18 Operators Matrices and Spin
We have already solved many problems in Quantum Mechanics using wavefunctions and differential
operators. Since the eigenfunctions of Hermitian operators are orthogonal (and we normalize them)
we can now use the standard linear algebra to solve quantum problems with vectors and matrices.
To include the spin of electrons and nuclei in our discussion of atomic energy levels, we will need
the matrix representation.
These topics are covered at very different levels inGasiorowicz Chapter 14, Griffiths Chapters
3, 4and, more rigorously, inCohen-Tannoudji et al. Chapters II, IV and IX.
18.1 The Matrix Representation of Operators and Wavefunctions
We will define our vectors and matrices using a complete set of, orthonormal basis states (See
Section8.1) ui, usually the set of eigenfunctions of a Hermitian operator. These basis states are
analogous to the orthonormal unit vectors in Euclidean space ˆxi.
〈ui|uj〉=δijDefine thecomponents of a state vectorψ(analogous toxi).
ψi≡〈ui|ψ〉 |ψ〉=∑
iψi|ui〉The wavefunctions are therefore represented asvectors.Define thematrix element
Oij≡〈ui|O|uj〉.We know that anoperator acting on a wavefunctiongives a wavefunction.
|Oψ〉=O|ψ〉=O∑
jψj|uj〉=∑
jψjO|uj〉If we dot〈ui|into this equation from the left, we get
(Oψ)i=〈ui|Oψ〉=∑
jψj〈ui|O|uj〉=∑
jOijψjThis is exactly the formula for a state vector equals amatrix operatortimes a state vector.
(Oψ) 1
(Oψ) 2
...
(Oψ)i
...
=
O 11 O 12 ... O 1 j ...
O 21 O 22 ... O 2 j ...
... ... ... ... ...
Oi 1 Oi 2 ... Oij ...
... ... ... ... ...
ψ 1
ψ 2
...
ψj
...
Similarly, we can look at theproduct of two operators(using the identity
∑
k|uk〉〈uk|= 1).(OP)ij=〈ui|OP|uj〉=∑
k〈ui|O|uk〉〈uk|P|uj〉=∑
kOikPkj