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7/28/2019 PChem Help

1/27

Chemical Bonding

Lecture 2

CHEM 6277-10

Lecturer: Hanning Chen, Ph.D.

08/29/2013

Quantum Mechanical Principles


2/27

Quiz 1

10 minutes

Please stop writing when the timer stops !


3/27

Quantum Mechanics Wavefunction

Wavefunction is the mathematical representation of a physical object.

r,t

( )r : position t : time

r,t( ) r,t( ) = r,t( ) 2 =* r,t( )can be a complexnumber

r,t( ) = a + bi

i : imaginary unit

: probability density(r,t)

must be a non-negative realnumb

spatialand temporal

distribution of a quantum state

*

r,t( )

* : comple

*

r,t( ) r,t( ) = a bi( ) a + bi( ) = a


4/27

Information in a Wavefunction

r,t( ) r : position t : time

Wavefunction is the only quantity needed to represent a quantum state

r,t( ) = r,t( )2

=*

r,t( ) r,t( )Probability Density:

Spatial distribution

x

x( )higher probability

lower probability

What else information? How about momentum ?


5/27

Master Equation of Quantum Mechanics

2

2m

d2

x

( )dx

2+V(x) x( ) = E x( )

Time-independent Schrdinger Equation:

Time-dependent Schrdinger Equation:

i x,t( )t = 2

2m

2

x,t( )x

2+V x,t( ) x,t( )

ordinary differential

(x) one independent

partial differential eq(x,t) two independent v

n-th orderlinear ordinary differential equation

An (x)y(n)+ An1(x)y

(n1)+ ...+ A

1(x)y

'+ A

0(x)y = g(x)

y(n)=

dny

dxn, y

(n) y

n for example y=y(0


6/27

Acceptable Wavefunctions

1. continuous

x(

)

x

NO!

2. smooth

x(

)

x

NO!

3. single-valued

x(

)

x

N

4. square-integrable

x(

)

x

2

2m

2 x( )x

2+V(x) x( ) = E x(

The Schrdinger equation is a second-orderpartial derivativ

*

r,t( ) r,t( )dr =1 normalization rule

NO!


7/27

Solution of the Homogeneous Second-order ODE

generic form:

c1y1''+ c2y2

''+ P x( )c1y1

'+ P x( )c2y2

'+Q x( )c1y1 +Q(x)c2y2 = 0

Let us start with and , both satisfying the generic formy1(x) y2 (x)

linear combination:

y = c1y1

x( ) + c2y2

x( )

y''+ P x( )y' + Q x( )y = 0

c1

y1

''+ P x( )y1

'+Q x( )y1 + c2 y2

''+ P x( )y2

'+Q(x)y

2 = 0

0 0

An arbitrary linear combination of the solutions is also a solution


8/27

What is the Solution Form?

tentative solution:

For

y1= e

k1x ,y2= e

k2x

y1

k1

2ek1x+ Pk

1ek1x+Qe

k1x= 0

k1

2+ Pk

1+Q = 0 (auxiliary quadratic equa

k1,2

x( ) =P x( ) P x( )

2 4Q x( )

2

Well-known solutions:

General solution of homogeneous second-order ODE:

y = c1e

k1x+ c

2e

k2x


9/27

Wavefunction of a Free Particle

Definition of afree particle:

V = 0 V : potential

2

2m

2 x( )

x2

+V(x) x( ) = E x( )

Time-independent Schrdinger equation:

2

2m

2

x( )x

2= E x( )

Standard solution:

x( ) = c1e

ik1x+ c

2e

ik2x

E=k2

2m

total energ

but also the kinet

c1,c2 : "arbitrary" constants *

r,t( ) r,t( )dr =1 what is k?


10/27

Momentum of a Free Particle

If thefree particle is treated by classicalmechanics:

Ek=

E=

p2

2m

p= 2

mEkp : momentum

kineticenergy given by Schrdinger equation:

Ek =k22

2m p = k

x( ) = Aeikx + Be

the wavefunction does ha

the momentum informat

Unfortunately, not every wavefunction can be easily expressed as a combination of pla

Formal quantum mechanics definition of momentum: x( ) : arbitrary wavefun

p = i

xmomentumoperator p =

*

x( ) p x( )dx

expectation valueinformation extractor


11/27

Operator

function: f(x) = x2+1 x : indepedent variab

A function has the magic to change the value of an independent variab

We define a rule, , so thatD

Df(x) =d

dx

f(x)

f(x) = x2+1

whenDf(x) =

d

dx(x

2+1) =

if we define g(x) = 2x Df(x) = 2x = g(x)

