130_notes.dvi

8.4 Eigenfunctions and Vector Space

Wavefunctions are analogous to vectors in 3D space. The unit vectors of our vector space are
eigenstates.

Innormal 3D space, we represent a vector by its components.

~r=xxˆ+yˆy+zzˆ=

∑^3

i=1

riuˆi

The unit vectors ˆuiare orthonormal,
ˆui·ˆuj=δij

whereδijis the usual Kroneker delta, equal to 1 ifi=jand otherwise equal to zero.

Eigenfunctions – the unit vectors of our space– are orthonormal.

〈ψi|ψj〉=δij

We represent ourwavefunctions – the vectors in our space– as linear combinations of the
eigenstates (unit vectors).

ψ=

∑∞

i=1

αiψi

φ=

∑∞

j=1

βjψj

In normal 3D space, we can compute thedot product between two vectorsusing the components.

~r 1 ·~r 2 =x 1 x 2 +y 1 y 2 +z 1 z 2

In our vector space, wedefine the dot productto be

〈ψ|φ〉 = 〈

∑∞

i=1

αiψi|

∑∞

j=1

βjψj〉=

∑∞

i=1

∑∞

j=1

α∗iβj〈ψi|ψj〉

=

∑∞

i=1

∑∞

j=1

α∗iβjδij=

∑∞

i=1

α∗iβi

We also can compute the dot product from the components of the vectors. Our vector space is a
little bit different because of the complex conjugate involved in the definition of our dot product.

From a more mathematical point of view, the square integrable functions form a (vector) Hilbert
Space. The scalar product is defined as above.

〈φ|ψ〉=

∫∞

−∞

d^3 rφ∗ψ

The properties of the scalar product are easy to derive from the integral.

〈φ|ψ〉=〈ψ|φ〉∗