1. Concept of Waves and Particles In the early part of the 20 th century all attempts to explain the behaviour of matter on the atomic level with the laws of classical physics were consistently unsuccessful. Various phenomena, such as the -electromagnetic radiation emitted by a heated object (blackbody radiation), -the emission of electrons by illuminated metals (the photoelectric effect), and -the emission of sharp spectral lines by gas atoms in an electric discharge tube, couldn’t be understood within the framework of classical physics

2. Between 1900 and 1930, however, a modern version of mechanics called quantum mechanics or wave mechanics was highly successful in explaining the behaviour of atoms, molecules, and nuclei. The earliest ideas of quantum theory were introduced by Planck, and most of the subsequent mathematical developments, interpretations, and improvements were made by a number of distinguished physicists, including Einstein, Bohr, Schrödinger, de Broglie, Heisenberg, Born, and Dirac.

3. BLACKBODY RADIATION AND PLANCK’S HYPOTHESIS The rate at which an object radiates energy is proportional to the fourth power of its absolute temperature. An object at any temperature emits electromagnetic radiation, called thermal radiation. Stefan’s law, (The rate at which an object radiates energy is proportional to the fourth power of its absolute temperature), describes the total power radiated

4. The spectrum of the radiation depends on the temperature and properties of the object. At low temperatures, the wavelengths of the thermal radiation are mainly in the infrared region and hence not observable by the eye. As the temperature of an object increases, the object eventually begins to glow red. At sufficiently high temperatures, it appears to be white, as in the glow of the hot tungsten filament of a light bulb. A careful study of thermal radiation shows that it consists of a continuous distribution of wavelengths from the infrared, visible, and ultraviolet portions of the spectrum.

5. From a classical viewpoint, thermal radiation originates from accelerated charged particles near the surface of an object; such charges emit radiation, much as small antennas do. The thermally agitated (exited) charges can have a distribution of frequencies, which accounts for the continuous spectrum of radiation emitted by the object. By the end of the 19th century.it was inadequate. The basic problem was in understanding the observed distribution energy as a function of wavelength in the radiation emitted by a blackbody.

6. By definition, a blackbody is an ideal system that absorbs all radiation incidents on it. The nature of the radiation emitted through the small hole leading to the cavity depends only on the temperat -ure of the cavity walls, and not at all on the material composition of the object, its shape, or other factors. Experimental data for the distribution of energy in blackbody radiation at three temperatures are shown in Active Figure 27.2.

7. The radiated energy varies with wavelength and temperature. As the temperature of the blackbody increases, the total amount of energy (area under the curve) it emits increases. Also, with increasing temperature, the peak of the distribution shifts to shorter wavelengths. This shift obeys the following relationship, called Wien’s displacement law, where λ max is the wavelength at which the curve peaks and T is the absolute temperature of the object emitting the radiation.

8. EXAMPLE: The temperature of the skin is approximately 35.0°C. At what wavelength does the radiation emitted from the skin reach its peak? Solution Apply Wien’s displacement law: Solve for λ max, noting that 35.0°C corresponds to an absolute temperature of 308 K:

9. Active Figure 27.3 shows an experimental plot of the blackbody radiation spectrum (red curve), together with the theoretical picture of what this curve should look like based on classical theories (blue curve). At long wavelengths, classical theory is in good agreement with the experimental data. As λ approaches zero, classical theory predicts that the amount of energy being radiated should increase.

10. In fact, the theory erroneously predicts that the intensity should be infinite, when the experimental data shows it should approach zero. This contradiction is called the ultraviolet catastrophe, because theory and experiment disagree strongly in the short-wavelength, ultraviolet region of the spectrum. In 1900 Planck developed a formula for blackbody radiation that was in complete agreement with experiments at all wavelengths, leading to a curve shown by the red line in Active Figure 27.3. Planck hypothesized that blackbody radiation was produced by submicroscopic charged oscillators, which he called resonators.

11. He assumed that the walls of a glowing cavity were composed of billions of these resonators, although their exact nature was unknown. The resonators were allowed to have only certain discrete energies E n, given by where n is a positive integer called a quantum number, f is the frequency of vibration of the resonator, and h is a constant known as Planck’s constant, which has the value The key point in Planck’s theory is the assumption of quantized energy states.

12. EXAMPLE: A 2.00-kg mass is attached to a spring having force constant k =25.0 N/m and negligible mass. The spring is stretched 0.400 m from its equilibrium position and released. (a) Find the total energy and frequency of oscillation according to classical calculations?

13. (b) Assume that Planck’s law of energy quantization applies to any oscillator, atomic or large scale, and find the quantum number n for this system. (c) How much energy would be carried away in a one-quantum change

14. THE PHOTOELECTRIC EFFECT AND THE PARTICLE THEORY OF LIGHT In the latter part of the 19th century, experiments showed that light incident on certain metallic surfaces caused the emission of electrons are emitted from the surfaces. This phenomenon is known as the photoelectric effect, and the emitted electrons are called photoelectrons. The first discovery of this phenomenon was made by Hertz, who was also the first to produce the electromagnetic waves predicted by Maxwell.

