Concise Physical Chemistry

c15 JWBS043-Rogers September 13, 2010 11:28 Printer Name: Yet to Come

EARLY QUANTUM THEORY 237

there should be a connection between momentum and wavelength, which he wrote as

p=h/λ. At the atomic level, particles must have a wave nature. Conversely, waves

have a particle nature. This was promptly experimentally verified. This mathematical

equivalence between waves and particles is referred to aswave-particle duality.

Schrodinger (1925, 1926) reasoned that if electrons have a wavelength, they should ̈

follow a wave equation. He wrote an equation for the electron in the hydrogen atom

employing awave function, and he arrived at the electromagnetic spectrum of

hydrogen just as Bohr had done but without making the arbitrary assumptions inherent

in Bohr theory. While it should be noted that the Schrodinger equation is itself an ̈

assumption and that quantum theory is based on a set of postulates, the implications

of Schrodinger’s theory are far wider and the results are far more general than earlier ̈

theories. Quantum mechanics now pervades virtually all of physical science from

molecular biology to string theory.

The Schrodinger equation can be written in equivalent forms with a rather in- ̈

timidating array of notations, but they are all manifestations of the same thing: the

postulate that astate vector|〉orstate functioncontains all the information we

can ever have about a mechanical system such as an atom or molecule.

TheHamiltonian functionwas known from classical mechanics:

E=H=T+V

whereTis the kinetic energy, written in terms of the velocitiesvin Cartesian 3-space

T=^12 m

(

v^2 x+v^2 y+v^2 z

)

, andV(x,y,z)is the potential energy. (Be careful not to

confuse velocityvwith frequencyν.)

Anoperatoris a mathematical symbol telling you to do something. For example,

Vtells you to multiply by the potential energy (a scalar) and∂^2 /∂x^2 tells you to

differentiate twice with respect toxwhile holding some other variable constant.

(It is a partial∂x^2 becauseyandzare also variables.) An operator must operate

on something,in the case we are interested in. If aHamiltonian operatorHˆ for

the hydrogen atom system is written in terms of mathematical operatorsVand

Hˆ =Tˆ−Vˆ in a way that is analogous to the classical Hamiltonian for the total

energy of a conservative system, we get the operator equation

Hˆ =Tˆ+Vˆ

The Schrodinger equation is a special case of the Hamiltonian form where ̈ Tˆis

the operator corresponding to thekinetic energyof the electron:

Tˆ=−

2

2 me

(

∂^2

∂x^2

+

∂^2

∂y^2

+

∂^2

∂z^2

)

andVˆ(x,y,z) is the potential energy operator that describes the Coulombic attraction

between the nucleus and the electron. (Vˆ(x,y,z)is usually written simplyV(x,y,z),

where we take its operator nature as obvious.)