130_notes.dvi

Since the eigenfunctions are orthogonal, we can easilycompute the coefficientsin the expansion
of an arbitrary wave functionψ.

αi=〈ψi|ψ〉=〈ψi|

∑

j

αjψj〉=

∑

j

〈ψi|ψj〉αj=

∑

j

δijαj=αi

We will later think of the eigenfunctions as unit vectors in a vector space (See section 8.4). The
arbitrary wave functionψis then a vector in that space.

ψ=







α 1

α 2

α 3







It is instructive to compute theexpectation value of the Hamiltonianusing the expansion of
ψand the orthonormality of the eigenfunctions.

〈ψ|H|ψ〉 =

∑

ij

〈αiψi|H|αjψj〉=

∑

ij

〈αiψi|αjHψj〉

=

∑

ij

α∗iαjEj〈ψi|ψj〉=

∑

ij

α∗iαjEjδij

=

∑

i

α∗iαiEi=

∑

i

|αi|^2 Ei

We can see that thecoefficients of the eigenstates represent probability amplitudes to be
in those states, since the absolute squares of the coefficientsα∗iαiobviously give the probability.

8.2 Hermitian Conjugate of an Operator

First let us define theHermitian Conjugateof an operatorH to beH†. The meaning of this
conjugate is given in the following equation.

〈ψ|H|ψ〉=

∫∞

−∞

ψ∗(x)Hψ(x)dx=〈ψ|Hψ〉≡〈H†ψ|ψ〉

That is,H†must operate on the conjugate ofψand give the same result for the integral as when
Hoperates onψ.

The definition of theHermitian Conjugate of an operatorcan be simply written in Bra-Ket
notation.

〈A†φ|ψ〉=〈φ|Aψ〉

Starting from this definition, we can prove some simple things. Takingthe complex conjugate

〈ψ|A†φ〉=〈Aψ|φ〉