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Lecture 6: Operators and Quantum
Mechanics
The material in this lecture covers the following in Atkins.
11.5 The informtion of a wavefunction
(c) Operators
Lecture on-line
Operators in quantum mechanics (PDF)
Operators in quantum mechanics (HTML)
Operators in Quantum mechanics (PowerPoint)
Handout (PDF)
Assigned Questions
Tutorials on-line
Reminder of the postulates of quantum mechanics
The postulates of quantum mechanics (This is the writeup for Dry-lab-II)(
This lecture has covered postulate 3)
Basic concepts of importance for the understanding of the postulates
Observables are Operators - Postulates of Quantum Mechanics
Expectation Values - More Postulates
Forming Operators
Hermitian Operators
Dirac Notation
Use of Matricies
Basic math background
Differential Equations
Operator Algebra
Eigenvalue Equations
Extensive account of Operators
Historic development of quantum mechanics from classical mechanics
The Development of Classical Mechanics
Experimental Background for Quantum mecahnics
Early Development of Quantum mechanics
Audio-visuals on-line
Early Development of Quantum mechanics
Audio-visuals on-line
Postulates of Quantum mechanics (PDF) (simplified version from Wilson)
Postulates of Quantum mechanics (HTML) (simplified version from Wilson)
Postulates of Quantum mechanics (PowerPoint ****)
(simplified version from Wilson)
Slides from the text book (From the CD included in Atkins ,**)
Operators and Quantum Mechanics
We now have
Re view
(Ia) A Quantum mechanical system is specified
by the statefunction (x)
(Ib) The state function (x) contains all
information about the system we can know
(Ic) A system described by the
state function H(x) = E(x)
has exactly the energy E
Operators and Quantum Mechanics
We have seen that a 'free' particle moving
Re view
in one dimension in a constant (zero) potential
has the Hamiltonian
2
2
X Hˆ  
O
The Schrodinger equation is
2 2
  (x)


E

(x)
2m x 2
with the general solution :
 (x)  Aexp
ikx
2 2
k
and energies E =
2m
 Bexp
ikx

2m x 2
Operators and Quantum Mechanics
How does the state function (x, t) give us
information about an observable other than
the energy such as the position or the momentum ?
Good question
Any observable ' ' can be expressed in classical physics
in terms of x,y, z and px ,p y ,p z .
Examples :
 = x, px , v x , p2
x , T, V(x), E
Operators and Quantum Mechanics
We can construct the corresponding operator
from the substitution:
Classical Mechanics Quantum Mechanics
x
px
y
py
z
pz
ˆ (x,y,z,
as 

xˆ   x ; pˆ x  
i x

yˆ   y ; pˆ y  
i y

zˆ   z ; pˆ z  
i z
d
d
d
,
,
)
i dx i dy i dz
Such as :
ˆ ˆ
ˆ xˆ , pˆ x , vˆx , pˆ 2
=
x , T, V(x), E
Re view
Operators and Quantum Mechanics
Im portan t news
For an observable  with the corresponding
ˆ we have the eigenvalue equation :
operator 
 n  nn
(IIIa). The meassurement of the quantity represented by 
has as the o n l y outcome one of the values
n n = 1, 2, 3 ....
(IIIb). If the system is in a state described by n
a meassurement of  will result in the
value n
Quantum mechanical principle.. Operators
ˆ
For any such operator 
Im portan t news
we can solve the eigenvalue problem
ˆ   

n
n n
We obtain
eigenfunctions and eigenvalues
The only possible values that can arise from measurements
of the physical observable  are the eigenvalues n
Postulate 3
Operators and Quantum Mechanics
Im portan t news
The x - component 'px ' of the linear momentum
p  pxex  pye y  pz ez
Is represented by the operator pˆ x 
With the eigenfunctions
Exp[ikx]

