Photoelectric Effect Photoelectric Effect (How Einstein really became famous!)

Photoelectric EffectPhotoelectric Effect

(How Einstein really became famous!)

(How Einstein really became famous!)


Metal Foil

Photoelectric Effect

Metal Foil


•As blue light strikes the metal foil, the foil emits electrons.

•As blue light strikes the metal foil, the foil emits electrons.


•When red light hits the metal foil, the foil does not emit electrons.

•Blue light has more energy than red light.

•How could we get more energy into the red light?

•Try increasing the brightness.

•When red light hits the metal foil, the foil does not emit electrons.

•Blue light has more energy than red light.

•How could we get more energy into the red light?

•Try increasing the brightness.


•Well, that didn’t work!•Maybe its still not bright

enough.

•Well, that didn’t work!•Maybe its still not bright

enough.


•Still not working.•What happens with brighter

blue light?

•Still not working.•What happens with brighter

blue light?


•More blue light means more electrons emitted, but that doesn’t work with red.

•More blue light means more electrons emitted, but that doesn’t work with red.


•Wave theory cannot explain these phenomena, as the energy depends on the intensity (brightness)

•According to wave theory bright red light should work!

•Wave theory cannot explain these phenomena, as the energy depends on the intensity (brightness)

•According to wave theory bright red light should work!

►BUT IT DOESN’T!BUT IT DOESN’T!


•Einstein said that light travels in tiny packets called quanta.

•The energy of each quanta is given by its frequency

•Einstein said that light travels in tiny packets called quanta.

•The energy of each quanta is given by its frequency

E=hfE=hfEnergyEnergy

Planck’s constantPlanck’s constant

frequencyfrequency


• Each metal has a minimum energy needed for an electron to be emitted.

• This is known as the work function, W.

• So, for an electron to be emitted, the energy of the photon, hf, must be greater than the work function, W.

• The excess energy is the kinetic energy, E of the emitted electron.

• Each metal has a minimum energy needed for an electron to be emitted.

• This is known as the work function, W.

• So, for an electron to be emitted, the energy of the photon, hf, must be greater than the work function, W.

• The excess energy is the kinetic energy, E of the emitted electron.

Most commonly observed phenomena with light can be explained by waves. But the photoelectric effect suggested a particle nature for light.

Photoelectric EffectPhotoelectric EffectEINSTEIN’S PHOTOELECTRIC EQUATION:-EINSTEIN’S PHOTOELECTRIC EQUATION:-

E= hf-W

Blackbody RadiationBlackbody Radiation

ContentsContents

• Definition of a Black-Body• Black-Body Radation Laws

*1- The Rayleigh-Jeans Law 2- The Wien Displacement Law3- The Stefan-Boltzmann Law*4- The Planck Law

• Application for Black Body • Conclusion• Summary

• Definition of a Black-Body• Black-Body Radation Laws

*1- The Rayleigh-Jeans Law 2- The Wien Displacement Law3- The Stefan-Boltzmann Law*4- The Planck Law

• Application for Black Body • Conclusion• Summary

Motivation

Motivation•The black body is importance in

thermal radiation theory and practice. •The ideal black body notion is

importance in studying thermal radiation and electromagnetic radiation transfer in all wavelength bands.

•The black body is used as a standard with which the absorption of real bodies is compared.

•The black body is importance in thermal radiation theory and practice.

•The ideal black body notion is importance in studying thermal radiation and electromagnetic radiation transfer in all wavelength bands.

•The black body is used as a standard with which the absorption of real bodies is compared.

Definition of a black bodyDefinition of a black bodyA black body is an ideal body which allows the whole of the incident radiation to pass into itself ( without reflecting the energy ) and absorbs within itself this whole incident radiation (without passing on the energy). This propety is valid for radiation corresponding to all wavelengths and to all angels of incidence. Therefore, the black body is an ideal absorber of incident radaition.

Black-Body Radiation LawsBlack-Body Radiation Laws

The Rayleigh-Jeans Law.* It agrees with experimental

measurements for long wavelengths.

* It predicts an energy output that diverges towards infinity as wavelengths grow smaller.

* The failure has become known as the ultraviolet catastrophe.

The Rayleigh-Jeans Law.* It agrees with experimental

measurements for long wavelengths.

* It predicts an energy output that diverges towards infinity as wavelengths grow smaller.

* The failure has become known as the ultraviolet catastrophe.

4

2),(

ckT

TI

• This formula also had a problem. The problem was the term in the denominator.

• For large wavelengths it fitted the experimental data but it had major problems at shorter wavelengths.

• This formula also had a problem. The problem was the term in the denominator.

• For large wavelengths it fitted the experimental data but it had major problems at shorter wavelengths.

Ultraviolet CatastropheUltraviolet Catastrophe4

2),(

ckT

TI

Planck Law- We have two forms. As a

function of wavelength.

And as a function of frequency

The Planck Law gives a distribution that peaks at a certain wavelength, the peak shifts to shorter wavelengths for higher temperatures, and the area under the curve grows rapidly with increasing temperature.

Planck Law- We have two forms. As a

function of wavelength.

