chapter 40: intro to quantum problems with...

Chapter 40: Intro to Quantum

Problems with Classical Physics

•Nature of Light- particle or wave?

•Nature of Spectra? Emission Lines?

•Blackbody Radiation? UV Catastrophe?

•Atoms exist? Structure of Atoms?

•Photoelectric Effect?

•Compton Effect?

Black Body Radiation and the

Dawn of Quantum Mechanics

https://www.youtube.com/watch?v=B7pACq_xWyw



Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.

Double Slit is VERY IMPORTANT because it is evidence

of waves. Only waves interfere like this.

Thomas Young 1804

sind m

REVIEW! Derive Fringe Equations

• For bright fringes

• For dark fringes

bright ( 0 1 2 ), ,λL

y m md

dark

1( 0 1 2 )

2, ,

λLy m m

d

Double Slit for Electrons

shows Wave Interference!

Key to Quantum Theory!


James Clerk Maxwell 1860s

Light is an electromagnetic wave.

The medium is the Ether.

8

0

13.0 10 /

o

c x m s

Heinrich Rudolf Hertz

•1857 – 1894

•German physicist

•First to generate and detect electromagnetic waves in a laboratory setting in 1887.

Section 34.2


The Electromagnetic Spectrum

Maxwell-Boltzmann Distribution: 1877

• The observed speed distribution of gas molecules in thermal equilibrium is shown at right

• NV is called the Maxwell-Boltzmann speed distribution function

• The distribution of speeds in N gas molecules is

• The probability of finding the molecule in a particular energy state varies exponentially as the negative of the energy divided by kBT

2

3 / 2

/ 22

B

42

Bmv k ToV

mN N v e

k T

nV (E ) = noe –E /kBT

Ludwig Boltzmann

1844 – 1906

• Temperature ~ Ave KE of each particle

• Particles have different speeds

• Gas Particles are in constant RANDOM motion

• Equipartition of Energy: Average KE of

each particle is: 3/2 kT

• Pressure is due to momentum transfer

Speed ‘Distribution’ at

CONSTANT Temperature

is given by the

Maxwell Speed Distribution

23/ 2 1/ 2 rmskT KE mv

k =1.38 x 10-23 J/K Boltzmann’s Constant

Equipartition of Energy

• Each translational degree of freedom contributes an equal amount to the energy of the gas

• Each degree of freedom contributes

½kBT to the energy of a system, where

possible degrees of freedom are those

associated with translation, rotation

and vibration of molecules

• Theory breaks down for black body

radiation – degrees of freedom get

frozen out.


Kirkoff’s Rules for Spectra: 1859

Bunsen

German physicist who developed the spectroscope and the science of

emission spectroscopy with Bunsen.

Kirkoff

* Rule 1 : A hot and opaque solid, liquid or highly compressed gas emits a continuous spectrum.

* Rule 2 : A hot, transparent gas produces an emission spectrum with bright lines.

* Rule 3 : If a continuous spectrum passes through a gas at a lower temperature, the transparent

cooler gas generates dark absorption lines.


Continuous vs Discrete

This is a continuous spectrum of colors: all colors are present.

This is a discrete spectrum of colors: only a few are present.


Compare absorption lines in a source with emission lines found in the laboratory!

Kirchhoff deduced that elements were present in the atmosphere of the Sun

and were absorbing their characteristic wavelengths, producing the absorption

lines in the solar spectrum. He published in 1861 the first atlas of the solar

spectrum, obtained with a prism ; however, these wavelengths were not very

precise : the dispersion of the prism was not linear at all.


Kirkoff’s Rules


Anders Jonas Ångström 1869

Ångström measured the wavelengths on the

four visible lines of the hydrogen spectrum,

obtained with a diffraction grating, whose

dispersion is linear, and replaced

Kirchhoff's arbitrary scale by the

wavelengths, expressed in the metric

system, using a small unit (10-10 m) with

which his name was to be associated.

Line color Wavelength

red 6562.852 Å

blue-green 4861.33 Å

violet 4340.47 Å

violet 4101.74 Å


Balmer Series: 1885Johann Balmer found an empirical equation that

correctly predicted the four visible emission

lines of hydrogen

H 2 2

1 1 1

2R

λ n

RH is the Rydberg constant

RH = 1.097 373 2 x 107 m-1

n is an integer, n = 3, 4, 5,…

The spectral lines correspond to different

values of n

Johannes Robert Rydberg generalized

it in 1888 for all transitions:

Hα is red, λ = 656.3 nm

Hβ is green, λ = 486.1 nm

Hγ is blue, λ = 434.1 nm

Hδ is violet, λ = 410.2 nm


When an object it heated it will

glow first in the infrared, then the

visible. Most solid materials break

down before they emit UV and

higher frequency EM waves.

