Chapter 5 Lecture 5 - Operators

5.1 Properties of Operators

AU_n=a_nU_n (from equation II.2)

where A is an operator, U_n is an eigenstate/eigenfunction and a_n is an eigenvalue. For an operator to be associated with a physical observable, these conditions must be true

eigenvalues ( a_n ) must be a real number
eigenfunctions ( U_n ) must be well behaved. This means the function is continuous, differentiable and single valued

These are ensured in equation II.1 if the operator A has the mathematical property that its Hermitian.

When we have two functions ( g(r) and f(r) ) that are well behaved functions then A is Hermitian if:

ó
õ

all of space

(Ag(r))^*f(r

)dV=

ó
õ

all of space

g^*(r)(Af(r))dV

For a Hermitian operator we will have three consequences:

a_n=a_n^* (5.1)
this means that a_n 's are real.
the eigenfunctions obey

ó
õ

all of space
U_n^*(r)U_n(r)dV=d_mn     (5.2)
where d_mn is the Kronecker delta, m and n are labels for two eigenstates of A . for example (orthogonality)

ó
õ

all of space
U_m^*U_ndV=0  (m¹ n)     (5.3)
and (orthonormal)

ó
õ

all of space
U_m^*U_ndV=1  (if m=n)     (5.4)
this is only true when the eigenstates are correctly normalised
Completeness Theorem: The spectrum of eigenfunctions ( U₁, U₂, ... ) of a Hermitian operator can be used to construct other functions. This means that any well behaved function can be expanded in terms of these eigenfunctions.

5.1.1 An Example of a Hermitian Function - The Infinite Square Well

The eigenfunctions of an infinite square well are a set of sines and cosines. Consequence two is satisfied as these functions are orthogonal. Consequence three is satified as its a Fourier Series:

f(x)=

n,m

é
ê
ê
ë

b_nsin

æ
ç
ç
è

ö
÷
÷
ø

+c_mcos

æ
ç
ç
è

ö
÷
÷
ø

ù
ú
ú
û

5.2 Construction of Operators

Starting from the free particle wavefunction

U(x,t)=Ae

(px-Et)

(5.5)

by inspection we can find an operator for energy

Eu(x,t)=Eu(x,t) (5.6)

this operator is

¶ t

(5.7)

the proof is:

¶ t

u(x,t)= x-

EAe

(px-Et)

=Eu(x,t)

similarly for momentum

p=-

¶ x

(5.8)

5.3 Operators and Measurements

if the state Y is an eigenstate of A then a unique eigenvalue a is determined
AY =aY
if state Y is a linear superposition of eigenstates

Y (r ,t)=

å

n
c_nU_n(r,t) (5.9)
the c_n 's are expansion coefficients (weighting factors) but also in Quantum Mechanics they can be interpreted as probability amplitudes for finding the system in eigenstate U_n .
P(U_n)=|C_n|²
To measure A

AY (r,t)=A

å

n
c_nU_n(r,t) (5.10)

AY (r ,t)=

å

n
c_na_nU_n(r,t)
We have a collapse when we make a measurement to a particular eigenvalue ( a_m ) with probability |c_m|² .