Art PowerPoints

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Art PowerPoints

CHAPTER NUMBER 7: Quantum Theory: Introduction and Principles

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• Quantum phenomena

1000 m 1 mm

100 m 10 m 1000 nm

100 nm 10 nm 1 nm

1 Å 10 Å

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7.1 Energy quantization

-Feature of classical physics

•Precisely specified motion and momenta at an instant

•Continuous vibrational, rotational, and translation modes of motion and enenrgy

-Failure of classical physics

•black body radiation & the Planck distribution

•Heat capacities

•Atomic and molecular spectra

classical physics doesn’t hold for a small world

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Physical Chemistry Fundamentals: Figure 7.1

Feature of

electromagnetic wave

-speed: light -wavelength -frequency -wavenumber -energy

-momentum

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One of blackbody: Nuclear Reactor

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Black body radiation







 T T d

d ( , )  ( , )



^

 0 ( , ) )

(   

 T T d

Energy density

Density of state

) ( )

(T V T

E  

4

) 8 ,

( 

 

 T  kT

Classical model:

Rayleigh-Jeans law

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-Energy distribution of black body (experiment)

-Wien displacement law

-Stefan-Boltzmann law

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Figure 7.6

-Rayleigh-Jeans law:

As  → 0,  ^→ : everything in the dark will glow (UV catastrophe)

As  → 0,  → 0 (why?)

based on classical mechanics

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Planck distribution: Figure 7.7

Long wave length limit

~Rayleigh-Jeans law Short wavelength limit:

No UV catastrophe



 



  

 

kT e hc

kT e

hc

kt hc kt

hc

 











1 ,

large for

8 )

1 (

8

4 5

Planck distribution:

based on quantization of energy

1. As  → 0,  ^→ 0 2. For long ,

) 1 (

) 8 ,

( ₅

 _hc _kt  e

T hc_



 



3. Stefan-Boltzmann Law



 



 

 





0^ 5 ⁴ ⁵ ⁴³

) ( 15

8 )

1 (

) 8

( hc

a k aT

e d

T _hchc_kt

 



 

_

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-Rayleigh-Jeans law: UV catastrophe

-Planck distribution:

based on quantization of energy E=nhv

(oscillators are excited only if they can acquire an energy of at least hv.)

based on classical mechanics

(all oscillators share equally in the energy)

Stefan-Boltzmann and Wien law is successfully derived from Planck distribution!!!

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-heat capacities (close to 0K)

C

_v

=3R (from U

_m

=3RT)

However, this deviates as T → 0

Einstein assumes

each atom oscillates with a single freq.

and the deviation can be successfully explained

(E is confined to discrete values: E=n hν )

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At high T (T>>_E),

same as classical theory

At low T (T<<_E),

as T → 0, f_E → 0

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Figure 7.9

Debye formula: atoms oscillate over a range of frequencies

Figure 7.8

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-Atomic and molecular spectra

-Atomic spectra:

Radiation emitted by excited iron atom

-Molecular spectra:

Absorbing radiation at definite frequencies due to electronic, vibrational, and

rotational excitation of SO₂.

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Spectroscopic transition

hv E 



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11.2 Wave particle duality

-Photoelectric effect:

particle character of electromagnetic radiation wave-like particle (photon)

-Diffraction:

wave character of particles

(de Broglie wave)

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-particle character of electromagnetic radiation

Photoelectric effect

1. No e^- are ejected below a threshold freq.

-> phtoelectric effects occur only when hv>

2. E_k of ejected e^- increases linearly w/ freq of incident radiation

->

3. Even at low light intensities, e^- are ejected above a threshold freq.

->e- appears once the collision happens

hv E 



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Kinetic energy of ejected electron Energy needed to

remove electron from a metal

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Figure 7.15

-wave character of particles: diffraction

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Figure 7.16

How short?

for electron, 10^-12 m See Ex 7.2

-De Broglie relation

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Scanning Probe Microscopy

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7.3 Shrödinger equation

For 1-d systems:

   V x   E  dx

d

m ( )

2

²

2



2













 V E

m

2 2

2



For 3-d systems:

In general,

H   E  ( )

2

2 2

x m V

H     

H : Hamiltonian operator Time-dep Shrödinger eqn.

i t

H 



 

 

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Table 7.1

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Since E-V is equal to E_k

Using the Schrödinger eqn to develop the de Broglie relation,

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7.4 The Born interpretation of the wavefunction

-Normalization

-Quantization

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Probability of finding a particle between x and x+dx

Figure 7.18

* 2

   

dx

²

*

  



Born interpretation of the wave function

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Figure 7.19

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Figure 7.20

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2 *

1

N    dx



Normalization

* 1 2

N

1

  dx

  







For a normalized wavefunction,

*

dx

1

 



   

^*

dxdydz

^¹

(or

   

*

d

1)

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Figure 7.21 Figure 7.22

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Figure 7.23

Since it is 2^nd derivative.