Frequency ~ Temperature

Long

Short

All objects radiate energy continuously

in the form of electromagnetic waves

due to thermal vibrations of their

molecules.


Blackbody Radiation

• A black body is an ideal system that

absorbs all radiation incident on it

• The electromagnetic radiation emitted by a

black body is called blackbody radiation


Why this shape? Why the drop?

Stefan’s Law: 1879

Rate of radiation of a Black Bodyis the rate of energy transfer, in Watts

– σ = 5.6696 x 10-8 W/m2 . K4

– A is the surface area of the object

– e is a constant called the emissivity• e varies from 0 to 1

• The emissivity is also equal to the absorptivity

– T is the temperature in Kelvins

– With his law Stefan determined the temperature of the Sun’s surface and he calculated a value of 5430C. This was the first sensible value for the temperature of the Sun.

– Boltzmann was his student and derived Stefan’s Law from Thermodynamics in 1884 and extended it to grey bodies.

Jožef Stefan

(1835–1893)

4P e T A


Blackbody Experiment Results

The total power of the radiation emitted from the

surface increases with temperature

– Stefan’s law: P = AeT4

– P is the power and is the Stefan-Boltzmann constant: = 5.670 x

10-8 W / m2 . K4 (0<e < 1, for a blackbody, e = 1)

Intensity is Power/Area so: I = eT4

The peak of the wavelength distribution shifts to

shorter wavelengths as the temperature increases

– Wien’s displacement law

(T must be in kelvin):


Sun Power

The radius of our Sun is 6.96 × 108 m, and its

total power output is 3.77 × 1026 W. (a)

Assuming that the Sun’s surface emits as a black

body, calculate its surface temperature. (b) Using

the result of part (a), find λmax for the Sun.


Sun vs Earth Black Body Emission Curves

Sun emits most strongly in the visible, the Earth in the Infrared which is trapped

by the Atmosphere and warms the planet– green house effect.

The heating effect of a medium such as glass or the Earth’s

atmosphere that is transparent to short wavelengths but opaque

to longer wavelengths: Short get in, longer are trapped!


Sun vs Earth Peak Wavelength


A brass plate at room temperature (300 K) radiates 10 W of

energy. If its temperature is raised to 600 K, the

wavelength of maximum radiated intensity

A. Increases.

B. Decreases.

C. Remains the same.

D. Not enough information to tell.

QuickCheck 37.2

Slide 37-31


A brass plate at room temperature (300 K) radiates 10 W of

energy. If its temperature is raised to 600 K, the

wavelength of maximum radiated intensity

A. Increases.

B. Decreases.

C. Remains the same.

D. Not enough information to tell.

QuickCheck 37.2

Slide 37-32


Intensity of Blackbody Radiation

• The intensity increases with increasing temperature

• The amount of radiation emitted increases with increasing temperature

– The area under the curve

• The peak wavelength decreases with increasing temperature

• Combining gives the Rayleigh-Jeans law:

I = P/A = T4

I , ~4

1λ T

λ

Stefan’s law: P = AeT4 Wein’s Law:


Problems with the Wein’s World

• At short wavelengths, there

was a major disagreement

between the Rayleigh-Jeans

law and experiment

• This mismatch became

known as the ultraviolet

catastrophe

– You would have infinite

energy as the wavelength

approaches zero

I , ~4

1λ T

λ

https://www.youtube.com/watch?v=FXfrncRey-4



https://www.youtube.com/watch?v=7BXvc9W97iU



Solves Black Body Mystery. Atomic Energy is quantized.

It comes in chunks of Planck’s constant, h.

34

, n= 0,1,2,3,...

6.626 10

E nhf

h x Js

2

5

2

1I ,

Bhc λk T

πhcλ T

λ e

Max Planck: Father of Quantum

• Introduced the concept of “quantum of action” in 1900 to solve the black body mystery

• In 1918 he was awarded the Nobel Prize for the discovery of the quantized nature of energy

The possible frequencies and energy states of a wave on

a string are quantized.

2

vf n

l

Strings are Quantized

Energy is Quantized

• An energy-level diagram

shows the quantized energy

levels and allowed

transitions

• Energy is on the vertical axis

• Horizontal lines represent

the allowed energy levels

• The double-headed arrows

indicate allowed transitions

Equipartition of Energy

• Each translational degree of freedom contributes an equal amount to the energy of the gas

• Each degree of freedom contributes

½kBT to the energy of a system, where

possible degrees of freedom are those

associated with translation, rotation

and vibration of molecules

• Theory breaks down for black body

radiation – degrees of freedom get

frozen out.