An operator is a function offunctions.

x = 1 f(x) = 2


12/27

Important Properties of Operators

An operator acting on more than one function:

D f(x,y),g(x,y)( )=

x f(x,y)+

g(x,y)( )+

y f(x,y)+

g(x,y( f(x,y) = x

2+ y

2,g(x,y) = x + y

D( f(x,y),g(x,y)) = 2x + 2y + 2 = h(x,y

predefined mixture of functions

Does the order of mixing matter ?

A B f(x)( ) = B A f(x)( )

?

??


13/27

Commutator of Two Operators

Definition of Commutator:

[ A, B] = AB BAdifference between two mixing orders

If forany function f x( )

[ A, B]f(x) = A Bf(x) B Af(x) = 0

The operators ofA and B are commute.

A simple example:

Af(x) =d

dxf(x), Bf(x) =

d2

dx2f(x)

[ A, B]f(x) = ddx

d2

dx2f(x) d

2

dx2ddx

f(x) = d3

dx3f(x) d

3

dx3f(x


14/27

One More Example

Af(x) =d

dxf(x), Bf(x) = xf(x)

ABf(x) =d

dxBf(x)( ) =

d

dxxf(x)( ) = xf '(x) + f

do they commute?

BAf(x) = x Af(x)( ) = x ddx

f(x)

= xf '(x)

A is afterB:

B is afterA:

[ A, B]f(x) = A Bf(x) B Af(x) = xf '(x) + f(x) xf '(x) = f

[ A, B] = 1 0 They DO NOT commute !

Wh D C M ?


15/27

Why Does Commutator Matter?

momentum operator:

p = i

x

position operator:

x = x

commutator ofmomentum and position operators:

[ p, x] x( ) = px x( ) xp x( ) = ix

x x( )( ) xix

(

[ p, x] x( ) = ix

x x( )( ) xix

x( )( ) = i

momentum and position operators do NOT commute !

There is no way to simultaneously determine the momentum and position of a p

residue

H i b U t i t P i i l


16/27

Heinsberg Uncertainty Principle

Cauchy-Schwartz Inequality:

A

2

B

2 AB BA

2

2

A

2

B

2 A, B 2

2

[ p, x] = iForposition and momentum,

p

2

x

2

p, x[ ]2

2

p

2

x

2

2

4

p

x

2

A= A

2 A

2Standard deviation: : average value

Forenergy and time,

Et

2It is impossible to exactly measure

a particles energy at any given time.

H ilt i O t


17/27

Hamiltonian Operator

2

2m

2 x( )

x2

+V(x) x( ) = E x( )


2

2m

2

x2+V(x)

x( ) = E x( )

H : Hamiltonian

operator

a systemstotal energy

Using the definition of momentum: p = i x Ek =p

2

2m=

2

2m

H = Ek+ V x( )

kinetic

energy

potentialenergy

H x( ) = E x( )eigenfunction-eigenvalue

problem

Ei f ti d Ei l


18/27

Eigenfunction and Eigenvalues

Definitions:

D f x( )( )=

cf x( )

For a given operator, only some particular functions satisfy the conditio

f x( ) is the eigenfunction of the operator, D, and c is the eigenvalue fo

For a given eigenfunction, we only have oneeigenvalue.

A simple example:D =

d

dxf(x) = e

3x d

dxf(x) = 3e

3x=

f(x)=

e

2x d

dxf(x) = 2e

2x=

Ph i l M i f Ei f ti d Ei l


19/27

Physical Meaning of Eigenfunction and Eigenvalue

H x( ) = E x( )


The eigenfunction of an operator has a constant observable value, which is the eige

The eigenfunction has a constant total energy E. x( )

For a free particle: x( ) = Aeikx + Beikx

p = i

xIs it an eigenfunction ofmomentum ?

p x( ) = i

xAe

ikx+ Be

ikx( ) = (i)(ik) Aeikx + Be ikx( ) = k x( A free particle has a well-defined momentum, k

Is it an eigenfunction ofposition ? x= x

x x( ) = x Aeikx + Beikx( ) = x x( )

