(a) A21 is the probability that an electron will spontaneously decay from energy state 2 to energy state 1, emitting a photon in the process. The longer the lifetime of the meta-stable state, the lower A21 will be.

B21 is the probability that an electron will be stimulated to decay from energy state 2 to energy state 1, emitting a stimulated photon in the process.

B12 is the probability that an electron will be stimulated to 'jump' from energy state 1 to energy state 2, absorbing a stimulated photon in the process.

For laser action to occur we require that the ratio A21B21

be as small as possible (or B21A21

be as big as

possible) because stimulate emission is what we want.

(b) To show that A21B21

=8 πhν3

c3 the key is that if the long equation above is true for all

temperatures, then varying the temperature (T) still makes the whole thing zero. The only way this is

possible is if the term in brackets that multiplies exp ( hνkT ) is zero, therefore

( A21B21 −8 πhν3

c3B12B21 )exp ( hνkT )=0. This just leaves the other term also equal to zero:

−( A21B21−8πh ν3

c3 )=0.

Rearranging the last expression gives: A21B21

=8 πhν3

c3.

So there isn't really much deriving here, just a couple of statements and a small bit of rearranging.

To show that B12=B21 the key is to use what you just stated, that both terms in the long equation above are zero:

A21B21

−8 πhν3

c3B12B21

=0(1 )∧A21B21

−8πhν3

c3=0 (2)

Just simply bring the minus term over to the right hand side (RHS) and re-state them:

A21B21

=8 πhν3

c3B12B21

(3 )∧A21B21

=8 πhν3

c3(4)

Substitute the RHS of (4) for the LHS of (3):

8πh ν3

c3=8 πhν

3

c3B12B21

Some rearranging just leaves B21B12

=1∨ ,B21=B12

Alternatively:

To show that B12=B21 the key is to use what you just stated, that both terms in the long equation above are zero:

A21B21

−8 πhν3

c3B12B21

=0(1 )∧A21B21

−8πhν3

c3=0 (2)

Just simply bring the minus term over to the right hand side (RHS) and re-state them:

A21B21

=8 πhν3

c3B12B21

(3 )∧A21B21

=8 πhν3

c3(4)

And then rearrange (3 ) a little: A21B21

B21B12

=8 πhν3

c3(5 )

We now have two expressions for 8πh ν3

c3 .

So if the RHS of equations (4 ) and (5 ) are equal then the LHS of both must be equal:

A21B21

=A21B21

B21B12

→B21B12

=A21B21

B21A21

=1

→B21B12

=1∨,B21=B12

(c) For laser action to occur we require that the ratio A21B21

be as small as possible (or B21A21

be as big

as possible) because stimulate emission is what we want. Since B21A21

∝ c3

8πh ν3 we see that the

probability of getting stimulated emission is inversely proportional to frequency cubed. Therefore as the frequency increases there is less chance of getting stimulated emission and more chance of getting spontaneous emission. Since UV is much higher frequency than IR it is much more difficult to get lasing to occur at UV than IR.