chem 373- lecture 1: classical mechanics and the schrödinger equation

8/3/2019 Chem 373- Lecture 1: Classical Mechanics and the Schrdinger Equation

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Welcome to the Chem 373

Sixth Edition

+ Lab Manual

http://www.cobalt.chem.ucalgary.ca/ziegler/Lec.chm373/index.html

It is all on the web !!


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Lecture 1: Classical Mechanics and the Schrdinger Equation

This lecture covers the following parts of Atkins

1. Further information 4. Classical mechanics (pp

911- 914 )

2. 11.3 The Schrdinger Equation (pp 294)

Lecture-on-line

Introduction to Classical mechanics and the Schrdinger

equation (PowerPoint)

Introduction to Classical mechanics and the Schrdinger

equation (PDF)

Handout.Lecture1 (PDF)Taylor Expansion (MS-WORD)


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Tutorials on-line

The postulates of quantum mechanics(This is the writeup for

Dry-lab-II)( This lecture has covered (briefly) postulates1-2)(You are not expected to understand

even postulates 1 and 2 fully after this lecture)

The Development of Classical Mechanics

Experimental Background for Quantum mecahnics

Early Development of Quantum mechanics

The Schrdinger Equation

The Time Independent Schrdinger Equation


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Audio-Visuals on-line

Quantum mechanics as the foundation of Chemistry (quick time

movie ****, 6 MB)Why Quantum Mechanics (quick time movie from the Wilson

page ****, 16 MB)

Why Quantum Mechanics (PowerPoint version without

animations)Slides from the text book (From the CD included in

Atkins ,**)


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Classical Mechanics

A particle in 3-D has the following attributes

X

Y

Z

1. Mass m

m

mass

r

Position

2. Positionrr

v = dr /dt

velocit y

3. Velocityrv

Rate of change of position with time


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Expression for total energy

ET Ekin Epot(rr )

The total energy of a particle with positionrr ,

mass m and velocity rv also has energy

Kinetic energy duto motion

Potential energydue to forces


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p

vv

small mass large velocity

v

v

large mass small velocity

or

Ek1

2mv

2

The kinetic energy can be written as :

r

p mvv

Or alternatively in terms of the

linear momentum:

as:

Ekp2

2m


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A particle moving in a potential energy field V is subject to a

force

V(x)

X

F=-dV/dx

Force in one dimension

Force in direction of

decreasing potential energy

The potential energy and force


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F dVdx

ex dVdy

eyPotential energy V


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vF (dV/dx)

rex (dV/dy)

vey (dV/dz)

vez

vF vV gradV


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The expression for the total energy in terms ofthe potential energy and the kinetic energy

given in terms of the linear momentum

The Hamiltonian will take on a special

importance in the transformation from

classical physics to quantum mechanics

E Ekin

Epot

p2

2mV(

rr )

is called the Hamiltonian

Hp

2

2m

V(r

r )

The Classical Hamiltonian


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Quantum Mechanics

The particle is moving in the potential V(x,y,z)

We consider a particle of mass m,

Linear momentumr

p mrv

and positionr

r

rr

rp =

X

Y

Z m

Positionmass

mvr

Linear Momentum


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rr

rp =

X

Y

Z m

Positionmass

mvr

Linear Momentum

The classical Hamiltonian is given by

H 12m

px2 py

2 pz2 V(x,y,z)

H1

2m

rp

rp V(

vr )

1

2m

p2

V(v

r )


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Quantum Mechanical Hamiltonian

The quantum mechanical Hamiltonian H is constructed by the

following transformations :

HClass H1

2mpx

2 py2 pz

2 V( x, y, z)

Classical Mechanics Quantum Mechanics

x px x x ; pxh

i x

y py y y ; pyh

i y

z pz z z ; pz hi z

Here h 'h-bar'=h

2is a modification of Plancks constanth

h 1.05457 1034 Js


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H

1

2m ( px2

py2

pz2

) V( x, y, z)

1

2m[(h

i x

h

i x) (

h

i y

h

i y) (

h

i z

h

i z)] V(x,y,z)

We have

h

i y

h

i y

h2

i2 y y

h2

2

y2

Thus

Hh

2

2m[

2

x2

2

y2

2

z2 ] V(x, y,z )

h2 2 2 2


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By introducing the Laplacian: 22

x2

2

y2

2

z2

we have

Hh

2m

2V(x,y,z)

