130_notes.dvi

This is exactly the formula for the product of two matrices.











(OP) 11 (OP) 12 ... (OP) 1 j ...

(OP) 21 (OP) 22 ... (OP) 2 j ...

(OP)i 1 (OP)i 2 ... (OP)ij ...











=     

O 11 O 12 ... O 1 j ...

O 21 O 22 ... O 2 j ...

Oi 1 Oi 2 ... Oij ...





















P 11 P 12 ... P 1 j ...

P 21 P 22 ... P 2 j ...

Pi 1 Pi 2 ... Pij ...











So, wave functions are represented by vectors and operatorsby matrices,all in the space
of orthonormal functions.

• See Example 18.10.1:The Harmonic Oscillator Hamiltonian Matrix.*


• See Example 18.10.2:The harmonic oscillator raising operator.*


• See Example 18.10.3:The harmonic oscillator lowering operator.*

Now compute the matrix for the Hermitian Conjugate (See Section8.2) of an operator.

(O†)ij=〈ui|O†|uj〉=〈Oui|uj〉=〈uj|Oui〉∗=O∗ji

The Hermitian Conjugate matrix is the (complex)conjugate transpose.

Check that this is true forAandA†.

We know that there is a difference between abra vectorand a ket vector. This becomes explicit
in the matrix representation. Ifψ=

∑

j

ψjujandφ=

∑

k

φkukthen, the dot product is

〈ψ|φ〉=

∑

j,k

ψ∗jφk〈uj|uk〉=

∑

j,k

ψj∗φkδjk=

∑

k

ψ∗kφk.

We can write this indot product in matrix notationas

〈ψ|φ〉= (ψ 1 ∗ ψ∗ 2 ψ∗ 3 ...)







φ 1

φ 2

φ 3







The bra vector is the conjugate transpose of the ket vector. The both represent the same state but
are different mathematical objects.

18.2 The Angular Momentum Matrices*.

An important case of the use of the matrix form of operators is that of Angular Momentum (See
Section14.1) Assume we have an atomic state withℓ= 1 (fixed) butmfree. We may use the