130_notes.dvi

Note that we can simply describe thejtheigenstate at|j〉.

Expanding the vectors|φ〉and|ψ〉,

|φ〉 =

∑

i

bi|i〉

|ψ〉 =

∑

i

ci|i〉

we can take the dot product by multiplying the components, as in 3D space.

〈φ|ψ〉=

∑

i

b∗ici

The expansion in energy eigenfunctions is a very nice way to do thetime development of a wave
function.
|ψ(t)〉=

∑

i

〈i|ψ(0)〉|i〉e−iEit/ ̄h

The basis ofdefinite momentum statesis not in the vector space, yet we can use this basis to
form any state in the vector space.

|ψ〉=

1

√

2 π ̄h

∫∞

−∞

dpφ(p)|p〉

Any of these amplitudes can be used to define the state.

ci=〈i|ψ〉

ψ(x) =〈x|ψ〉

φ(p) =〈p|ψ〉

11.1.2 Projection Operators|j〉〈j|and Completeness

Now we move on a little with our understanding of operators. A ket vector followed by a bra vector
is an example of an operator. For example theoperator which projects a vector onto thejth
eigenstateis
|j〉〈j|

First the bra vector dots into the state, giving the coefficient of|j〉in the state, then its multiplied
by the unit vector|j〉, turning it back into a vector, with the right length to be a projection. An
operator maps one vector into another vector, so this is an operator.

The sum of the projection operators is 1, if wesum over a complete set of states, like the
eigenstates of a Hermitian operator.

∑

i

|i〉〈i|= 1