The time independent Schr ̈odinger equation is an example of an eigenvalue equation (See section
8.1).
Hψi(~x) =Eiψi(~x)
If we operate onψiwithH, we get back the same functionψitimes some constant. In this case
ψiwould be called and Eigenfunction, andEiwould be called an Eigenvalue. There are usually an
infinite number of solutions, indicated by the indexihere.
Operators for physical variables must have real eigenvalues. They are called Hermitian operators
(See section 8.3). We can show that the eigenfunctions of Hermitianoperators are orthogonal (and
can be normalized).
〈ψi|ψj〉=δij
(In the case of eigenfunctions with the same eigenvalue, called degenerate eigenfunctions, we can
must choose linear combinations which are orthogonal to each other.) We will assume that the
eigenfunctions also form a complete set so that any wavefunction can be expanded in them,
φ(~x) =∑
iαiψi(~x)where theαiare coefficients which can be easily computed (due to orthonormality) by
αi=〈ψi|φ〉.So now we have another way to represent a state (in addition to position space and momentum space).
We can represent a state by giving the coefficients in sum above. (Note thatψp(x) =ei(px−Et)/ ̄his
just an eigenfunction of the momentum operator andφx(p) =e−i(px−Et)/ ̄his just an eigenfunction
of the position operator (in p-space) so they also represent and expansion of the state in terms of
eigenfunctions.)
Since theψiform an orthonormal, complete set, they can be thought of as theunit vectors of a
vector space (See section 8.4). The arbitrary wavefunctionφwould then be a vector in that space
and could be represented by its coefficients.
φ=
α 1
α 2
α 3
...
The bra-ket〈φ|ψi〉can be thought of as a dot product between the arbitrary vectorφand one of
the unit vectors. We can use the expansion in terms of energy eigenstates to compute many things.
In particular, since the time development of the energy eigenstates is very simple,
ψ(~x,t) =ψ(~x)e−iEit/ ̄hwe can use these eigenstates to follow the time development of an arbitrary stateφ
φ(t) =
α 1 e−iE^1 t/ ̄h
α 2 e−iE^2 t/ ̄h
α 3 e−iE^3 t/ ̄h
...
simply by computing the coefficientsαiatt= 0.
We can define the Hermitian conjugate (See section 8.2) O†of the operatorOby
〈ψ|O|ψ〉=〈ψ|Oψ〉=〈O†ψ|ψ〉.Hermitian operatorsHhave the property thatH†=H.