130_notes.dvi

This is an extremely useful identity for solving problems. We could already see this in the decom-
position of|ψ〉above.

|ψ〉=

∑

i

|i〉〈i|ψ〉.

The same is true for definite momentum states.

∫∞

−∞

|p〉〈p|dp= 1

We can form a projection operator into asubspace.

P=

∑

subspace

|i〉〈i|

We could use this to project out the odd parity states, for example.

11.1.3 Unitary Operators

Unitary operatorspreserve a scalar product.

〈φ|ψ〉=〈Uφ|Uψ〉=〈φ|U†Uψ〉

This means that
U†U= 1.

Unitary operators will be important for the matrix representationof operators. The will allow us to
change from one orthonormal basis to another.

11.2 A Complete Set of Mutually Commuting Operators

If an operator commutes withH, we can makesimultaneous eigenfunctionsof energy and that
operator. This is an important tool both for solving the problem andfor labeling the eigenfunctions.

Acomplete set of mutually commuting operatorswill allow us to define a state in terms of
the quantum numbers of those operators. Usually, we will need onequantum number for each degree
of freedom in the problem.

For example, the Hydrogen atom in three dimensions has 3 coordinates for the internal problem, (the
vector displacement between the proton and the electron). We willneed three quantum numbers
to describe the state. We will use an energy index, and two angular momentum quantum numbers
to describe Hydrogen states. The operators will all commute with each other. The Hydrogen atom
also has 3 coordinates for the position of the atom. We will might usepx,pyandpzto describe that
state. The operators commute with each other.