Transcript No Slide Title

Lecture 5: Eigenvalue Equations and
Operators
The material in this lecture covers the following in Atkins.
11.5 The informtion of a wavefunction
(b) eigenvalues and eigenfunctions
(c) operators
Lecture on-line
Eigenvalue Equations and Operators (PDF)
Eigenevalue Equations and Operators (PowerPoint)
Handout for lecture 5 (PDF)
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 4)
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
Basic math background
Differential Equations
Operator Algebra
Eigenvalue Equations
Extensive account of Operators
Audio-visuals on-line
Postulates of Quantum mechanics (PDF)
Postulates of Quantum mechanics (HTML)
Postulates of quantum mechanics (PowerPoint ****)
Slides from the text book (From the CD included in Atkins ,**)
Quantum mechanical principles..Eigenfunctions
The Schrödinger equation
2
2
 (x)



(x)V(x)

E

(x)
2m x 2
can be rewritten as
 2  2


(x)  E(x)

V(x)
 2m 2

x


or :
ˆ (x)  E(x); H
ˆ =
H
2
2


V(x)
2m x 2
ˆ is the quantum mechanical Hamiltonian
where H
Quantum mechanical principles..Eigenfunctions
ˆ   E is an example
The Schrödinger equation H
of an eigenfunction equation
(operator)(function)  (cons tant )(samefunction)
  
(operator)(eigenfunction) = (eigenvalue)(eigenfunction)
Quantum mechanical principles..Eigenfunctions
ˆ we have a function f(x) such that
If for an operator A
ˆ f(x) = kf(x) (where k is a constant)
A
ˆ
than f(x) is said to be an Eigenfunction of A
with the eigenvalue k
e.g.
d
exp[2x ]  2 exp[2x ]
dx
d
thus exp[2x] is an eigenfunction to
dx
with eigenvalue 2
Quantum mechanical principle.. Operators
Aˆ f (x )  g(x ) : general definition of operator
An operator is a rule that transforms
a given function f into another function.
We indicate an operator with a
circumflex '^' also called 'hat'.
ˆ
Operator A
d
dx
3
cos()
Function f
ˆ f(x)
A
f
f'(x)
f
x
3f
cosx
x
x
Quantum mechanical principle.. Operators
Rules for operators :
(Aˆ  Bˆ )f (x )  Aˆ f (x )  Bˆ f (x ) : Sum of operators
(Aˆ  Bˆ )f (x )  Aˆ f (x )  Bˆ f (x ) : Dif. of operators
d
ˆ =
Example D
dx
3
3
3
ˆ
ˆ
ˆ
(D  3)(x  5)  D (x  5)  3(x  5)
 3x  (3x  15)
2
3
 3x  3x  15
2
3
Quantum mechanical principle.. Operators
Aˆ Bˆ f (x )  Aˆ [Bˆ f (x )] : product of operators
ˆ'
We first operate on f with the operator 'B
on the right of the operator product, and
ˆ f) and
then take the resulting function (B
ˆ on the left
operate on it with the operator A
of the operator product.
d
ˆ
Example D =
; xˆ  x
dx
Dˆ xˆf (x )  Dˆ (xf (x ))  f (x )  xf '(x )
xˆDˆ f (x )  xˆ (Dˆ f (x ))  xf '(x )
Quantum mechanical principle.. Operators
Operators do not necessarily obey the commutative law :
Aˆ Bˆ  Bˆ Aˆ  0 : Aˆ Bˆ  Bˆ Aˆ  [ Aˆ , Bˆ ]  0
ˆ = X 2; B
ˆ =
Example : A
Cummutator :
d
dx
2f)
df
d(x
2
2 df
ˆ
ˆ
ˆ
ˆ
ABf  x
: BAf =
 2xf  x
dx
dx
dx
ˆ ,B
ˆ ]f  2xf
[A
Quantum mechanical principle.. Operators
The square of an operator is defined as the product of
ˆ2 = A
ˆA
ˆ
the operator with itself: A
ˆ= d
Examples : D
dx
ˆD
ˆ f(x) = D
ˆ (Dˆ f(x)) = D
ˆ f'(x)  f "(x )
D
d2
D 
dx 2
ˆ2
Quantum mechanical principle.. Operators
ˆ ,B
ˆ ,C
ˆ , etc.
We shall be dealing with linear operators A
where the follow rules apply
Aˆ {f (x )  g(x )}  Aˆ f (x )  Aˆ g(x)
Aˆ {kf (x )}  kAˆ f (x )
Some linear operators:
d d2
x;x ; ; 2
dx dx
2
Multiplicative
Some non - linear operators:
Differential
cos;
:

