Page 97 - Electrical Properties of Materials

Feynman’s coupled mode approach 79

What do we mean by weak coupling? It means that even in the presence of
coupling, it is still meaningful to talk about one or the other state. The states
influence each other but may preserve their separate entities.
Let us return now to the solution of equations (5.25) and (5.26). As we are
going to investigate symmetric cases only, we may introduce the simplifications

H 11 = H 22 = E 0 , H 12 = H 21 =–A, (5.30)

leading to

dw 1
i = E 0 w 1 – Aw 2 , (5.31)
dt
dw 2
i =–Aw 1 + E 0 w 2 . (5.32)
dt
Following the usual recipe, the solution may be attempted in the form
E E

w 1 = K 1 exp –i t , w 2 = K 2 exp –i t . (5.33)

Substituting eqn (5.33) into equations (5.31) and (5.32) we get

K 1 E = E 0 K 1 – AK 2 (5.34)

and

K 2 E =–AK 1 + E 0 K 2 , (5.35)

which have a solution only if

E 0 – E –A

= 0. (5.36)
–A E 0 – E

Expanding the determinant we get
2
2
(E 0 – E) = A , (5.37)
whence

E = E 0 ± A. (5.38)
If there is no coupling between the two states, then E = E 0 ; that is, both
states have the same energy. If there is coupling, the energy level is split. There
are two new energy levels E 0 + A and E 0 – A. This is a very important phe-
nomenon that you will meet again and again. Whenever there is coupling, the
energy splits.
The energies E 0 ± A may be defined as the energies of so-called stationary
states obtainable from linear combinations of the original states. For our pur-
pose it will suffice to know that we can have states with energies E 0 + A and
E 0 – A.