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 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

2mδ2ψ(x)

δx2 +ψ(x)V(x)=Eψ(x)

The Schrödinger equation

can be rewritten as

−h2

2mδ2

δx2+V(x)⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ Ψ(x)=EΨ(x)

or :

ˆ H Ψ(x)=EΨ(x); ˆ H =−h2

2mδ2

δx2+V(x)

where ˆ H is the quantum mechanical HamiltonianQuantum mechanical principles..Eigenfunctions

The Schrödinger equation ˆ H Ψ =EΨ is an exampleof an eigenfunction equation

Ωψ =ϖψ

(operator )(function) =(cons tant)(samefunction )

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

Quantum mechanical principles..Eigenfunctions

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

Quantum mechanical principles..Eigenfunctions

e.g.

ddx

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

thus exp[2x] is an eigenfunction to ddx

with eigenϖalue 2

ˆ A f (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 ˆ A Function f ˆ A f(x)d

dx f f' (x)

3 f 3fcos() x cosx

x x

( ˆ A + ˆ B )f(x ) = ˆ A f(x )+ ˆ B f(x ) : Sum of operators

( ˆ A − ˆ B )f(x ) = ˆ A f(x )− ˆ B f(x ) : Dif. of operators

Rules for operators :

Quantum mechanical principle.. Operators

Example ˆ D = ddx

( ˆ D + ˆ 3 )(x 3 −5)= ˆ D (x 3 −5)+ 3(x 3 −5)

=3x 2 + (3x 3 −15 )=3x 2 + 3x 3 −15

ˆ A ˆ B f (x ) = ˆ A [ ˆ B f(x )] : product of operators

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

Quantum mechanical principle.. Operators

Example ˆ D = ddx

; ˆ x =x

ˆ D x f(x ) = ˆ D (xf(x ))=f(x )+ xf'(x )ˆ x D f(x ) = ˆ x ( ˆ D f(x ))=xf'(x )

Example : ˆ A = X2; ˆ B = ddx

ˆ A B f =x2 dfdx: ˆ B A f =d(x2f)

dx =2xf+x2 dfdx

[ˆ A , ˆ B ]f =−2xf

Operators do not necessarily obey the commutative law :ˆ A ˆ B − ˆ B A ≠0 : ˆ A B − ˆ B A =[ ˆ A , ˆ B ] ≠0 Cummutator :

Quantum mechanical principle.. Operators

The square of an operator is defined as the product of the operator with itself: ˆ A 2 = ˆ A A

Examples : ˆ D = ddx

ˆ D D f(x) = ˆ D ( ˆ D f(x)) = ˆ D f' (x) =f"(x )

ˆ D 2 =d 2

dx 2

Quantum mechanical principle.. Operators

Some linear operators : x;x 2; ddx

; d 2

dx 2

Multiplicative Differential

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

For linear operators the following identities apply :( ˆ A + ˆ B ) ˆ C = ˆ A C + ˆ B ˜ C ; ˆ A ( ˆ B + ˆ C ) = ˆ A ˆ B + ˆ A C

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

ˆ A {f (x ) +g (x )} = ˆ A f(x )+ ˆ A g (x )

ˆ A {kf (x )} =k ˆ A f(x )

Quantum mechanical principle.. Operators

Quantum mechanical principles..Eigenfunctions

with the eigenfunction f and the eigenvalue k**.

** ˆ A f =kf

ˆ A (cf ) =k (cf)

Must show

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

proof:

ˆ A is a Linear operator

ˆ A (cf)=cˆ A f

* ˆ A (cf)=cˆ A f

Let A be a linear operator*

c is a constantf is a functione.g. A = d

dx

=ckf

f is an eigenfunction of ˆ A

Rearrangement of constantfactors and QED

=k(cf)

General Commutation RelationsGeneral Commutation Relations The following relations are readily shown

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

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

[k A ,B ] = [A ,kB ] = k[A ,B ]

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

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

Quantum mechanical principle.. Operators

[ A ,BC ] = [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 {fn(x ),n =1..} is orthonormal :

fi(x )all space

∫ fj(x )dx =dij=o if i ≠ j= 1 if i = j

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

ˆ A fn(x ) =knfn (x )

Quantum mechanical principles..Eigenfunctions

Example Operator Eigenfunction Eigenvalue

1 d

d x

exp[ ikx ] ik

2δ

2

δ x

exp[ ikx ]

− k

2

3δ

2

δ x

cos kx

− k

2

4δ

2

δ x

sin kx

− k

2

Examples of operators and their eigenfunctions

Quantum mechanical principles..Eigenfunctions

What you should learn from this lecture1. In an eigenvalue equation :Ωψ=ϖψ; an operatorΩ works on a function ψ to give the function back timesa constant ϖ. The function ψ is called an eigenfunction and the constant ϖ.2. An operator ( ˆ A ) is a rule that transforms a given function f into another function g as ˆ A f=g.We indicate an operator with a circumflex '' also called 'hat'.

ˆ A (ˆ B C )f(x)=(ˆ A B )ˆ C f(x): associative law of multiplication

( ˆ A + ˆ B )f(x ) = ˆ A f(x )+ ˆ B f(x ) : Sum of operators( ˆ A − ˆ B )f(x ) = ˆ A f(x )− ˆ B f(x ) : Dif. of operatorsˆ A ˆ B f (x ) = ˆ A [ ˆ B f(x )] : product of operators

3. Oprators obays:

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

What you should learn from this lecture

ˆ A {f (x ) +g(x )} = ˆ A f(x )+ ˆ A g (x )ˆ A {kf (x )} =k ˆ A f(x )

Some linear operators are : x;x 2; ddx

; d 2

dx 2

4. Linear operators obey:

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

The set of eigenfunction {fn(x),n=1..} is orthonormal :

fi(x)(all space

∫ fj(x))*dx=δij