130_notes.dvi

8 Eigenfunctions, Eigenvalues and Vector Spaces

8.1 Eigenvalue Equations

The time independent Schr ̈odinger Equation is an example of an Eigenvalue equation.

H u(x) =E u(x)

The Hamiltonian operates onu(x)the eigenfunction, giving a constantEthe eigenvalue, times
the same function. (Eigen just means the same in German.)

Usually, for bound states, there aremany eigenfunction solutions(denoted here by the indexi).

H ψi=Eiψi

For states representing one particle (particularly bound states)we mustrequire that the solutions
be normalizable. Solutions that are not normalizable must be discarded. A normalizable wave
function must go to zero at infinity.
ψ(∞)→ 0

In fact, all the derivatives ofψmust go to zero at infinity in order for the wave function to stay at
zero.

We will prove later that the eigenfunctions are orthogonal (See section 8.7.1) to each other.

〈ψi|ψj〉=δij

∫∞

−∞

ψ∗(x)ψ(x)dx= 1

We will assume that theeigenfunctions form a complete setso that any function can be written
as a linear combination of them.

ψ=α 1 ψ 1 +α 2 ψ 2 +α 3 ψ 3 +...

ψ=

∑∞

i=1

αiψi

(This can be proven for many of the eigenfunctions we will use.)