ix
and eigenvalue k
 Exp[ikx]
= kExp[ikx]
i
x
We note that k can take any value
 > k > 
Operators and Quantum Mechanics
 (x)  Aexp
ikx
For A = 0
 Bexp
ikx
New insight
2 2
k
and energies E =
2m
ikx
  (x )  B exp
ˆx
this wavefunction is also an eigenfunction to p
With eigenvalue for pˆ x of - k
2 2
k
Thus - (x ) describes a particle of energy E=
2m
Px2
and momentum p x   k ; note E =
as it must be.
2m
This system corresponds to a particle
moving with constant velocity
p
We know nothing about its position
vx  x  - k/m
m
since |  (x) |2  B
Operators and Quantum Mechanics
 (x)  Aexp
ikx
For B = 0
New insight
ikx and energies E =
 Bexp
2 2
k
2m
  (x )  A exp
ikx
ˆx
this wavefunction is also an eigenfunction to p
With eigenvalue for pˆ x of k
Thus  (x ) describes a particle of energy E=
2 2
k
2m
Px2
and momentum p x  k ; note E =
as it must be.
2m
This system corresponds to a particle
moving with constant velocity
px
We know nothing about its position
vx 
 k/m
m
since |  (x) |2  B
Operators and Quantum Mechanics
What about : (x )  A exp
ikx
d
d
ikx
ikx
ˆ
p x (x) = A
exp  B
exp
i dx
i dx
 A k expikx  B k expikx
How can we find
px in this case ?
ikx
 B exp
ˆ x since:
It is not an eigenfunction to p
New insight
?
Quantum mechanical principles..Eigenfunctions
ˆ will have a set of
A linear operator A
eigenfunctions fn (x ) {n = 1,2,3..etc}
and associated eigenvalues kn such that :
ˆ fn (x )  k n fn (x )
A
The set of eigenfunction {fn (x),n  1..}
is orthonormal :
* f (x)dx  
f
(x)
i
j
ij
all space
 o if i  j
 1 if i= j
Quantum mechanical principles..Eigenfunctions
An example of an orthonormal set is the Cartesian unit vectors
ei
ei  e j   ij
ei
ei
An example of an orthonormal function set is
 n (x) =
1 nx 
sin
L  L 
n = 1, 2, 3,4, 5....
L
*
 n (x) m (x)  nm
o
Quantum mechanical principles..Eigenfunctions
The set of eigenfunction {fn (x ),n  1..}
forms a complete set.
That is, any function g(x) that
depends on the same variables
as the eigenfunctions can be written
ei ; i = 1, 2,3 form a complete set
all
g(x) =  anfn (x )
ei
i=1
ei
where
an 
ei
*
 fn (x) g(x)dx
all space
For any vector v
v  (v  e1 )e1  (v  e 2 )e2  (v  e 3 )e3
Quantum mechanical principles..Eigenfunctions
all
In the expansion :
g(x) =  aifi (x )
(1)
i=1
we can show that : an   fn (x)* g(x)dx
V
*
from the orthonormality :  fi (x) fj (x)dx  ij
V
A multiplication by fn (x) on both sides followed by
integration affords
all
*
all
g(x) =  aifi (x)   fn (x) g(x)dx =  ai  fn (x)* fi (x)dx
i=1
or : : aann 
or
V
*
g(x)f
(x
)dx
g(x)f
(x)
dx

nn
all space
space
all
i=1 V
ij
Operators and Quantum Mechanics
(x)  Aexpikx  Bexp ikx is a linear combination
of two eigenfunctions to pˆ x
px  k
How can we find
px in this case ?
px   k
What you should learn from this lecture
1. Postulate 3
ˆ
For an observable  with the corresponding operator 
we have the eigenvalue equation :   n   n  n
(i) The meassurement of the quantity represented by 
has as the o n l y outcome one of the values  n n = 1, 2, 3 ....
(ii) If the system is in a state described by  n
a meassurement of  will result in the value n
Illustrations :
  (x)  A expikx is an eigenfunction to pˆ x with eigenvalue k
  (x)  A expikx is an eigenfunction to pˆ x with eigenvalue - k
Both are eigenfunctions to the Hamiltonian for a free particle
2 (ˆp )2
2 k2
x
H=
with eigenvalues E =
2m
2m
  (x) represents a free particle of momentum k
  (x) represents a free particle of momentum - k
What you should learn from this lecture
2. Postulate 4.
The set of eigenfunction {fn (x),n  1..}
forms a complete set.
That is, any function g(x) that depends on the same
variables as the eigenfunctions can be written :
all
g(x) =  anfn (x) where
i=1
an 
 g(x)fn (x)dx
all space