"COMPLETENESS RELATION"
If we have an operator with its eigenvalues and eigenvectors
The eigenvectors form a basis
This will be very useful
We will study the requirements to be a basis
The requirements to have a basis are: (1) be orthonormal (2) verify a completeness relation
We remember the orthonormal product for basis elements
It is a Kronecker delta
This is the completeness relation which is analogous to the clousure relation
We shall use a Pauli matrix as an example of this question
PROBLEM: Prove that Pauli matrix eigenvectors form a basis
Prove the eigenvectors are orthonormal and the completeness relation
Firstly, we solve the determinant to obtain the eigenvalues
We write the Pauli matrix minus "lambda" times the identity matrix
We will write the equation
A second order equation to obtain "lambda"
We expand and rearrange
We obtain both "lambda" values
Secondly, we obtain the eigenvectors
We write "u_1" and "u_2"
We substitute
We multiply to obtain "a" and "b"
Rows by columns
"a=a"
We have obtained that "b=0"
We know the form of the eigenvector
We normalize
We write "u_1" ket
Now we solve the next eigenvector
The eigenvalue is now -1
"a=0"
We normalize again
Now we write "u_2"
Now we check it the basis is orthonormal
It is clear that is orthonormal
In any case, we will check it
Equal to 1
Equal to 0
They are normalized
It is orthonormal
Now we calculate the completeness relation
We write the outer products
We make rows by columns...
We have obtained two matrices
We can add both matrices
It is proved
