# Lec

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```					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 ,**)
sno itcnufnegiE. .selpicnirp lacinahcem mutnauQ
The Schrödinger equation
2  2 (x)

2m x   2   (x)V(x)  E(x)
can be rewritten as
 2  2       

 2m 2   V(x)(x)  E(x)

   x        
or :
2 2
ˆ              ˆ
H(x)  E(x); H =           V(x)
2m x 2
ˆ
where H is the quantum mechanical Hamiltonian
Quantum mechanical principles..Eigenfunctions

ˆ
The Schrödinger equationH  E is an example
of an eigenfunction equation

(operator )(function )  (cons tant )(samefunction )

  

(operator)(eigenfunction)= (eigenvalue)(eigenfunction)
Quantum mechanical principles..Eigenfunctions

ˆ
If for an operator A we have a function f(x) such that
ˆ
Af(x) = kf(x) (where k is a constant)
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
ˆ
Af (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        Function f           ˆ
Af(x)
d
f                       f'
(x)
dx
3                   f                     3f
cos()              x                      cosx
x                       x
Quantum mechanical principle.. Operators
Rules for operators:

ˆ ˆ             ˆ         ˆ
(A  B )f (x )  Af (x )  Bf (x) : Sum of operators
ˆ ˆ             ˆ         ˆ
(A  B )f (x )  Af (x )  Bf (x ) : Dif. of operators

ˆ    d
Example D =
dx
ˆ  3)(x 3  5)  D (x 3  5)  3(x 3  5)
(D ˆ               ˆ
 3x  (3x  15)
2        3

 3x  3x  15
2       3
Quantum mechanical principle.. Operators
ˆ ˆ        ˆ ˆ
ABf (x )  A[Bf (x )] : product of operators

We first operate on f with the operator B''ˆ
on the right of the operator product,and
then take the resulting function (ˆ f) and
B
ˆ
operate on it with the operatorA on the left
of the operator product.

Example Dˆ = d ;x  x
ˆ
dx
ˆˆ         ˆ
Dxf (x )  D (xf (x ))  f (x )  xf '(x )
ˆˆ         ˆ ˆ
xDf (x )  x (Df (x ))  xf '(x )
Quantum mechanical principle.. Operators

Operators do not necessarily obey the commutative law:
ˆ ˆ ˆˆ        ˆ ˆ ˆˆ      ˆ ˆ
AB  BA  0 : AB  BA  [ A, B ]  0   Cummutator :

ˆ        ˆ       d
Example : A = X 2; B =
dx

ˆ Bf  x2
ˆ       df ˆ ˆ     d(x2f)           2 df
A            : BAf =         2xf  x
dx          dx                dx
ˆ ˆ
[ A, B]f  2xf
Quantum mechanical principle.. Operators
The square of an operator is defined as the product of
ˆ    ˆ ˆ
the operator with itself: A2 = AA

ˆ d
Examples : D = dx
ˆˆ       ˆ ˆ        ˆ
DDf(x) = D(Df(x)) = Df'(x)  f "(x )
ˆ2  d2
D 
dx 2
Quantum mechanical principle.. Operators
ˆ ˆ ˆ
We shall be dealing with linear operatorsA, B, C, etc.
where the follow rules apply

ˆ                   ˆ         ˆ
A{f (x )  g(x )}  Af (x )  Ag(x)
ˆ             ˆ
A{kf (x )}  kAf (x )

d d2
Some linear operators:            x;x 2 ; ; 2
dx dx
Multiplicative                        Differential