2
For linear operators the following identities apply:
ˆ +B
ˆ )C
ˆ =A
ˆC
ˆ +B
ˆC
˜; A
ˆ (B
ˆ +C
ˆ) =A
ˆB
ˆ +A
ˆC
ˆ
(A
Quantum mechanical principles..Eigenfunctions
ˆ be a linear operator*
Let A
with the eigenfunction f and the eigenvalue k** .
Demonstrate that cf also is an eigenfunction to
with the same eigenvalue k if c is a constant
Must show
Aˆ (cf )  k (cf )
proof :
ˆ (cf )  cA
ˆf
A
ˆ
A
 ckf  k(cf )
ˆ is a Linear operator
A
Rearrangement of constant
factors and QED
ˆ
f is an eigenfunction of A
*A
ˆ (cf )  cAˆ f
c is a constant
d
f is a function e.g. A = dx
** A
ˆ f  kf
Quantum mechanical principle.. Operators
General Commutation Relations
The following relations are readily shown
[ A
^
^
,B
] = - [ B ,A
^
^
,A
n]
[ A
^
[k
A
^
[ A
^
[ A
^
^
,B
= 0
^
]=[ A
^
^
, B +C
^
^
^
]
n=1,2,3,.......
^
,k B
^
^
^
,B
]=[ A
^
+B ,C ] = [ A
^
,B
] = k[ A
^
]+[ A
^
^
,C
^
,B ] + [ A
]
]
^
,C ]
Quantum mechanical principle.. Operators
^
[ A
^^
, BC
^ ^
^
]=[ A
^
^
^
^
,
[ A B , C ]=[ A
The operators
are differential or
^
A
^
, B ]C
,
^
B
^
^
C ] B
,
^
+ B [A
^
,C ]
^
+ A
^
[ B
^
C
multiplicative operators
,
^
C ]
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 :
 fi (x )fj (x )dx  ij
all space
 o if i  j
 1 if i= j
Quantum mechanical principles..Eigenfunctions
Examples of operators and their
eigenfunctions
Example
1
2
3
4
Operator
Eigenfunction
Eigenvalue

x
2
x
2
x
2
x
exp[ikx ]
ik
exp[ikx ]
k 2
coskx
k 2
sinkx
k 2
What you should learn from this lecture
1. In an eigenvalue equation :   ; an operator
 works on a function  to give the function back times
a constant . The function  is called an eigenfunction and
the constant .
ˆ ) is a rule that transforms a given
2. An operator ( A
ˆ f = g.
function f into another function g as A
We indicate an operator with a
circumflex '^' also called 'hat'.
3. Oprators obays :
(Aˆ  Bˆ )f (x )  Aˆ f (x )  Bˆ f (x ) : Sum of operators
(Aˆ  Bˆ )f (x )  Aˆ f (x )  Bˆ f (x ) : Dif. of operators
Aˆ Bˆ f (x )  Aˆ [Bˆ f (x )] : product of operators
ˆ (B
ˆC
ˆ )f(x) = (A
ˆB
ˆ )C
ˆ f(x): associative law of multiplication
A
ˆB
ˆ B
ˆA
ˆ = [A
ˆ ,B
ˆ ]  0; Operators do not commute,
A
ˆ ,B
ˆ ] is call the commutator.
order of operators matters. [A
What you should learn from this lecture
4. Linear operators obey :
Aˆ {f (x )  g(x )}  Aˆ f (x )  Aˆ g(x)
Aˆ {kf (x )}  kAˆ f (x )
2
d
d
Some linear operators are : x;x 2 ; ;
dx dx 2
ˆ will have a set of eigenfunctions
5. A linear operator A
fn (x) {n = 1,2,3..etc} and associated eigenvalues kn
ˆ f (x)  k f (x)
such that : A
n
nn
The set of eigenfunction {fn (x),n  1..}
is orthonormal :
*
 fi (x)(fj (x)) dx   ij
all space