15. An evacuated glass tube known as a photocell contains a metal plate E (the emitter) connected to the negative terminal of a variable power supply. Another metal plate, C (the collector), is maintained at a positive potential by the power supply. When the tube is kept in the dark, the ammeter reads zero, indicating that there is no current in the circuit. when plate E is illuminated by light having a wavelength shorter than some particular wavelength that depends on the material which the plate E made of, a current is detected by the ammeter, indicating a flow of charges across the gap between E and C. This current arises from photoelectrons emitted from the negative plate E and collected at the positive plate C.

16. Active Figure 27.5 is a plot of the photoelectric current versus the potential difference ΔV between E and C for two light intensities. At large values of ΔV, the current reaches a maximum value. In addition, the current increases as the incident light intensity increases, as you might expect. Finally, when ΔV is negative—that is, when the power supply in the circuit is reversed to make E positive and C negative—the current drops to a low value because most of the emitted photoelectrons are repelled by the now negative plate C. In this situation, only those electrons having a kinetic energy greater than the magnitude of eΔV reach C, where e is the charge on the electron.

17. When ΔV is equal to or more negative than ΔV s, the stopping potential, no electrons reach C and the current is zero. The stopping potential is independent of the radiation intensity. The maximum kinetic energy of the photoelectrons is related to the stopping potential through the relationship Several features of the photoelectric effect can’t be explained with classical physics or with the wave theory of light:

18. No electrons are emitted if the incident light frequency falls below some cut-off frequency f c, which is characteristic of the material being illuminated. This is inconsistent with the wave theory, which predicts that the photoelectric effect should occur at any frequency, provided the light intensity is sufficiently high. The maximum kinetic energy of the photoelectrons is independent of light intensity. According to wave theory, light of higher intensity should carry more energy into the metal per unit time and therefore eject photoelectrons having higher kinetic energies.

19. The maximum kinetic energy of the photoelectrons increases with increasing light frequency. The wave theory predicts no relationship between photoelectron energy and incident light frequency. Electrons are emitted from the surface almost instantaneously (less than 10 -9 s after the surface is illuminated), even at low light intensities. Classically, we expect the photoelectrons to require some time to absorb the incident radiation before they acquire enough kinetic energy to escape from the metal.

20. A successful explanation of the photoelectric effect was given by Einstein in1905, the same year he published his special theory of relativity. As part of a general paper on electromagnetic radiation, for which he received the Nobel Prize in 1921, Einstein extended Planck’s concept of quantization to electromagnetic waves. He suggested that a tiny packet of light energy or 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. Conservation of energy would require the decrease in oscillator energy, hf, to be equal to the photon’s energy E, so that

21. The key point here is that the light energy lost by the emitter, hf, stays sharply localized in a tiny packet or particle called a photon. In Einstein’s model, a photon is so localized that it can give all its energy hf to a single electron in the metal. According to Einstein, the maximum kinetic energy for these liberated photoelectrons is where ϕ is called the work function of the metal The work function, which represents the minimum energy with which an electron is bound in the metal, is on the order of a few electron volts. Table 27.1 lists work functions for various metals.

22. With the photon theory of light, we can explain the previously mentioned features of the photoelectric effect that cannot be understood using concepts of classical physics: Photoelectrons are created by absorption of a single photon, so the energy of that photon must be greater than or equal to the work function, else no photoelectrons will be produced. This explains the cut-off frequency. From Equation 27.6, KE max depends only on the frequency of the light and the value of the work function. Light intensity is immaterial, because absorption of a single photon is responsible for the electron’s change in kinetic energy. Equation 27.6 is linear in the frequency, so KE max increases with increasing frequency

23. Electrons are emitted almost instantaneously, regardless of intensity, because the light energy is concentrated in packets rather than spread out in waves. If the frequency is high enough, no time is needed for the electron to gradually acquire sufficient energy to escape the metal Experimentally, a linear relationship is observed between f and KE max, as sketched in Figure 27.6. The intercept on the horizontal axis, corresponding to KE max = 0, gives the cutoff frequency below

24. which no photoelectrons are emitted, regardless of light intensity. The cutoff wavelength λ c can be derived from Equation 27.6: where c is the speed of light. Wavelengths greater than λ c incident on a material with work function ϕ don’t result in the emission of photoelectrons.

25. EXAMPLE: A sodium surface is illuminated with light of wavelength 0.300 _m. The work function for sodium is 2.46 eV. (a) Calculate the energy of each photon in electron volts, Solution

26. (b) the maximum kinetic energy of the ejected photoelectrons, and (c) the cutoff wavelength for sodium. The cutoff wavelength is in the yellow-green region of the visible spectrum.

27. Photocells The photoelectric effect has many interesting applications using a device called the photocell. The photocell produces a current in the circuit when light of sufficiently high frequency falls on the cell, but it doesn’t allow a current in the dark. This device is used in streetlights: a photoelectric control unit in the base of the light activates a switch that turns off the streetlight when ambient light strikes it. Many garage-door systems and elevators use a light beam and a photocell as a safety feature in their design. When the light beam strikes the photocell, the electric current generated is sufficiently large to maintain a closed circuit. When an object or a person blocks the light beam, the current is interrupted, which signals the door to open.