bei48482_FM

218 Chapter Six

states are not. The pronounced lobe patterns characteristic of many of the states turn

out to be significant in chemistry since these patterns help determine the manner in

which adjacent atoms in a molecule interact.

A look at Figure 6.12 also reveals quantum-mechanical states that resemble these

of the Bohr model. The electron probability-density distribution for a 2pstate with

ml1, for instance, is like a doughnut in the equatorial plane centered at the nu-

cleus. Calculation shows the most probable distance of such an electron from the nu-

cleus to be 4a 0 —precisely the radius of the Bohr orbit for the same principal quantum

number n2. Similar correspondences exist for 3dstates with ml2, 4fstates

with ml3, and so on. In each of these cases the angular momentum is the high-

est possible for that energy level, and the angular-momentum vector is as near the zaxis

as possible so that the probability density is close to the equatorial plane. Thus the

Bohr model predicts the most probable location of the electron in oneof the several

possible states in each energy level.

6.8 RADIATIVE TRANSITIONS

What happens when an electron goes from one state to another

In formulating his theory of the hydrogen atom, Bohr was obliged to postulate that the

frequency of the radiation emitted by an atom dropping from an energy level Emto

a lower level Enis



It is not hard to show that this relationship arises naturally in quantum mechanics.

For simplicity we shall consider a system in which an electron moves only in the

xdirection.

From Sec. 5.7 we know that the time-dependent wave function nof an electron

in a state of quantum number nand energy Enis the product of a time-independent

wave function nand a time-varying function whose frequency is

n

Hence nne(iEn^ h)t *n  *ne(iEn^ )t (6.26)

The expectation value x of the position of such an electron is

x 





x*nndx





x*nne[(iEn^ )(iEn^ )]tdx







x*nndx (6.27)

The expectation value x is constant in time since nand *nare, by definition, functions

of position only. The electron does not oscillate, and no radiation occurs. Thus quan-

tum mechanics predicts that a system in a specific quantum state does not radiate, as

observed.

En



h

EmEn



h

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