It is now a postulate of quantum mechanics that :

the solutions (x,y,z) to the Schrdinger equation

H (x,y,z) E (x,y,z)

h2

2m

2(r

r ) V(r

r ) (r

r ) E (r

r )

h2

2m[

2

x2

2

y2

2

y2 ] V(x,y,z) E

Contains all kinetic information about a particle

moving in the Potential V(x,y,z)

Hh

2m[

x2

y2

z2 ] V(x, y,z )


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What you should learn from this lecture

Definition of :

Linear momentum (pm),

kinetic energy(p2

2m);

Potential Energy

Relation between force F

and potential energy V (rF = -

rV)

The definition of the Hamiltonian (H)

as the sum of kinetic and potential energ

with the potential energy written in terms

of the linear momentum

For single particle: Hp2

2m

V(rr )


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You must know that : The quantum mechanical Hamiltonian His constructed from the classical Hamiltonian H by

the transformation

HClassH

1

2mpx

2 py2 pz

2 V( x, y, z)

Classical Mechanics Quantum Mechanics

x px x x ; pxh

i x

y py y y ; pyh

i y

z pz z z ; pzh

i z

Appendix A


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The position of the particle is a function of time.

Let us assume that the particle at t to

has the position

rr (to )

and the velocity

r

v(to ) (d

r

r/dt )t to

What isv

r (to t) =v

r (t1) = ?

vr (to t) =

vr ( to)+(d

vr / dt)t to t +

1

2(d2

vr / dt2)t to t

2

vr (to t) =

vr (to) +

vv(to ) t +

1

2(d2

vr / dt2 )t t

o

t2

By Taylor expansion around

rr (to )

or

Newton's Equation and determination of position..cont

vr (t o )

v

r (to+

t)

(d2

vr/dt2 )t t

o

t2

(d

vr/dt )t t

ot

ppe d

Appendix A


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vr (t o )

vr (to + t)

vv (t

o) t

(d

2vr /dt

2)t t

o

t2

vr (to t) =

vr (to) +

vv(to ) t +

1

2(d2

vr / dt2 )t to t

2

vF(to )

vV gradV m(d2 vr /dt2 )t to

However from Newtons law:

vr (t

o

t) =vr (t

o

) +vv(t

o

) t -1

2m(gradV)

t=t0t2

Thus :

Newton's Equation and determination of position..contpp

Newton's Equation and determination of position cont


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vr (t

o)

vr (to

+ t)

v

v(to )

t

-1

m(gradV)

t = t ot

vr (to t) =

vr (to) +

vv(to ) t -

1

2m(gradV)t=t0 t

2

Newton s Equation and determination of position..cont

Appendix A


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At the later time : t1

to

t we have

vr (t1 t) =vr (t1 )+(d

vr / dt)t t1 t +1

2(d

2vr /dt

2)t t1 t

2(1)

The last term on the right hand side of eq(1)

can again be determined from Newtons equation

vF(t1 )

vV gradV m(d

2vr /dt

2)t t1

as

(d

2vr /dt2

)t t11

m(gradV)t t1


pp

Appendix A


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We can determine the first term on the right side of

eq(1) By a Taylor expansion of the velocity

vr (t1 t) =

vr (t1 )+(d

vr / dt)t t1 t +

1

2m

(gradV)t t1 t2(1)

(dvr / dt)t t1 (d

vr / dt)t t0

1

2

(d2vr / dt2 )t t0 t

or

(d

vr / dt)t t1

vv(to )

1

2m(gradV)t to t

Where both: vv(to ) and

1

m(gradV)t to are known


Newton's Equation and determination of position cont

Appendix A


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The position of a particle is determined at all

times from the position and velocity at to

vv(t2 ) (d

vr / dt)t t2

vv(t1)

1

m(gradV) t t1 t


v

r (t2 t) =v

r (t2 ) +v

v(t2 ) t +1

2 (d2v

r /dt2

)t t2 t2

(d2

vr /dt2)t t2

1

m(gradV)t t2

At t2 t0 2 t what aboutvr (t2 t) ?

r (t2 )

r(t2 t)

v(t2 ) t-1

m(gradV)

t = t 2

t

chem 373- lecture 1: classical mechanics and the schrödinger equation

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