Chapter 27:Quantum PhysicsBlackbody Radiation and Planck’s Hypothesis

Homework : Read and understand the lecture note.

Thermal radiation• An object at any temperature emits electromagnetic radiation called thermal radiation.

Sample homework problems : 1,18,24,30,37,46

• The spectrum of the radiation depends on the temperature and properties of the object.• From a classical point of view, thermal radiation originates from accelerated charged particles near the surface of an object.

Blackbody• Is an ideal system that absorbs all radiation incident on it.

• An opening in the cavity of a body is a good approximation of a blackbody.

Blackbody radiation• The nature of the blackbody radiation depends only on the temperature of the body, not on the material composition of the object.

Blackbody Radiation and Planck’s Hypothesis

• The distribution of energy in blackbody radiation varies with wavelength and temperature.

- The total amount of energy (area under the curve) it emits increase with the temperature.- The peak of the distribution shifts to shorter wavelengths. This shift obeys Wien’s displace- ment law: Km 102898.0 2

max T• The classical theory of thermal radiation at the end of 19th century failed to explain the distribution of energy of the blackbody radiation.

Planck’s hypothesis• To solve the discrepancy between the classical physics prediction and the observation of the blackbody radiation spectrum, in 1900 Planck developed a formula for the spectrum that explains the observed spectrum behavior.• Planck’s hypothesis: - Blackbody radiation is produced by submicroscopic charged oscillation (resonators). - The resonators are allowed to have only certain discrete energies given by:

Blackbody Radiation and Planck’s Hypothesis

nhfEn n= quantum number (positive integer)f = frequency of vibration of the resonatorsh= Planck’s constant 6.626 x 10-34 J s

• Energy is quantized.• each discrete energy value represents a different quantum state, where the quantum number n specifies the quantum state.

Photoelectric effect• Light incident on certain metallic surfaces causes the emission of electrons from the surfaces.• This phenomenon is called photoelectric effect and the emitted electrons are called photoelectrons.

Photoelectric Effect and Particle Theory of Light

• For an electron to reach Plate C when V<0, its kinetic energy must be at least eV.• When V is equal to or more negative than –Vs, the stopping potential, no electrons reach C and the current is zero.• The maximum kinetic energy of the photo- electrons is :• The stopping potential is independent of the radiation intensity.

sVeKE max

V=VC-VE

Photoelectric effect (cont’d)

Photoelectric Effect and Particle Theory of Light

Observations Predictions by wave theory

No electrons are emitted if theincident light frequency falls belowa cutoff freq. fc.

Wave theory predicts that this effectshould occur at any frequency,provided the intensity is enough.

The max. kinetic energy of thephotoelectrons is independent oflight intensity.

Light of higher intensity carries moreenergy into the metal per unit timeand eject photoelectrons with higherenergies.

The max. kinetic energy of thephotoelectrons increases withlight frequency.

No relation between photoelectronenergy and incident light frequencyis predicted.

Electrons are emitted from thesurface almost instantaneouslyeven at low intensities (10-9 s) .

It is expected that the photoelectronsneed some time to absorb the incidentradiation before they acquire enoughkinetic energy to escape.

Einstein’s particle theory of light

Photoelectric Effect and Particle Theory of Light

• Einstein successfully resolve the mystery in 1905 by extending Planck’s idea of quantization to electromagnetic waves.• Einstein’s theory: - A localized packet of light energy (photon) would be emitted when a quantized oscillator made a jump from an energy state En=nhf to the next lower state En-1=(n-1)hf. - From conservation of energy, the photon’s energy is E=hf. - A well localized photon can give all its energy hf to a single electron in the metal. - An electron in the metal is bound by electromagnetic force and it needs to gain a certain energy (work function ) to be liberated:

hfKEmax

Einstein’s particle theory of light

Photoelectric Effect and Particle Theory of Light

• Predictions of Einstein’s theory:

Cutoff frequency Photoelectrons are created by absorption of a single photon that has enough energy to overcome the work function.Independence of KEmax of light intensity KEmax depends on only the frequency of light and the work function.Linear dependence of KEmax on light frequency KEmax=hf- explains it.Instantaneous production of photoelectrons The light energy is concentrated in packets. If the light has enough energy (frequency), no time is need to knock-off a photoelectron.

00max

c

c

chhKE

hcc

Roentgen’s discovery of X-rays

X-Rays

• Roentgen noticed that a fluorescent screen glowed even when placed several meters from the gas discharge tube and when black cardboard was placed between the tube and the screen. Discovery of x-rays• It was found that x-rays were beams of uncharged particles.• von Laue suggested that if x-rays were electromagnetic waves with very short wavelengths (~0.1 nm), it should be possible to diffract them using the regular atomic spacings of crystal lattice as a diffraction grating. • Experiments demonstrated that his suggestion was indeed valid.

Mechanism of x-ray production

X-Rays

• X-rays are produced when high-speed electrons are suddenly slowed down. This happens when a metal target is struck by electrons that are accelerated by a potential difference.• The spectrum of x-rays produced by an x-ray tube shows two components in the intensity vs. wavelength curve – a continuous broad spectrum and a series of sharp intense lines.

- The continuous spectrum is produced by bremsstrahlung, radiation by electrons when they undergo acceleration under influence of electromagnetic field of nuclei of atoms.- The line spectrum is produced by transitions from one quantum state to another and it depends on the target material.

