Assume {|ui>} and {|ti>} are orthonormal bases for a Hilbert space.
Let
{|ui>} = old basis,
{|ti>} = new basis,
and let
U|ui> = |ti> be the unitary transformation from
the old
to new basis.
Then
<ui|U|uj> = <ui|tj> = Uij = matrix elements of U.
Assume |ψ> = ∑jaj|uj> = ∑jdj|tj> is an arbitary vector in the vector space.
What is the relationship between the components of this vector in the two bases?
<ti|ψ> =
<ti|∑jdj|tj> = di
=
∑jaj<ti|uj> = ∑jU†ijaj.
di = ∑jU†ijaj
= components in new basis in terms of the components in the old basis.
ai = ∑jUijdj =
components in old basis in terms of the components in the new basis.
Let |Φ> = Ω|ψ>,
|ψ> = ∑jaj|uj>,
|Φ> = ∑jaj'|uj>.
∑jaj'|uj> = ∑jajΩ|uj>.
ai' = ∑j<ui|Ω|uj>aj = ∑jΩijaj.
In a given representation, the matrix of Ω is defined by the action of Ω on the
basis vectors.
What is the relationship between the matrix elements of an operator Ω in the two bases
{|ui>} and
{|ti>}?
<ti|Ω|tj> = <ti|UU†ΩUU†|tj>
= <ui|U†ΩU|uj>, or
Ω(new basis)ij = (U†ΩU)(old basis)ij.
<ui|Ω|uj> = <ui|U†UΩU†U|uj>
= <ti|UΩU†|tj>, or
Ω(old basis)ij = (UΩU†)(new basis)ij.
Assume that
Ω(old basis)ij is given in the the
{|ui>} basis, and it is not diagonal.
Assume
Ω(new basis)ij is diagonal, the
{|ti>} are eigenvectors of Ω.
To find the matrix of U we find the eigenvalues and expansion coefficient of the
eigenvectors of
Ω in the old basis. This is the typical eigenvalue problem.
Let aij be the ith expansion coefficient of the jth
eigenvector in the
{|ui>} bases.
In the
{|ti>} the ith expansion coefficient of the jth eigenvector is just δij.
Then
ai = ∑kUikdk -->
aij = ∑kUikδkj
= Uij,
which shows that the columns of the matrix of U are the expansion coefficients
of the different eigenvectors in the
{|ui>} basis, i.e. Uij = aij.
The matrix of Ω in the {|ti>} basis is diagonal. It has the eigenvalues on its diagonal.
Note: What you call U and what you call U† depends on which
basis you consider the old basis and which basis the new basis.
If the old basis is the eigenbasis and the new basis is a different basis, then
U and U† are reversed.