5.3.1 Solution 2state-a
The effectiveness of mixing states was already shown by the hydrogen molecule and molecular ion examples. But the generalized story above restricts the
basis states to be orthogonal, and the states used in the hydrogen examples were not.
Show that if and are not orthogonal states, but are normalized and produce a real and positive value for , like in the hydrogen examples, then orthogonal states can be found in the form
For normalized and the Cauchy-Schwartz inequality implies that will be less than one. If the states do not overlap much, it will be much less than one and will be small.
(If and do not meet the stated requirements, you can always redefine them by factors and , with , , and real, to get states that do.)
The inner product of and must be zero for them to be orthogonal:
and this can be multiplied out, dropping the common factor and noting that and are one, as
for which the smallest root can be written as
That is less than , hence is small if the overlap is small.
The constant follows from the requirement that the new states must still be normalized, and is found to be
Note that the denominator is nonzero; the argument of the square root exceeds .