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Chapter 8 Quantum Theory

- Failures of classical physics
- Black-body radiation
- Heat capacities
- Photoelectric effects
- Diffraction of electrons
- Atomic spectra
- Quantum mechanics
- Schroedinger equation
- Born interpretation
- Uncertainty principle

- Applications of quantum mechanics
- Translation motion a particle in a box
- Rotation motion a particle on a ring
- Vibration motion the harmonic oscillator

- Failures of classical physics

- Around the turn of 19th century, 3 principles of

classical physics were challenged by new

findings. - A particle travels in a trajectory
- -location and velocity (momentum) of a particle

are known at all time (Newtons law) - A particle can possess any arbitary energy
- -There exists continuous excited states
- Waves and particles are distinct concepts

- Black-body radiation

Black-body is a body that can emit and absorb

all frequencies of electromagnetic

radiations. Fig 12.3 p 287, Atkins

All substances contain black-bodies. It was

found that radiation emitted or absorbed by

black-body changes with temperature Iron turns

from red to white (redblue) when heated. Fig

12.3 p 287, Atkins

Not all frequencies are emitted or absorbed with

the same energy density. There exists a maximum

corresponding to a particular wavelenght, lmax,

in the curve. This lmax changes with

temperature As temperature is raised, lmax moves

toward smaller wavelenght (from red to blue)

called blue shift Wiens displacement

law c2, second radiation constant, 1.44 cm K

Another features of black-body radiation given by

Josef Stefan called Stephan-Boltzmann law is

E aT4

E ? total energy density or M ?T4

M emittance i.e. brightness of emission ?

5.67 x 10-8 wm-2k-4

Rayleigh-Jeans proposed that black-body consists

of oscillators which can emit or absorb light at

any frequency Using classical physics and

equipartition principle, they arrived at

Rayleigh-Jeans Law

dE ? d? where ? 8pkT/ ?4

Rayleigh-Jeans law suggested there can be

emission of very short wavelength even at room

temperature. In other word, there should in fact

be no darkness. (objects should glow in the

dark.) Failure of Reyleigh-Jeans law are called

UV-catastrophe Planck took Rayleigh-Jeanss

idea but instead of allowing oscillators to be

able to emit or absorb light at any frequency

each oscillator possesses only discrete values of

energy

E nhn

h Plancks constant 6.626 x 10-34

Js Plancks hypothesis also implies that energy

is quantized and is regarded as Quantum theory

By using statistical interpretation Planck

proposed that

dE ? d?

where

For very large ?

Rayleigh-Jeans Law

By integrating ? from 0 to ?, one obtains

2p5k4

-Stefan-Boltzman law

By differentiating

-Wiens displacement law

5k

c2 1.439 cm K

Example 8.1 Find the surface temperature of sun

where the maximum emission occurs at 490 nm.

Wiens displacement law

Example 8.2 Calculate the number of photons

emitted by a 100 W yellow lamp in 10.0 s. Take

the wavelength of yellow light as 560 nm and

assume 100 percent efficiency.

Energy emitted by lamp Eemitted

(Power)(time) (100 W) (10.0 s) 1,000 J

Photon energy Ephoton hn h

c/l (6.626x10-34 J s)(2.998x108

ms-1) (560x10-9 m) 3.547x10-9 J

Nphoton Eemitted/Ephoton 1,000/3.547x10-9

J 2.82x1021

- Heat capacity

- In 1819, Dulong Petit proposed that for all

monoatomic solids, heat capacities are about 25

J/Kmol (3R) at all range of temperature - Dulong Petit arrived at this conclusion by

assuming the concept of classical physics - heat capacities of solid is the energy used to

oscillate or vibrate atom at the lattice position - Each atom can vibrate in 3 orthogonal directions
- For N atoms solid, their will be 3N motions
- From equipartition theory, each motions uses

energy of kT

The total vibrational energy U 3NkT For 1

mole of atoms

3NA kT

25 J/Kmol

Later on with the advance in measurement, it was

found that at low temperature all substances have

heat capacities lower than 25 J/Kmol. Diamond

has Cm 6.1 J/K mol at 25C

Dulong-Petit

Albert Einstein borrowed Plancks idea. He

proposed that each atom can possess only a

discrete amount of energy which is an integral

multiple of hn

Debye corrected Einsteins formula by allowing

atom to oscillate at more than one frequency value

- Photoelectric effects

In late 1800s, it was demonstrated that electrons

can be ejected from surface of certain metals

when exposed to light of at least a certain

minimum frequency.

This phenomena is called photoelectric effect

In 1905, Albert Einstein using Plancks quantum

concept was able to explain the photoelectric

effect

Einsteins proposal 1) Light is composed of

light particle called photon. 2)

Each photon has discrete value of energy E

hn h c/l 3) Let F, work function of

metal, be the minimum energy that required to

knock electrons out of metal (binding energy,

each metal has different F).

If photon energy is equal to F, electrons will be

ejected from metal when exposed to light.

If photon energy is greater than F, electrons

will be ejected from metal and carry kinetic

energy EK when exposed to light.

EK hn - F

EK could be measured, F could then be determined

Example 8.3 The work function for metallic

caesium is 2.14 eV. Calculate the kinetic energy

and the speed of the electron ejected by light of

wavelength 250 nm.

- The diffraction of electrons

- In 1925, Davisson and Germer observed the

diffraction of electrons by a crystal - Davisson-Germers experiment has been repeated

using other particles such as H2 - particles have wave-like properties
- photoelectric effect suggests that
- waves have particle-like properties
- This phenomena is called
- wave-particle duality

In 1924, de Broglie suggested that any particle

traveling with a linear momentum, p , should have

a wavelength ? according to ? h/p faster

object will have smaller wavelength heavier

object will have smaller wavelength

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Example 8.4 Estimate the wavelength of electrons

that have been accelerated from rest through a

potential difference of 1.00 kV.