Some non- linear operators:           cos;   :   
2

:
For linear operators the following identities apply
ˆ ˆ ˆ ˆˆ ˆ ˜ ˆ ˆ ˆ            ˆ ˆ ˆ ˆ
(A + B)C = AC + BC; A(B + C) = AB + AC
Quantum mechanical principles..Eigenfunctions
ˆ
Let A be a linear operator*
**
with the eigenfunction f and the eigenvalue k .
Demonstrate that cf also is an eigenfunction to        ˆ
A
with the same eigenvalue k if c is a constant
wohs tsuM
ˆ
A(cf )  k (cf )
proof :
ˆ         ˆ
A(cf )  cAf       ckf  k(cf )
Rearrangement of constant
factors and QED
ˆ
A is a Linear operator                               ˆ
f is an eigenfunction ofA
* ˆ        ˆ
A(cf )  cAf                           ** ˆ
Af  kf
c is a constant          d
f is a function e.g. A = dx
Quantum mechanical principle.. Operators

General Commutation Relations
The following relations are readily shown

^    ^
,B                  ^       ^
[ A             ] = - [ B ,A                   ]

^    ^
[ A       ,A       n]   = 0 n=1,2,3,.......

^     ^             ^      ^                      ^        ^
[k       A    ,B   ] =[ A        ,k B             ] = k[ A         ,B       ]

^         ^  ^
, B +C              ^        ^
,B            ^        ^
,C
[ A                        ] =[ A                 ] +[ A                ]

^        ^        ^             ^        ^                ^        ^
[ A          +B ,C ] = [ A                ,B ] + [ A                ,C ]
Quantum mechanical principle.. Operators

^     ^^
, BC       ^     ^        ^      ^   ^       ^
[ A             ] =[ A   , B ]C          + B [A   ,C ]

^ ^        ^        ^         ^      ^       ^        ^        ^
[ A B , C ] =[ A           ,   C ] B      + A         [ B   ,   C ]

^     ^
, B         ^
, C
The   operators       A

are differential or            multiplicative operators
Quantum mechanical principles..Eigenfunctions
ˆ
A linear operator A will have a set of
eigenfunctions f (x ) {n = 1,2,3..etc}
n
and associated eigenvalues kn such that :

ˆ
Afn (x )  k n fn (x )

The set of eigenfunction {f (x ),n  1..}
n
is orthonormal :
 o if i  j
 fi (x )fj (x )dx  ij
all space                         1 if i = j
Quantum mechanical principles..Eigenfunctions

Examples of operators and their
eigenfunctions
Example   Operator   Eigenfunction   Eigenvalue

1                   exp[ikx ]       ik
x
2         2         exp[ikx ]
k 2
x
3         2         coskx
k 2
x
4         2         sinkx
k 2
x
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
2. An operator ( A) is a rule that transforms a given
ˆ
function f into another function g as Af = g.
We indicate an operator with a
circumflex '^' also called 'hat'.
3. Oprators obays :
ˆ ˆ            ˆ          ˆ
(A  B )f (x )  Af (x )  Bf (x) : Sum of operators
ˆ ˆ            ˆ         ˆ
(A  B )f (x )  Af (x )  Bf (x ) : Dif. of operators
ˆ ˆ         ˆ ˆ
ABf (x )  A[Bf (x )] : product of operators
ˆ ˆ ˆ        ˆˆ ˆ
A(BC)f(x) = (AB)Cf(x) : associative law of multiplication
ˆˆ ˆˆ       ˆ ˆ
AB  BA = [A, B]  0; Operators do not commute,
ˆ ˆ
order of operators matters. [A, B] is call the commutator.
What you should learn from this lecture
4. Linear operators obey:
ˆ                   ˆ         ˆ
A{f (x )  g(x )}  Af (x )  Ag(x)
ˆ             ˆ
A{kf (x )}  kAf (x )
d d2
Some linear operators are : x;x 2 ; ;
dx dx 2

ˆ
5. A linear operator A will have a set of eigenfunctions
fn (x) {n = 1,2,3..etc} and associated eigenvalues kn
ˆ
such that : Afn (x)  kn fn (x)

The set of eigenfunction {f (x),n  1..}
n
is orthonormal :
*
 fi (x)(fj (x)) dx   ij
all space

```
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