Chapter 7

Assume that there is a lowest energy state

Use the raising operator on the state once and label this state .

Commute the creation operator one time an get an . The operation on the 0-state is zero for .

Appling the operator n times

The raising and lowering operators move us up or down through this set of states. Raising find an energy eigenstate that has an greater and therefore we name the state . While the loweing operator reduces the energy by so we label this states as .

Also

ASIDE

,

Because the states are not necessarily normalized we define the state as the state that is raised without any additional factor and as the state lowered without any additional factors. A full exposition of the relationships between these and the normalized states can be found in the l9-26_notation correcton. All of these states are energy eigenstates and also eigenstates of the number operator with eigenvalue n.

Now the next question to address is: are these normalized? Any state will satisfy the eigenvalue equation with the same eigenvalue as the state . We haven’t established that the states are properly normalized.

so if we start with a ground state that is normalized and we want to use the raising and lowering operators to generate the normalized states. Then we need a factor of .