1. Chapter 27:Quantum Physics Blackbody 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.

2.  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: The classical theory of thermal radiation at the end of 19 th century failed to explain the distribution of energy of the blackbody radiation.

3.  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 n= quantum number (positive integer) f = frequency of vibration of the resonators h= 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.

4.  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 e  V. When  V is equal to or more negative than –  V s, 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.  V=V C -V E

5.  Photoelectric effect (cont’d) Photoelectric Effect and Particle Theory of Light ObservationsPredictions by wave theory No electrons are emitted if the incident light frequency falls below a cutoff freq. f c. Wave theory predicts that this effect should occur at any frequency, provided the intensity is enough. The max. kinetic energy of the photoelectrons is independent of light intensity. Light of higher intensity carries more energy into the metal per unit time and eject photoelectrons with higher energies. The max. kinetic energy of the photoelectrons increases with light frequency. No relation between photoelectron energy and incident light frequency is predicted. Electrons are emitted from the surface almost instantaneously even at low intensities (10 -9 s). It is expected that the photoelectrons need some time to absorb the incident radiation before they acquire enough kinetic energy to escape.

6.  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 E n =nhf to the next lower state E n-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:

7.  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 KE max of light intensity KE max depends on only the frequency of light and the work function. Linear dependence of KE max on light frequency KE max =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.

8.  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.

9.  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.

10.  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), 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…

11.  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: NaCl crystal Blue spheres : Cl - Red spheres : Na +

12.  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 structure of DNA was determined by x-ray crystallography.

13.  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 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 : m e :electron mass  : scattering angle h/m e c = 0.00243 nm Compton wavelength

14.  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 45.0 o to the incident beam. (a) Calculate the wavelength of the scattered x-rays at this angle. (b) Compute the fractional change in the energy of a photon in the collision.

15.  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) : His postulate was verified by diffraction pattern produced by electrons. An application: electron microscope

16.  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. Don’t worry. Just to be 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.

17.  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 precision  x and a simultaneous measurement of linear momentum is made with precision  p x, then the product of the two uncertainties can never be smaller than h/4  It is physically impossible to measure simultaneously the exact position and exact linear momentum of a particle.

18.  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 photon is, the maximum uncertainty in the electron’s momentum after collision is. It is a reasonable guess from the wave property of photon, you can determine the position of the electron with an accuracy of. Then. Another form of the uncertainty principle:

19.  Example 27.8 : Locating an electron Uncertainty Principle The speed of an electron is measured to be 5x10 3 m/s to an accuracy of 0.00300%. Find the minimum uncertainty in determining the position of this electron.  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?