Thermodynamics, Statistical Physics, and Quantum Mechanics

QUANTUM MECHANICS 249

Matching at either gives the pair of equations:

Eliminating the constants A and B gives the final equation for the unknown
constant

For large values of the hyperbolic tangent is unity, and we have the
approximate result that which gives for large P the eigenvalue
For small values of we see that and
This is always the lowest eigenvalue.
The other possible eigenstate is antisymmetric: it has odd parity. When
the separate bound states from the two delta functions overlap, they com-
bine into bonding and antibonding states. The bonding state is the sym-
metricstate we calculated above. Now we calculate the antibonding state,
which is antisymmetric:

Using the same matchingconditions, we find the two equations, which are
reduced to the final equation for

For large values of the hyperbolic cotangentfunction (coth) approaches
unity, and again we find and At small values
of the factor of coth Here we have so we find at small
values of that The antisymmetric mode only exists for
For the only bound state is the symmetric one. For there are
two bound states, symmetric and antisymmetric.