Lecture 5: Eigenvalue Equations andOperators

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 OperatorsBasic 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 ,**)

− + =h2 2

22m xV x E

δ ψδ

ψ ψ(( ( ) (

x)x) x)

The Schrödinger equation

can be rewritten as

− +

=h2 2

22m xV x x E

δδ

( ) ( ) (Ψ Ψ x)

or :

ˆ ( ) ( ˆ ( )H x); H =Ψ Ψx E

m xV x= − +h2 2

22δδ

where H is the quantum mechanical Hamiltonianˆ

Quantum mechanical principles..Eigenfunctions

The Schrödinger equation H is an exampleof an eigenfunction equation

ˆ Ψ Ψ= E

Ωψ ϖψ=

( )( ) ( tan )( )operator function cons t samefunction=

(operator)(eigenfunction) = (eigenvalue)(eigenfunction)

Quantum mechanical principles..Eigenfunctions

If for an operator A we have a function f(x) such that Af(x) = kf(x) (where k is a constant)than f(x) is said to be an of A with the eigenvalue k

ˆˆ

ˆEigenfunction

Quantum mechanical principles..Eigenfunctions

e.g.

d

dx

thus exp[2x] is an eigenfunction to d

dxwith eigenvalue 2

exp[ ] exp[ ]2 2 2x x=

ˆ ( ) ( )Af x g x= : general definition of operator

An operator is a rule that transformsa given function f into another function. We indicate an operator with a circumflex '^' also called 'hat'.

Quantum mechanical principle Operators ..

Operator A Function f Af(x)d

dx f f'(x)

f 3f x cosx

x x

ˆ ˆ

cos()3

( 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

Rules for operators :

Quantum mechanical principle Operators ..

Example D =d

dxˆ

( ˆ ˆ )( ) ˆ ( ) ( )

( )

D x D x x

x x

x x

+ − = − + −= + −= + −

3 5 5 3 5

3 3 15

3 3 15

3 3 3

2 3

2 3

ˆ ˆ ( ) ˆ [ ˆ ( )] :ABf x A Bf x= product of operators

We first operate on f with the operator 'B'on the right of the operator product, and

then take the resulting function (Bf) and

operate on it with the operator A on the leftof the operator product.

ˆ

ˆ

ˆ

Quantum mechanical principle Operators ..

Example D =d

dxˆ ; ˆ

ˆ ˆ ( ) ˆ ( ( )) ( ) ' ( )

ˆ ˆ ( ) ˆ ( ˆ ( )) ' ( )

x x

Dxf x D xf x f x xf x

xDf x x Df x xf x

=

= = += =

Example : ˆ ˆ A = X ; B = d

dx2

ˆ ˆ : ˆ ˆ )ABf x

dfdx

fxf x

dfdx

= = +2 22 BAf =d(x

dx

2

[ ˆ , ˆ ]A B f xf= −2

Operators do not necessarily obey the commutative law :

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ [ ˆ , ˆ ]AB BA AB BA A B− ≠ − = ≠0 0: ummutator :

Quantum mechanical principle Operators ..

The square of an operator is defined as the product of

the operator with itself : A = AA 2ˆ ˆ ˆ

Examples : D =

DDf(x) = D(Df(x)) = Df' (x)

ddx

ˆ

ˆ ˆ ˆ ˆ ˆ "( )

ˆ

=

=

f x

Dd

dx2

2

2

Quantum mechanical principle Operators ..

Some linear operators : x xd

dxd

dx; ; ;2

2

2

Multiplicative Differential

cos; : [ ]2Some non - linear operators :

For linear operators the following identities apply :

(A +B)C = AC +BC; A(B + C) = AB + ACˆ ˆ ˆ ˆ ˆ ˆ ˜ ˆ ˆ ˆ ˆ ˆ ˆ ˆ

We shall be dealing with linear operators A,B,C, etc. where the follow rules apply

ˆ ˆ ˆ

ˆ ( ) ( ) ˆ ( ) ˆ ( )A f x g x Af x Ag x+ = +

ˆ ( ) ˆ ( )A kf x kAf x=

Quantum mechanical principle Operators ..

Quantum mechanical principles..Eigenfunctions

with the eigenfunction f and the eigenvalue k .**

** Af kf=

ˆ ( ) ( )A cf k cf= Must show

Demonstrate that cf also is an eigenfunction to A with the same eigenvalue k if c is a constant

ˆ

proof :

A is a Linear operator

ˆ ( ) ˆA cf cAf=

* ˆ( ) ˆA cf cAf=

Let A be a linear operator *ˆ

c is a constantf is a function e g. . A =

ddx

= ckf

f is an eigenfunction of A

Rearrangement of constantfactors and QED

= k cf( )

GG ee nn ee rr aa ll CC oo mm mm uu tt aa tt ii oo nn RR ee ll aa tt ii oo nn ss The following relations are readily shown

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

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

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

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

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

Quantum mechanical principle Operators ..

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

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

The operators A^ , B^ , C^

are differential or multiplicative operators

Quantum mechanical principle Operators ..

The set of eigenfunction f is orthonormal :

f f

n

i space

j

( ), ..

( ) ( )

x n

x x dxall

ij

=

∫ =

1

δ= ≠o if i j

= 1 if i = j

A linear operator A will have a set of eigenfunctions f n = 1,2,3..etcand associated eigenvalues k such that :

n

n

ˆ

( )x

ˆ ( ) ( )Af fn nx k xn=

Quantum mechanical principles..Eigenfunctions

Example Operator Eigenfunction Eigenvalue

1 δδ

x

exp[ ]ikx ik

2 δδ x

2 exp[ ]ikx −k 2

3 δδ x

2 coskx −k 2

4 δδ x

2 sinkx −k 2

Examples of operators and their eigenfunctions

Quantum mechanical principles..Eigenfunctions

What you should learn from this lecture1. ;

..

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

In ΩΩ

ψ ϖψψ

ϖ ψϖ

=

2. An operator ( A) is a rule that transforms a given function f into another function g as Af = g.We indicate an operator with a circumflex '^' also called 'hat'.

ˆˆ

A(BC)f(x) = (AB)Cf(x) : associative law of multiplicationˆ ˆ ˆ ˆ ˆ ˆ

( 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 operatorsˆ ˆ ( ) ˆ [ ˆ ( )] :ABf x A Bf x= product of operators

3. obays :Oprators

ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ

AB BA− ≠ = [A,B] 0; Operators do not commute,order of operators matters. [A,B] is call the commutator.

What you should learn from this lecture

ˆ ( ) ( ) ˆ ( ) ˆ ( )A f x g x Af x Ag x+ = +ˆ ( ) ˆ ( )A kf x kAf x=

Some linear operators are : x xd

dxd

dx; ; ;2

2

2

4. Linear operators obey :

5. A linear operator A will have a set of eigenfunctions f n = 1,2,3..etc and associated eigenvalues such that : Af f

The set of eigenfunction f is orthonormal :

f f

n

n n

n

i space

j

ˆ( )

ˆ ( ) ( )

( ), ..

( )( ( ))*

x kx k x

x n

x x dx

n

n

allij

=

=

∫ =

1

δ