And as a function of frequency

The Planck Law gives a distribution that peaks at a certain wavelength, the peak shifts to shorter wavelengths for higher temperatures, and the area under the curve grows rapidly with increasing temperature.

Black-Body Radiation LawsBlack-Body Radiation Laws

1

12),(

2

3

kTh

e

c

hTI

1

15

22),(

kThc

e

hcTI

Comparison between Classical and Quantum viewpoint

Comparison between Classical and Quantum viewpoint

http://upload.wikimedia.org/wikipedia/commons/a/a1/Blackbody-lg.png

There is a good fit at long wavelengths, but at short wavlengths there is a major disagreement. Rayleigh-Jeans ∞, but Black-body 0.

Radiation Curves

ConclusionConclusion• As the temperature

increases, the peak wavelength emitted by the black body decreases.

• As temperature increases, the total energy emitted increases, because the total area under the curve increases.

• The curve gets infinitely close to the x-axis but never touches it.

• As the temperature increases, the peak wavelength emitted by the black body decreases.

• As temperature increases, the total energy emitted increases, because the total area under the curve increases.

• The curve gets infinitely close to the x-axis but never touches it.

The Birth of Quantum Mechanics___________________________

• At the turn of the last century, there were several experimental observations which could not be explained by the established laws of classical physics and called for a radically different way of thinking

• This led to the development of Quantum Mechanics which is today regarded as the fundamental theory of Nature.

Some key events/observations that led to the development of quantum mechanics…_________________________________

Black body radiation spectrum (Planck, 1901)

• Photoelectric effect (Einstein, 1905)

• Model of the atom (Rutherford, 1911)

• Quantum Theory of Spectra (Bohr, 1913)

Some key events/observations that led to the development of quantum mechanics…_________________________________

• Matter Waves (de Broglie 1925)

EinsteinPlanck

Rutherford

Bohr

de Broglie

The basic hydrogen energy level structure is in agreement with the Bohr model. Common pictures are those of a shell structure with each main shell associated with a value of the principal quantum number n.

This Bohr model picture of the orbits has some usefulness for visualization so long as it is realized that the "orbits" and the "orbit radius" just represent the most probable values of a considerable range of values.

Hydrogen Energy Levels

The Bohr model for an electron transition in hydrogen between quantized energy levels with different quantum numbers n yields a photon by emission, with quantum energy

This is often expressed in terms of the inverse wavelength or "wave number" as follows:

eV 11

6.1311

8 22

21

22

21

220

4

nnnnh

meh

320

4

8 ch

meRH

17 m 10 097.1 HR

Uncertainty Principle

Uncertainty Principle

• We assume that if we measure something we will have some small errors

• You know this from your lab work• With better instruments and

techniques you can reduce these errors• Heisenberg showed that there is a limit

to how small you can make the error!

• We assume that if we measure something we will have some small errors

• You know this from your lab work• With better instruments and

techniques you can reduce these errors• Heisenberg showed that there is a limit

to how small you can make the error!

Uncertainty PrincipleUncertainty Principle

Recall the diffraction limit of light. We can measure position to about a wavelength of the light we use. To get more accurate position, shorten the wavelength which ups the frequency. But E=hf, so the photon has higher energy. Whacks the electron harder and you don’t know where it goes or how fast it is moving. Lower the frequency and you get more uncertainty in position.

Uncertainty PrincipleUncertainty Principle

x p h /xp hxph /2

The Schrödinger equation plays the role of Newton's laws and conservation of energy in classical mechanics - i.e., it predicts the future behavior of a dynamic system. It is a wave equation in terms of the wavefunction which predicts analytically and precisely the probability of events or outcome. The detailed outcome is not strictly determined, but given a large number of events, the Schrödinger equation will predict the distribution of results.

The kinetic and potential energies are transformed into the Hamiltonian which acts upon the wavefunction to generate the evolution of the wavefunction in time and space. The Schrödinger equation gives the quantized energies of the system and gives the form of the wavefunction so that other properties may be calculated.

Schrödinger Equation

For a generic potential energy U the 1-dimensional time-independent Schrodinger equation is

In three dimensions, it takes the form

for cartesian coordinates. This can be written in a more compact form by making use of the Laplacian operator

The Schrodinger equation can then be written:

Time-independent Schrödinger Equation

HΨ = EΨ

The time dependent Schrödinger equation for one spatial dimension is of the form

For a free particle where U(x) =0 the wavefunction solution can be put in the form of a plane wave

For other problems, the potential U(x) serves to set boundary conditions on the spatial part of the wavefunction and it is helpful to separate the equation into the time-independent Schrödinger equation and the relationship for time evolution of the wavefunction

Time Dependent Schrödinger Equation

EhT

πp

hk

2

22

22

Time-Independent Schrödinger Equation

Time-Independent Schrödinger Equation

• Schrödinger developed the equation from which we can find the wavefunction

• Below is time-independent Schrödinger equation, which describes stationary states

– the energy of such states does not change with time

• ψ(x) is often called eigenfunctions or eigenstate

• Here U is a potential function, representing forces acting upon particle (particle’s interaction with environment)

