ABSTRACT
Ψ=Ψ Iˆ In view of the expansion (29.28), which holds for an arbitrary state function in the form
Ψ = Ψ Ψ Ψ Ψ = Ψ Ψ Ψ∑ ∑ we may define the unit or identity operator as
i iI ΨΨ=∑
=1 ˆ (33.4)
Conversely, for every linear operator we can form a square matrix, in terms of a complete set of functions by multiplying the left-hand side of Eq.(33.3) by and integrating, which gives
fˆ ,iΨ ∗Ψi
ΨΨ=ΨΨ=ΦΨ=ΦΨ∫ ∗ Iffd iiii ˆˆˆV
ΨΨ=ΨΨΨΨ= n
ˆˆ
Since ii b=ΦΨ this result is identical with that given by Eq.(33.1), provided we set
Vˆˆ dfff kikiik ΨΨ=ΨΨ= ∫ ∗ (33.5)
which are called the matrix elements of the linear operator in terms of the particular complete set of functions It is a straightforward matter to show that the rules of matrix algebra for addition, multiplication, associativity and distributivity (see Appendix II) apply to the matrix elements (33.5), following from their definition. Two properties of
fˆ .iΨ
matrix elements that are needed in quantum mechanics can be readily derived from Eq.(33.5). Using Eq.(29.19) we obtain
∗∗ =ΨΨ=ΨΨ= kiikkiik ffff ˆˆ (33.6)
which means that if is a Hermitian operator, then its matrix representation in a complete set of functions is Hermitian. Hence, the elements in the principal diagonal of a Hermitian matrix must be real. When all the elements are real, the matrix must be symmetric. From Eq.(29.20) we have
fˆ
ikiikiik f δλλ =ΨΨ=ΨΨ ˆ (33.7)
that is: if the functions are the eigenfunctions of the Hermitian operator the matrix representation is diagonal, and the matrix elements are the eigenvalues of the operator. A quantum mechanical problem can be stated in a differential equation form or in a matrix form. In both cases the solution consists of finding the eigenvalues and the eigenfunctions of an operator associated with a physical observable. In view of a matrix representation in any complete set of functions, the eigenvalue equation (29.20) must be formulated as
iΨ ,fˆ
(33.8) Ψ=Ψ λfˆ where the arbitrary eigenfunction Ψ can be expanded, as given by Eq.(29.4), in terms of a complete set of functions If we multiply Eq.(33.8) by and integrate, we obtain .iΨ ∗Ψi
ˆ ˆ ˆV V ori i if d d f Iλ λ∗ ∗Ψ Ψ = Ψ Ψ Ψ Ψ = Ψ Ψ∫ ∫ i or also, in view of Eq.(33.4)
k kki f λ
ˆ
This can be rewritten as
= n
1 λ
which reads
(33.9)
⎟⎟ ⎟⎟ ⎟
⎠
⎞
⎜⎜ ⎜⎜ ⎜
⎝
⎛ =
⎟⎟ ⎟⎟ ⎟
⎠
⎞
⎜⎜ ⎜⎜ ⎜
⎝
⎛
⎟⎟ ⎟⎟ ⎟
⎠
⎞
⎜⎜ ⎜⎜ ⎜
⎝
⎛
a
a a
a
a a
fff
fff fff
…… …
……… … …
λ
Equation (33.6) is called the matrix eigenvalue equation and consists of n linear
homogenous equations for the given by ,ka ( ) 0... 1212111 =+++− nnafafaf λ ( ) 0... 2222121 =++−+ nnafafaf λ (33.10) ………
( ) 0...2211 =−+++ nnnnn afafaf λ This system of simultaneous equations has a nontrivial solution for the if and only if the characteristic determinant vanishes
( ) ( )
( ) ( )
01ˆˆdet
= ⎟⎟ ⎟⎟ ⎟
⎠
⎞
⎜⎜ ⎜⎜ ⎜
⎝
⎛
−
− −
=− λ
λ λ
λ nnnn
fff
fff fff
f
… ………
… …
(33.11)
If expanded, the determinant gives a polynominal of nth degree called the characteristic polynominal of fˆ with n roots which are the desired eigenvalues. For each eigenvalue Eqs.(33.10) can be solved for the giving the corresponding eigenfunction that is a column matrix.