Example 8.4 Estimate the wavelength of electrons

that have been accelerated from rest through a

potential difference of 1.00 kV

- Atomic spectra

Radiation is emitted (or absorbed) at a series of

discrete frequencies

This phenomena can be explained if we assume

energy of atomic state is quantized and light

omits or absorbs as electron transfer from lower

to higher state or vice versa and releases photon

Quantum Mechanics

Failures of classical physics made scientists

realized that new kind Of mechanics was needed.

This new mechanics should not give accurate

trajectories of particles. This new mechanics was

later called Quantum mechanics

Uncertainty principle

In 1927, Heisenberg pointed out that for very

small particles (quantum particles) like

electrons it is not possible to measure

accurately their positions (x) and linear

momentum (px) at the same time. This statement is

now known as the uncertainty principle.

For us to find the position of electron, light

(h?) must hit electron and reflect back to the

lens of the microscope.

The accuracy of measurement can be determined

from

where ? is the wavelength of light and 2sin? is

the dimension of the lens

From de Broglies relation,

p

p

As photon hits electron, momentum is

transferred. The leaving electron carries

momentum of mv while leaving photon has momentum

of h/? (or energy h?)

From law of conservation

incoming momentum outgoing momentum

Along x-axis

(1)

Along y-axis

(2)

From (1)

Generally ? gt ? but for simplicity assuming ? ?

? then

px

90 - ? lt ? lt- 90 ?

Thus

lt px lt

lt px lt

The accuracy of measurement of momentum is then

Dpx

The accuracy of measurement of position is

DxDp

DxDp

uncertainty principle

-Schrödinger equation

Wave concept was introduced to explain quantum

particle. Schrödinger adapted Maxwells equation

for electromagnetic radiation for describing

behavior of quantum particle.

Maxwells equation

Wamplitude of wave

where ? is de Broglie wavelength of the particle

velocity of the particle can be given by

Replacing waves amplitude by wave function ?

(psi) which gives the information about

positions of electron

?(x,y,z,t) is not trajectory, it gives

information about position but not exactly like

the amplitude of the wave

For localized or standing wave

Thus

Kinetic energy Tp2/2m and total kinetic

potential (ETV)

p2

-2m4p2

-8p2m

8p2m

? Laplacian operator

operator

Schrodinger equation or SE

SE is differential equation SE is an eigen-value

equation E is an eigen-value which represents

energy of particle Solution of SE is wave

function ?

SE can be solved using methods of differential

equation

Born Interpretation

In wave theory, square amplitude of wave is

interpreted as intensity. Using the same

analogy, Born gave interpretation of wave

function as

The probability of finding a particle in a small

region of space dV is proportional to l?l2dV,

where ? is the value of wave function in the

region. In other word, l?l2 is the probability

density

Applications of quantum mechanics

- Solution to SE

2

For 1-dimensional system

2

gt Free electron V 0

2

y

2

-k2

i2-1

-k2

-k2

gt 1D Square potential well

Area 1 prob.

Probability

III

II

I

We can substitute

At x 0

yII yIII

If D 0, trivial solution is obtained. Then D

must not be equal to 0 and sinkL must be equal to

0. sinkL 0 only when kL np

k np/L

npx

Using Borns interpretation

From cos (AB) cosAcosB - sinAsinB cos2x

cosxcosx - sinxsinx cos2x

- sin2x (1-sin2x) - sin2x

cos2x 1- 2sin2x sin2x

(1 cos2x)/2

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The lowest energy is achieved when n 1, and

equal to h2/8mL2. The lowest energy is non-zero

called zero-point energy. Excitation or

transition to adjacent wall could be calculated as

gtExpanding to 3D particle in a

box

c

b

a

Particle in a box is a model representing

translation motion

Particle on a ring

Particle is held on the ring, hence travels in 2

dimension with V0 Thus

To solve this equation, one must transform from

cartesian (x,y) to polar coordinate (r,?)

r is fixed

I ?moment of inertia mr2

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From boundary condition

Thus, 2m must be positive or negative even

integer

From classical interpretation

Angular momentum j pr

Linear momentum

p2

from Schrödinger equation

From deBroglies relation ph/?

Particle on a ring represents rotation motion

Harmonic oscillator

V 1/2kx2

(1)

(2)

(a - b2x2)

(3)

(4)

which

The first 5 polynomial are

H0(x) 1

H1(x) 2x

H2(x) 4x2-2

H3(x) 8x3-12x

H4(x) 16x4-48x212

If we set 2n, then (4) is the

Hermites equation.

Equation (4) has Hermite polynomials for

solution. The wave function for the harmonic

oscillator is then given by,

v

N0,N1 are normalizing factor

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Take definition of a and ß

( )

From classical physics (vibration frequency)

Thus

v 0,1,2,

Harmonic oscillator represents mode of

vibration

Example 8.5 Show whether (a) or (b)

is eigenfunction of the operator d/dx and

find the corresponding eigenvalue

f(x) would be an eigenfunction of an operator

only when

where is an eigenvalue

(a)

( )

(b)

( )

is an eigenfunction of d/dx with

eigenvalue a is not an eigenfunction of

d/dx

Example 8.6 Proof that sinx and sin2x

which are eigenfunction of d2/dx2 are mutually

orthogonal.

Orthogonal means

(2x x)-

eigenfunctions of the same operator must be

orthogonal

Example 8.7 Find the normalizing constant for

ground state and first excited state of harmonic

oscillator

even n0

0

y1(x) y1(x) dx

even n1

0