Lecture 6: Operators and QuantumMechanics 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 ,**)

(Ia) A Quantum mechanical system is specifiedby the statefunction Ψ(x)

(Ib) The state function Ψ(x) contains allinformation about the system we can know

We now have Operators and Quantum Mechanics

(Ic) A system described by the state function HΨ(x)=EΨ(x)has exactly the energy E

Review

O

X

We have seen that a ' free' particle movingin one dimension in a constant (zero) potentialhas the Hamiltonian

−

h2

2mδ2ψ(x)

δx 2 = Eψ(x)

ˆ H =−

h2

2mδ2

δx 2

The Schrodinger equation is

with the general solution :

ψ(x) = Aexpikx + Bexp−ikx

and energies E =

h2k 2

2m

Operators and Quantum MechanicsReview

How does the state function Ψ(x,t) give us information about an observable other than the energy such as the position or the momentum ?

Any observable 'Ω' can be expressed in classical physicsin terms of x,y,z and px,py,pz.

Examples:Ω=x, px, vx, px

2, T, V(x), E

Operators and Quantum MechanicsGood question

Classical Mechanics Quantum Mechanics

x px ˆ x −> x ; ̂ px−>hiδδx

y py ˆ y−> y ; ̂ py−>hiδδy

z pz ˆ z−> z ; ̂ pz−>hiδδz

We can construct the corresponding operatorfrom the substitution:

as ˆ Ω (x,y,z,

hi

ddx

,hi

ddy

,hi

ddz

)

Such as:ˆ Ω =ˆ x , ˆ p x, ˆ v x, ˆ p x

2, ˆ T , ˆ V (x), E

Operators and Quantum MechanicsReview

Ω ψn = ϖnψn

For an observable Ω with the correspondingoperator ̂ Ω we have the eigenvalue equation :

Operators and Quantum Mechanics

(IIIa). The meassurement of the quantity represented by Ωhas as the o n l y outcome one of the values

ϖn n = 1,2,3 ....

(IIIb). If the system is in a state described by ψn a meassurement ofΩ will result in the

valueϖn

Important news

Quantum mechanical principle.. Operators

we can solve the eigenvalue problem

ˆ Ω ψn =ϖnψn

For any such operator ˆ Ω

We obtain eigenfunctionsand eigenvalues

The only possible values that can arise from measurements of the physical observable Ω are the eigenvalues ϖn

Postulate 3

Important news

The x - component 'px ' of the linear momentum

r p =px

r ex + py r ey + pz

r ez

Is represented by the operator ˆ p x =

hiδδ x

With the eigenfunctions Exp[ikx] and eigenvalue hk

hiδ Exp[ikx]

δ x =hkExp[ikx]

We note that k can take any value −∞ > k > ∞

Operators and Quantum MechanicsImportant news

ψ(x) = Aexpikx + Bexp−ikx and energies E =

h2k 2

2m

For A = 0 ψ−(x ) = B exp−ikx

this wavefunction is also an eigenfunction to ˆ p x

With eigenvalue for ˆ p x of - hk

Thus ψ- (x ) describes a particle of energy E=h2k 2

2m

and momentum px =−hk ; note E= Px2

2m .as it must be

This system corresponds to a particle moving with constant velocity

vx = pxm

=-h /k m We know nothing about its positionsince |ψ( )x |2=B

Operators and Quantum MechanicsNew insight

ψ(x) = Aexpikx + Bexp−ikx and energies E =

h2k 2

2m

For B = 0 ψ+(x ) = A expikx

this wavefunction is also an eigenfunction to ˆ p x

With eigenvalue for ˆ p x of hk

Thus ψ (x ) describes a particle of energy E=h2k 2

2m

and momentum px =hk ; note E= Px2

2m .as it must be

This system corresponds to a particle moving with constant velocity

vx = pxm

=h /k m We know nothing about its positionsince |ψ( )x |2=B

Operators and Quantum MechanicsNew insight

What about : ψ(x ) =A expikx+B exp−ikx ?It is not an eigenfunction to ˆ p x since :

ˆ p xψ(x)=Ahi

ddx

expikx+Bhi

ddx

exp−ikx

=Ahkexpikx−Bhkexp−ikx

How can we find px in this case ?

Operators and Quantum MechanicsNew insight

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

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

fi(x)*

all space∫ fj(x)dx=δij

ˆ A fn(x ) =knfn(x )

=o if i ≠ j

= 1 if i = j

Quantum mechanical principles..Eigenfunctions

Quantum mechanical principles..Eigenfunctions

ei •ej =δij

ei

ei

ei

An example of an orthonormal set is the Cartesian unit vectors

An example of an orthonormal function set is

ψn(x)= 1L

sinnπxL

⎛ ⎝

⎞ ⎠

n=1,2,3,4,5....

ψn(x)*

o

L∫ ψm(x)=∂nm

That is, any function g(x) thatdepends on the same variables as the eigenfunctions can be written

g(x) = anfn (x )i=1

all∑

where

an = fn(x)*g(x)dxall space

∫

Quantum mechanical principles..Eigenfunctions

The set of eigenfunction {fn(x ),n =1..} .forms a complete set

ei

ei

ei

r e i ; i=1,2,3 form a complete set

For any vector r v

v=(r v •

r e 1)

r e 1+(

r v •

r e 2)

r e 2+(

r v •

r e 3)

r e 3

we can show that : an = fn(x)*g(x)dxV∫

In the expansion : g(x) = aifi (x )i=1

all∑ (1)

from the orthonormality: fi(x)*

V∫ fj(x)dx=δij

Quantum mechanical principles..Eigenfunctions

g(x)= aifi(x)i=1

all∑ ⇒

V∫ fn(x)*g(x)dx= ai

V∫ fn(x)*fi(x)

i=1

all∑ dx

A multiplication by fn (x) on both sides followed byintegration affords

or : an = ( )g x fn(x )dxall space

∫ δij

A multiplication by fn (x) on both sides followed byintegration affords

or : an = g(x)fn(x)*dxall space

∫

ψ(x)=Aexpikx+Bexp−ikx is a linear combination of two eigenfunctions to ˆ p x

How can we find px in this case ?

Operators and Quantum Mechanics

px =hk px =−hk

1. Postulate 3For an observable Ω with the corresponding operator ˆ Ω we have the eigenvalue equation : Ω ψn =ϖnψn(i) The meassurement of the quantity represented by Ωhas as the o n l y outcome one of the values ϖn n=1,2,3 ....(ii) If the system is in a state described by ψn a meassurement of Ω will result in the value ϖn

What you should learn from this lecture

Illustrations:

ψ+(x)=Aexpikx is an eigenfunction to ˆ p x with eigenvalue hk

ψ−(x)=Aexp−ikx is an eigenfunction to ˆ p x with eigenvalue -hkBoth are eigenfunctions to the Hamiltonian for a free particle

H=h2(ˆ p x)2

2m with eigenvalues E=

h2k2

2m ψ+(x) represents a free particle of momentum hk

ψ−(x) represents a free particle of momentum -hk

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 :

g(x)= anfn(x)i=1

all∑ where

an = g(x)fn(x)dxall space

∫