YES

NO

, but not a well-definedposi

x ,+

(

Expectation Values


20/27

Expectation Values

A = *x( ) A x( )dxexpectation value

average value of a physical observable, which is represented by an operator

E = *x( ) H x( )dxaverage total energy H : Hamiltonian o

p = * x( )

p x( )dxaverage momentum p : momentum o

average position x = *x( ) x x( )dx x : position oper

If A x( ) = a x( ) eigenfunction-eigenvalue

A = * x( ) A x( )dx = * x( ) a x( )dx = a

* x( ) x( )dx=1

Special case:

Hermitian Operators


21/27

Hermitian Operators

Hermiticity:

i*(x)xj x( )

dx = i

*(x)xj x( )

dx( )

*

*

= i (x)x*j

*x( )

dx( )

= i (x)xj*x( ) dx( )

*

= j*x( )xi (x) dx( )

*

A is a Hermitian operator

If a physical property is observable, its corresponding operator must beHerm

For example, x = x

i*(x) Aj x( ) dx = j

*(x) Ai x( ) dx( )

*

Q.E.D.

H, p, x are all Hermitian operators because they are measurable.

a + bi( )*

= a

Hermicity of Quantum Operators


22/27

Hermicity of Quantum Operators

Let us say A x( ) = a x( )a : complex numb

a = f+ giIfA is a Hermitian operator

A = *x( ) A x( )dx =

*x( ) A x( )dx( )

*

*x( ) a x( )dx =

*x( ) a x( )dx( )

*

a *x( ) x( )dx = a

*

*x( ) x( )dx( )

*

a 1= a* 1( )

*

a = a*

a : real nu

The Hermicity of quantum operators is used to ensure realobservable val

Superposition of Hermitian Eigenfunctions


23/27

Superposition of Hermitian Eigenfunctions

eigenfunction set:

A x( ) = a x( )

1(x),2(

x),...,N(

x){ }eigenvalue set: a

1,a

2,...,a

N{ }

IfA is a Hermitian operator

i* x( ) j x( )dx = ij =

1 if i = j

0 if i j

spatial overlap between wavefunctions

normalizat

orthogonal

x( ) = ci

i

i=1

N

superimposed

wavefunctionlinear combina

A = * x( ) A x( )dx

Expectation Value for a Superimposed Wavefuncti


24/27

Expectation Value for a Superimposed Wavefuncti

A = * x( ) A x( )dx x( )

=

ciii=1

N

*

x( )=

cj*

j*

j=1

N

A = cj*j

*

j=1

N

A ciii=1

N

dx

A = cj*ci

i=1

N

j=1

N

j* Ai dx j

*aii d

ai j*i dx

A=

cj*

cii=1

N

j=1

N

ai

ij

j= i

A=

cici*

aii=1

N

A=

i=

N

An Example


25/27

An Example

x( ) = c1

1+ c

2

2

two-component decomposition:

1 and 2 are the eigenfunctions of Hamiltonian operator, H

E1 and E2 are the corresponding eigenvalues

what is the expectation value of energy for x( )?

E = c1

2

E1+ c

2

2

E2

c1

2and c2

2: Projection of x( ) onto 1 x( ) and 2 x( )

If x

( )=

1x

( )c

1

2

= 1 and c2

2

= 0 If x

( )=

2x

( )c

2

2

= 1 and c1

In general, 0< c12


26/27

Review of Homework 1

1.11 A one-particle, one-dimensional system has the state function

= sinat( ) 2 /c2( )1/4

ex

2 /c2+ cosat( ) 32 /c6( )

1/4xe

x2 /c2

where a is a constant and c =2.0. If the particle's position is measured at t=0,estimate the probability that the result will lie between 2.000 and 2.001

t= 0 = 32 /c6( )1/4

xex2/c2

x( ) = 2= 32 /c6

( )

1/2

x2e2x2 /c2

P x( ) x=2.000x=2.001

= 32 /c6( )1/2

x2e2x2/c2 dx

x=2.000

x=2.001

= 32 /c6( )1/2 c2

4erf(

x

c) c 2xex

2/c2

c = 2.0

P x( ) x=2.000x=2.001 0.000216

Homework 2


27/27

Homework 2

Reading assignment: Chapter 3

Homework assignment: Problems 3.8 and 3.25

Homework assignments must be turned in by 5:00 PM, August 30th, Frida

to my mailbox in the Department Main Office

located at Room 107, Corcoran Hall

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