Simple Nature - Light and Matter

Experimentally, we do observe these two states in ammonia. The

difference in energy is extremely small — e.g., for the state with

angular momentum 1~it is about 9.8× 10 −^5 eV, so that if a photon

is emitted or absorbed in a transition between the states, it lies in

the microwave spectrum. This energy difference equals 2|f|, and its

smallness indicates that the tunneling probability is small.

Let’s find the states of definite energy for this system. For the

ground state, whose energy is−|f|, we need to look for a state of

the form|g.s.〉= (...)| 1 〉+ (...)| 2 〉such thatHˆ|g.s.〉=−|f||g.s.〉=

f|g.s.〉. If we don’t worry about normalization or an over-all phase,

we are free to take the first (...) equal to 1, so that|g.s.〉=| 1 〉+α| 2 〉,

for some complex numberα. We then have

Hˆ|g.s.〉=Hˆ(| 1 〉+α| 2 〉)

=f| 2 〉+αf| 1 〉,

and setting this equal tof|g.s.〉givesα= 1, so that

|g.s.〉=| 1 〉+| 2 〉.

The coefficients (...) that we set out to find are both equal to +1.

Their equal magnitudes tell us that the ground state is one in which

the molecule has anequalprobability of existing in either inversion.

Since the two coefficients are both positive, and we have defined| 1 〉

and| 2 〉such that their phases agree when they overlap inside the

barrier, this is a state of positive parity. The determination of the

excited state is left as an exercise, problem 10 on p. 1010.

From a classical point of view, we would think of the set of states

{ | 1 〉, | 2 〉 }

as the natural way of describing the possible states of the system.

These two states are the ones that we can draw pictures of, a/1 and

a/2. But part of the structure of quantum mechanics is that there

isno preferred basis(p. 987), and there is nothing wrong with using

the ground state and first excited state to form the basis

{ |g.s.〉,|ex.s.〉 }

instead. In the language of the completeness principle (p. 987),

one possible choice of a complete set of compatible observables for

this molecule is the set consisting of a single observable, the energy.

The{ground-state,excited-state}basis just happens to be the one

associated with this particular observable. If the ammonia molecule

had just broken off from some larger molecule, then it would be

oriented in a specific direction, and we would probably find it more

convenient to describe it in the{1,2}basis.

994 Chapter 14 Additional Topics in Quantum Physics