Planck’s Two Assumptions

• The energy of an oscillator can have only certain discretevalues En= nhƒ, n=interger

– This says the energy is quantized

– Each discrete energy value corresponds to a different quantum state

• The oscillators emit or absorb energy when making a transition from one quantum state to another

– The entire energy difference between the initial and final states in the transition is emitted or absorbed as a single quantum of radiation

2

5

2

1I ,

Bhc λk T

πhcλ T

λ e

The Turn Around in Planck’s

Model

• The average energy of a wave is the average

energy difference between levels of the oscillator,

weighted according to the probability of the wave

being emitted

• This weighting is described by the Boltzmann

distribution law and gives the probability of a state

being occupied as being proportional to

BE k Te


Planck had seen the turn over before:

Maxwell-Boltzmann Distribution: 1877• The observed speed

distribution of gas molecules in thermal equilibrium is shown at right

• NV is called the Maxwell-Boltzmann speed distribution function

• The distribution of speeds in N gas molecules is

• The probability of finding the molecule in a particular energy state varies exponentially as the negative of the energy divided by kBT

2

3 / 2

/ 22

B

42

Bmv k ToV

mN N v e

k T

nV (E ) = noe –E /kBT

Ludwig Boltzmann

1844 – 1906

Planck’s Wavelength

Distribution Function

• Planck generated a theoretical expression

for the wavelength distribution

– h = 6.626 x 10-34 J.s

– h is a fundamental constant of nature

2

5

2

1I ,

Bhc λk T

πhcλ T

λ e


Solves Black Body Mystery. Atomic Energy is quantized.

It comes in chunks of Planck’s constant, h.

34

, n= 0,1,2,3,...

6.626 10

E nhf

h x Js

2

5

2

1I ,

Bhc λk T

πhcλ T

λ e

Max Planck: Father

of Quantum

• In 1900 Planck developed a theory of blackbody radiation that leads to an equation for the intensity of the radiation which is in complete agreement with experimental observations.

• From his radiation equations Stefan’s Law is Derived!!

• Introduced the concept of “quantum of action” in 1900

• In 1918 he was awarded the Nobel Prize for the discovery of the quantized nature of energy

34

, n= 0,1,2,3,...

6.626 10

E nhf

h x Js

2

5

2

1I ,

Bhc λk T

πhcλ T

λ e

Intensity of Blackbody RadiationP40.61

The total power per unit area radiated by a black body at a temperature T is the area under the I(λ, T)-versus-λ curve, as shown in Figure 40.3. (a) Show that this power per unit area is

where I(λ, T) is given by Planck’s radiation law and σ is a constant independent of T.

This result is Stefan’s law.

To carry out the integration, you should make

the change of variable x = hc/λkT and use the

fact that

4

0λ λ, TdTI

0

43

151

xe

dxx

2

5

2

1I ,

Bhc λk T

πhcλ T

λ e

Particle Wave Duality

What is a particle? What is a wave?

The Photoelectric Effect• In 1886 Hertz noticed, in the

course of his investigations, that a

negatively charged electroscope

could be discharged by shining

ultraviolet light on it.

• In 1899, Thomson showed that the

emitted charges were electrons.

• Around 1900, Phillip Lenard built

an apparatus which produced an

electric current when ultraviolet

light was shining on the cathode.

• This phenomenon is called the

photoelectric effect.

The Problem with Waves:

Increasing the intensity of a low frequency

light beam doesn’t eject electrons. This

didn’t agree with wave picture of light

which predicts that the energy of waves

add so that if you increase the intensity of

low frequency light (bright red light)

eventually electrons would be ejected –

but they don’t! There is a cut off

frequency, below which no electrons will

be ejected no matter how bright the beam!

Also there is no time delay in the ejection

of electrons as the waves build up!

The PROBLEM with the

Photoelectric Effect

The Problem with Waves:

Increasing the intensity of a low frequency

light beam doesn’t eject electrons. This

didn’t agree with wave picture of light

which predicts that the energy of waves

add so that if you increase the intensity of

low frequency light (bright red light)

eventually electrons would be ejected –

but they don’t! There is a cut off

frequency, below which no electrons will

be ejected no matter how bright the beam!

Also there is no time delay in the ejection

of electrons as the waves build up!

The Photoelectric EffectProof that Light is a Particle

https://phet.colorado.edu/en/simulation/legacy/photoelectr

ic

https://phet.colorado.edu/en/simulation/legacy/photoelectric

https://phet.colorado.edu/en/simulation/legacy/photoelectric

chapter 40: intro to quantum problems with...

Documents