• Schrödinger developed the equation from which we can find the wavefunction

• Below is time-independent Schrödinger equation, which describes stationary states

– the energy of such states does not change with time

• ψ(x) is often called eigenfunctions or eigenstate

• Here U is a potential function, representing forces acting upon particle (particle’s interaction with environment)

xExxUxx

m

)(

2 2

22

Particle in a box with “Infinite Barriers”


• A particle is confined to a one-dimensional region of space between two impenetrable walls separated by distance L– This is a one- dimensional “box”

• The particle is bouncing elastically back and forth between the walls– As long as the particle is inside the box,

the potential energy does not depend on its location. We can choose this energy value to be zero

• U(x) = 0, 0 < x < L, U(x) , x ≤ 0 and x ≥ L

• Since walls are impenetrable, we say that this models a box (potential well) has infinite barriers

• A particle is confined to a one-dimensional region of space between two impenetrable walls separated by distance L– This is a one- dimensional “box”

• The particle is bouncing elastically back and forth between the walls– As long as the particle is inside the box,

the potential energy does not depend on its location. We can choose this energy value to be zero

• U(x) = 0, 0 < x < L, U(x) , x ≤ 0 and x ≥ L

• Since walls are impenetrable, we say that this models a box (potential well) has infinite barriers



• Since the walls are impenetrable, there is zero probability of finding the particle outside the box. Zero probability means that ψ(x) = 0, for x < 0 and x > L

• The wave function must also be 0 at the walls (x = 0 and x = L), since the wavefunction must be continuous– Mathematically, ψ(0) = 0

and ψ(L) = 0

• Since the walls are impenetrable, there is zero probability of finding the particle outside the box. Zero probability means that ψ(x) = 0, for x < 0 and x > L

• The wave function must also be 0 at the walls (x = 0 and x = L), since the wavefunction must be continuous– Mathematically, ψ(0) = 0

and ψ(L) = 0

Schrödinger Equation Applied to a Particle in a “Infinite” Box


• In the region 0 < x < L, where U(x) = 0, the Schrödinger equation can be expressed in the form

• We can re-write it as

• In the region 0 < x < L, where U(x) = 0, the Schrödinger equation can be expressed in the form

• We can re-write it as

xExxUxx

m

)(

2 2

22

xEx

xm

2

22

2

22

22

2

22

2

2

2

mEk

xkx

x

xmE

x

x

Schrödinger Equation Applied to a Particle in a “Infinite”

Box

Schrödinger Equation Applied to a Particle in a “Infinite”

Box

• The most general solution to this differential equation is

ψ(x) = A sin kx + B cos kx– A and B are constants determined by the

properties of the wavefunction as well as boundary and normalization conditions

• The most general solution to this differential equation is

ψ(x) = A sin kx + B cos kx– A and B are constants determined by the

properties of the wavefunction as well as boundary and normalization conditions

xkx

x 22

2



1. Sin(x) and Cos(x) are finite and single-valued functions

2. Continuity: ψ(0) = ψ(L) = 0• ψ(0) = A sin(k0) + B cos(k0) = 0 B = 0

ψ(x) = A sin(kx)• ψ(L) = A sin(kL) = 0 sin(kL) = 0 kL

= πn, n = ±1, ± 2, ± 3, …

1. Sin(x) and Cos(x) are finite and single-valued functions

2. Continuity: ψ(0) = ψ(L) = 0• ψ(0) = A sin(k0) + B cos(k0) = 0 B = 0

ψ(x) = A sin(kx)• ψ(L) = A sin(kL) = 0 sin(kL) = 0 kL

= πn, n = ±1, ± 2, ± 3, …

22

22

2

22

22

22

82

2

nmL

hn

mLE

nL

mEkn

Lk

n

nnn

• The allowed wave functions are given by

– After, the normalization, the normalized wave function



x

Ln

A(x) ψn

sin

x

Ln

L (x) ψn

sin

2

Particle in the Well with Infinite Barriers

Particle in the Well with Infinite Barriers

01

2

22

02

02

2

22

, )1( 2

,2

EEenergyn stategroundmL

EwithnEnmL

En

Probability to Find particleProbability to Find particle

Finite Potential WellGraphical Results for ψ (x)

Finite Potential WellGraphical Results for ψ (x)

• Outside the potential well, classical physics forbids the presence of the particle

• Quantum mechanics shows the wave function decays exponentially to approach zero

• Outside the potential well, classical physics forbids the presence of the particle

• Quantum mechanics shows the wave function decays exponentially to approach zero

Finite Potential WellGraphical Results for

Probability Density, | ψ (x) |2

Finite Potential WellGraphical Results for

Probability Density, | ψ (x) |2

• The probability densities for the lowest three states are shown

• The functions are smooth at the boundaries

• Outside the box, the probability to find the particle decreases exponentially, but it is not zero!

• The probability densities for the lowest three states are shown

• The functions are smooth at the boundaries

• Outside the box, the probability to find the particle decreases exponentially, but it is not zero!

Photoelectric Effect Photoelectric Effect (How Einstein really became famous!)

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