So the eigenstates are
E=E 0 −A
( 1
√
2
√^1
2)
E=E 0 +A
( 1
√
2
−√^12)
The states are split by the interaction term.
Feynman goes on to further split the states by putting the molecules in an electric field. This makes
the diagonal terms of the Hamiltonian slightly different, like a magneticfield does in the case of
spin.
Finally, Feynman studies the effect of Ammonia in an oscillating Electric field, the Ammonia Maser.
18.9.2 The Neutral Kaon System*
The neutral Kaons,K^0 andK ̄^0 form a very interesting two state system. As in the Ammonia
molecule, there is a small amplitude to make a transition form one to the other. The Energy (mass)
eigenstates are similar to those in the example above, but the CP (Charge conjugation times Parity)
eigenstates are important because they determine how the particles can decay. A violation of CP
symmetry is seen in the decays of these particles.
18.10Examples
18.10.1Harmonic Oscillator Hamiltonian Matrix
We wish to find the matrix form of the Hamiltonian for a 1D harmonic oscillator.
The basis states are the harmonic oscillator energy eigenstates. We know the eigenvalues ofH.
Huj=Ejuj〈i|H|j〉=Ejδij=(
j+1
2
)
̄hωδijThe Kronecker delta gives us a diagonal matrix.
H= ̄hω
1
2 0 0 0 ...
0 32 0 0 ...
0 0^520 ...
0 0 0^72 ...
18.10.2Harmonic Oscillator Raising Operator
We wish to find the matrix representing the 1D harmonic oscillator raising operator.