Hamiltonian
functionH=T+V(Chapter 15). The kinetic and potential energy operators areTˆ
andVˆ,soHˆ =Tˆ+Vˆ. (Even thoughVˆis an operator, it is often writtenV.) These
notational changes give the concise form of the Schrodinger equation: ̈

Hˆ=E

where the total energyEis ascalareigenvalue. It is distinguished from the opera-

tors by not having a circumflex notation. Clearly energy is a scalar because it has

magnitude but not direction.

16.4 QUANTUM MECHANICAL SYSTEMS

Asystemis a collection of mechanical entities governed by physical laws. If we know

thestateof a system, we know every physical property it can have. It is astonishing

but true that all this information can be known by specifying a very few fundamental

variables (degrees of freedom) and a small number of postulates.

The wave function satisfies all of the properties of a vector; therefore it is written

as a vector|〉or〈|. We shall use either the vector form|〉or the functional form

, according to which is more convenient. When it is useful to specify the variables

as degrees of freedom, we shall do that:|(x 1 ,x 2 ,...)〉or(x 1 ,x 2 ,...), but usually

the simpler notation|〉oris preferred.