bei48482_FM

Quantum Theory of the Hydrogen Atom 219

We next consider an electron that shifts from one energy state to another. A system

might be in its ground state nwhen an excitation process of some kind (a beam of

radiation, say, or collisions with other particles) begins to act upon it. Subsequently

we find that the system emits radiation corresponding to a transition from an excited

state of energy Emto the ground state. We conclude that at some time during the

intervening period the system existed in the state m. What is the frequency of the

radiation?

The wave function of an electron that can exist in both states nand mis

anbm (6.28)

where a*ais the probability that the electron is in state nand b*bthe probability that

it is in state m.Of course, it must always be true that a*a b*b1. Initially a 1

and b0; when the electron is in the excited state, a0 and b1; and ultimately

a1 and b0 once more. While the electron is in either state, there is no radiation,

but when it is in the midst of the transition from mto n(that is, when both aand b

have nonvanishing values), electromagnetic waves are produced.

The expectation value x that corresponds to the composite wave function of

Eq. (6.28) is

x 





x(a**nb**m)(anbm) dx







x(a^2 *nnb*a*mna*b*nmb^2 *mm) dx (6.29)

Here, as before, we let a*a a^2 andb*b b^2. The first and last integrals do not vary

with time, so the second and third integrals are the only ones able to contribute to a

time variation in x.

With the help of Eqs. (6.26) we expand Eq. (6.29) to give

x a^2





xnndxba





x*me(iEm^ )t ne(iEn^ )t dx

a*b





x*ne(iEn^ )t me(iEm^ )t dxb^2





x*mmdx (6.30)

Because

eicos isin  and eicos isin 

the two middle terms of Eq. (6.30), which are functions of time, become

cos t





x[b*a*mna*b*nm] dx

isin t





x[b*a*mna*b*nm) dx (6.31)

The real part of this result varies with time as

cos tcos 2tcos 2t (6.32)

EmEn



h

EmEn





EmEn





EmEn





bei48482_ch06 1/23/02 8:16 AM Page 219