Bremsstrahlung

X-Rays

• When an electron passes close to positively charged nucleus in the target material, it is deflected and accelerated and at the same time it radiates a photon.• If the electron loses all its energy to the emitted photon (x-ray in this case),

minmax

hchfVe

Ve

hc

min

• The reason for the continuous spectrum is that many of electrons do not lose all their energy at once in a single collision.

Application of x-rays

Crystallography, imaging of organs, radiation therapy, security screening…

Principle

Diffraction of X-Rays by Crystals

• If atoms are aligned in a regular manner, they act as a diffraction grating device. This is realized in a crystal.• When the incident beam of x-ray are reflected at the upper and lower plane, the path difference produces interference pattern.• If Bragg’s law is satisfied, constructed interference is produced:

Na

Cl c

ryst

al

Blu

e s

ph

ere

s : C

l-

Re

d s

phe

res

: Na+

,...)3,2,1( sin2 mmd

Principle (cont’d)

Diffraction of X-Rays by Crystals

• If well-collimated beam of x-ray is used, each configuration of layers made by atoms in a crystal produces a point on the screen.• A pattern of spots (Laue pattern) created on the screen including the positions and intensities of the spots gives information about the structure of the crystal.

Double-helix structureof DNA was determined byx-ray crystallography.

Photon nature of light

Compton Effect

• Compton’s experiment:- X-ray beam of wavelength 0

directed to a lock of graphite.- Compton observed that the scattered x-rays had a slightly longer wavelength (lower energy).

- The amount of reduction in energy depends on the amount of angle x-rays are deflected.

Compton shift 0'

• Compton’s explanation :

- If a photon behaves like a particle, its collision with other particles is similar to a collision between two billiard balls.- A photon collides with an electron at rest and transfer part of its energy and momentum to the electron. Then from conservation of energy and momentum he derived a formula :

)cos1( cm

h

e

me:electron mass : scattering angleh/mec = 0.00243 nm Compton wavelength

Example 27.6 : Scattering x-rays

Compton Effect

• X-rays of wavelength 0 =0.200000 nm are scattered from a block of material. The scattered x-rays are observed at an angle of 45.0o to the incident beam. (a) Calculate the wavelength of the scattered x-rays at this angle.

m 1011.7)0.45cos1(m/s) 10kg)(3.00 1011.9(

sJ 1063.6)cos1( 13

831

34

cm

h

e

(b) Compute the fractional change in the energy of a photon in the collision.

nm 200711.00

c

hhfE

31054.3nm 200711.0

nm 000711.0

/

//

ii

if

i

if

hc

hchc

E

EE

E

E

Light and electromagnetic radiation

Dual Nature of Light and Matter

• The photoelectric effect and the Compton scattering suggest particle nature of light with energy hf and momentum h/. • At the same time light behaves like wave.

Light has a dual nature, exhibiting both wave and particle characteristics.

Wave properties of particles• De Broglie’s idea :

All forms of matter have both properties - wave and particle characteristics

• According to de Broglie’s idea, electrons like light have a dual particle-wave nature with the following relation among the momentum, energy and wavelength (de Broglie wavelength of a particle) :

)/(/ mvhph

hEf /

• His postulate was verified by diffraction pattern produced by electrons.

An application: electron microscope

Light and electromagnetic radiation

Wave Function

• In 1926, Schroedinger proposed a wave equation that described how wave change in space and time : The Schroedinger wave equation• Solving Schroedinger’s equation determines a quantity called the wave function.

),,(2

2

zyxUmt

i

Don’t worry. Just tobe cool.

• Putting aside historical controversies over the interpretation of a wave function this wave function describes a single particle, the value of ||2 at some location at a given time is proportional to the probabilities per unit volume of finding the particle at that location at that time.

• Adding up all the values of ||2 in a given region gives the probability of finding the particle in that region.

Uncertainty principle

Uncertainty Principle

• In 1927, Heisenberg answered this question and introduced a surprising principle:

• How accurately can we measure the position and speed of a particle at any instance? Is there any limit?

Heisenberg’s uncertainty principle:

If a measurement of the position of a particle is made with precisionx and a simultaneous measurement of linear momentum is madewith precision px, then the product of the two uncertainties can neverbe smaller than h/4

4

hpx x

It is physically impossible to measure simultaneously the exactposition and exact linear momentum of a particle.

Interpretation of uncertainty principle

Uncertainty Principle

• Consider a thought experiment: You are trying to measure the position and momentum of an electron as accurate as possible using a powerful microscope.

As the momentum of incoming photonis , the maximum uncertainty inthe electron’s momentum after collisionis ./hpx It is a reasonable guess from the waveproperty of photon, you can determinethe position of the electron with anaccuracy of . Then .

/h

x htx • Another form of the uncertainty principle:

4

htE

Example 27.8 : Locating an electron

Uncertainty Principle

• The speed of an electron is measured to be 5x103 m/s to an accuracy of 0.00300%. Find the minimum uncertainty in determining the position of this electron.

m/skg 104.56m/s) 10kg)(5.00 1011.9( -27331 vmp ex

m/skg 1037.10000300.0 31 ppx

mm 0.384m 10384.044

3

xx p

hx

hpx

Example 27.9 : Excited states of atoms

• Electrons in atoms can be found in certain high states of energy called excited states for short periods of time. If the average time that an electron exists in one of these states is 1.00x10-8 s, what is the minimum uncertainty in energy of the excited state?

eV. 103.30J 1028.5s) 1000.1(4

sJ 1063.6

48-27

8

34

t

hE