But, delta ftn is valid!

Quantization [1]

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Quantization [2]

A particle may possess only certain energies,

for otherwise its wave function would be physically unacceptable

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7.5 The information in a wavefunction -the probability density distribution

-Eigenvalues and eigenfunctions -Construction of operators

-Hermittian operators

-Superpositions and expectation values

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The information in a wave function

Solution:

2 2

d E

m dx

 

 

When V=0,

2 2

, 2

ikx ikx

k

Ae Be E



  ^ 

m

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Then, where is the particle?

if B=0,

Ae

ikx





  

^*

   

2 ikx ikx * ikx ikx 2

Ae Ae A e Ae A



  ^ 

Equal probability of finding the particles

(a) The probability density

Figure 7.24 a

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Then, where is the particle?

if A=B,



^ikx ^ikx



^{2 cos}

A e e A kx



  ^ 

   

2 * 2 2

2 cos

A kx

2 cos

A kx

4

A

cos

kx



 

Node: particles will never be found

Figure 7.24 b

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

2



2 2

wavefunction waveamplitude

 

The probability density

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(b) Operators, Eigenvalues and eigenfunctions

•operator

2 2

2 ( )

2

H d V x

 

m dx



H 



E 

_H: hamiltonian (operator) -total E of a system

Eigenvalue

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•Eigenvalue equation

(Operator)(same function)=(constant factor)X(same function)

(Operator corresponding to an observable) ψ

=(constant factor) ψ Ex. (Energy Operator) Xψ=(energy)Xψ

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Illustration for orthogonality of eigen functions

) 2 )(sin (sin

)

( x x x

f 

Area = 0

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(c) Construction of Operators



 x x ˆ

Observables are represented by operators,

dx d p

_x

 i

 ˆ

Position operators Momentum operators

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ˆ 2

2 V  kx 

2

2 V  kx

2

x k

E p

 m 1 ² ²₂

2 2

k

d d d

E m idx idx m dx

  

     

2 2

ˆ ˆ ˆ( ) 2 ˆ( )

k 2

H E V x d V x

   

m dx



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KE of a particle is an average contribution from the entire space.

Figure 7.26

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Wave ftn of a particle in a potential decreasing towards right

Figure 7.27

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(d) Hermitian operators



^*



^*

*



 

ⁱ

 

^j

d   

^j

 

ⁱ

d 

Hermiticity

*

0

i j

d

   



Orthogonality…

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(e) Superpositions and expectation values

Ex. if ψ A cos kx

When the wavefunction of a particle is not an eigen function of an

operator, the property to which the operator corresponds does not have a definite value.

not an eigenfunction any more!!!

Linear combination!

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Weighted mean of a series of observations

: the expectation value is the sum of the two eigencalues weighted by the probabilities that

each one will be found in a series of measurements

Weighted mean of a series of observations

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Expectation value

2 2

* *

2 2

k k

E E d d d

m dx

     





 



Ex. Mean kinetic energy

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1. When the momentum is measured, one of the eigenvalue will be measured

2. Measuring an eigenvalue ~ Proportional to the square of the modulus in the linear combinations

3. Average value = Expectation value

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Summary of Operators

•position

•momentum

•Kinetic E

•Potential E

•Note) Expectation value

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7.6 The uncertainty principle

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In Quantum Mechanics, Position and momentum cannot be predicted simultaneously

•Heisenberg uncertainty principle

Ae

ikx

 

p

x

 k

Ex. Particle travelling to the right -> position is unpredictable But, the momentum is definite

But, in Classical Mechanics,

Position and momentum can be predicted simultaneously

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Wave function for a particle at a well- defined location

An infinite number of waves is needed to construct the wavefunction of perfectly localized partile.

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1 p q 2

  

Quantitative version of uncertainty principle



² ²



^{1 2}

, 

² ²



^{1 2}

p p p q q q

     

Note) p and q are the same direction

Standard deviation

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1 2 1 2 2 1

ˆ ˆ, ˆ ˆ ˆ ˆ

       

 

Complementary observables~ they do not commute

commutator

1 2 2 1

ˆ ˆ(  ) ˆ ( ˆ  )

    

Uncertainty principle

1 2 1 2

ˆ ˆ 1 ˆ , ˆ

2  

      

More general version of uncertainty principle



x p^{ˆ ˆ}^, x



 i

Ex.

1

p q 2

  

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The uncertainty principle should be applied!!!

•If there are a pair of complementary observables, (non-commuting)

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Physical Chemistry Fundamentals: Table 7.2

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7.7 Postulates of quantum mechanics -wave function

-Born interpretation

-Acceptable wave function -Observables

-Uncertainty relation

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Physical Chemistry Fundamentals: Mathematic Background 7.1

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