In deriving the results for the eigenstates of the annihilation operator I discovered my derivation had an overall factor of was missing. I traced this to a bad definition in my previous notation and a misunderstanding on my part (carelessness). To correct the notation I will introduce the following:

	states that are built using the creation operator
	states generated by the annihilation operator
	Normalized statesè

These states are all proportional to each other.

For convenience we define states that are simply related by the raising and lowering operators with no additional constants and . Starting with these simple properties we can find the the way these states behave with both operators, , and find a normalized state its properties under the operations.

When introducing these operators the book simply states “apart from normalization factors”. Using these operators one can show that the creation operator produces a state that is next higher energy and the annihilation operator produces the next lower energy state. The operator

the number operator, has these states as eigenvectors and the eigenvalue is n.

Remember that if

then is an eigentate of . Notice also that

is also an eigenstate of with the same eigenvalue. The constant doesn’t affect this result. Any vector proportional to an eigenvector is also and eigenvector with the sane eigenvalue.

The characteristics of the state don’t depend on these multiplicative factors but of course we prefer to use normalized states so that one can write down probabilities and probability densities.

	In my original development I assumed incorrectly that notice
so the state has a factor of n when lowered. It has an overall multiplicative factor after being lowered. Thus .

Find the factors that relates the states.

Remember that the definition of the states is

Similarly

Same as above. Thus an acceptable value for the coefficients

We have already seen that a definition of the normalized states is

The inpact of on these states is

With the correct definitions for the states and the correct results for the and on the states, we can find the eigenvectors for .