Title 25.Dual Nature of Radiation and Matter.pdf ✅

NEET-2022 Ultimate Crash Course PHYSICS Dual Nature of Radiation and Matter

POINTS TO REMEMBER 1. The failure of the wave theory of light to explain photoelectric effect can be better understood with the help of the following mathematical resonating 2. (a)According to wave theory, the energy in a light wave is spread out uniformly and continuously over its wavefront. Thus, each atom in the top atomic layer absorbs an equal amount of energy which is proportional to its cross-sectional area A and the light intensity I. If the atom gives this energy to one of its electrons, energy absorbed by the electron in time t, i.e., K = IAt-----(1) (as intensity energy area time =  ) Since 0 , is the energy required to liberate the electron from the metal surface, maximum kinetic energy acquired by the electron, i.e., K K IAt max 0 0 = − = −   -----------(2) Clearly, Kmax depends on I (intensity of light). Further, no electrons should be emitted at low intensity, which contradicts Law II. (b) According to wave theory, the intensity of a light wave is proportional to the square of the amplitude of the electric field, i.e., 2 0 0 I E  Thus, from equation (2) 2 K E At max 0 0 = − -----(3) Eqn (3) shows that Kmax should not depend on frequency of light which contradicts Law III. 3. Thus, even though a typical light bulb gives off roughly 1018 photons per second, we need only about 103 photons per second to see. Our eyes are extraordinary instruments, sensitive to an incredibly wide range of intensities. 4. Suppose an astronomer views a dim, distant galaxy with scotopic vision. The 2870 photons that enter the astronomer's eye each second are separated from each other by about 65 miles (as 2870 photons are separated 3 x 105 km, distance between two 3 x105km — 105 km or 65 miles). consecutive photons is 5 3 10 105 2870 km  = km or 65 miles. It follows that only one photon at a time from the distant galaxy, traverses the astronomer's telescope. Difference between Light waves and Matter waves: We know that p = h /  , and E hv = Also, we know that for a particle of mass (m) and moving with a velocity (v), de Broglie wavelength is given by h mv  = or mv h = /  Further, 1 2 2 E mv hv = = Thus, ( ) 2 2 2 2 / 1 2 m v h hv mv  = or 2 2 h m  v = or 2 h v m   = v being the product of wavelength and frequency, is equal to the velocity of the matter waves. This velocity is called the wave velocity (vw ) and is different from the particle velocity (v). Thus, 2 w h v m = It is clear that w v depends upon the wavelength even if the particle is moving in a vacuum. This is very much different from the behaviour of light which moves in vacuum with the same velocity c regardless of the wavelength. The dependence of wave velocity (vw ) on wavelength in vacuum is one of the basic differences between matter waves and light waves. 5. A photon of 1242 nm wavelength has an energy of 1 eV. A light of wavelength (1242/4) = 310.5 nm has an energy of 4 eV and so on. 6. Angular momentum of an atom changes when a photon is emitted or absorbed. Since angular momentum must be conserved, we conclude that the photon involved in the process must carry angular momentum. Hence, a photon has energy, linear momentum and angular momentum.
7. It is tempting to remember the classical definition of momentum, mu and write p = mc for the photon, but we do not have a value for the photon's mass. In fact, we can show that the rest mass of a photon must be zero. 8. Since a photon travels with speed c in vacuum ( ) 0 0 0 2 1 1 0 1 / m m m m v c = = = − If m0 were anything but zero, the photon must have infinite mass. Since E = mc2 , however, infinite mass implies infinite photon energy; and we know this to be untrue. Therefore, we must conclude that 0 m = 0 . If this seems strange, remember that a photo never at rest. It is both emitted and absorbed at the speed of light. A photon moving through vacuum never travels at any speed other than c. The only mass such a particle has is due to its kinetic energy. Thus, ( ) 2 2 0 / E m m c mc hc photon = − = =  Photon momentum 2 mc hc h Ephoton / p mc c c c   = = = = = = 9. Photons travel with the speed alight, which means that all observers see them as having the same speed. It is not possible to stop a photon and hold it in your hand. In contrast, particles with a finite rest mass can never attain the speed alight. It follows, then, that photons must have zero mass. 10. If h were sufficiently large, the wavelike properties of matter would be apparent even on the macroscopic level and our experience of the nature would be very different indeed. 11. The ratio of the Planck's constant/Clip the charge of the electron (e) is called photoelectric constant. 12. It is interesting to note that when Einstein was awarded the Nobel Prize in 1922, the Swedish Academy judged his greatest contribution to physics to have been the theory of photoelectric effect. No mention was made at all of his theory of relativity! 13. To try to force light and electrons into categories like waves and particles is to miss the essence of their existence because they are neither one not the other, though they have characteristics of both. One is reminded of the saying "The universe is not only stranger than we imagine, it is stranger than we can imagine". 14. According to de Broglie's hypothesis : (i) The motion of a particle of momentum p is guided by a wave of wavelength  , given by  = h /p. (ii) The square of the amplitude of a wave of wavelength  at a given point in s ace is proportional to the probability of finding a particle of momentum p at that point, where p = h/  k. In both equations p and  are related through Planck's constant (h). 15. Although the wavelength of matter waves can be experimentally determined, it is important to understand that they are not just like other waves because their frequency and phase velocity cannot be directly measured. In particular: The phase velocity of an individual matter wave is greater than the velocity of light and varies with wavelength. 16. The de Broglie wavelength of a typical macroscopic object (a 0.13 kg apple moving with a speed of 5m/s) is 33 1.0 10−  m/s. Clearly, this wavelength, which is much smaller than the diameter of an atom by a factor of 1023, is much too small to be observed in any macroscopic experiment. Thus, an apple could have wavelength as given by the de Broglie relation, and we would never notice it. In contrast, an electron with a kinetic energy of 10 eV has de Broglie wavelength of 3.88 A, which is comparable to the size of an atom or molecule and would clearly be significant. 17. Therefore, the de Broglie wavelength may be unobservable in macroscopic systems but is all-important in atomic systems. 18. To represent a particle velocity, a superposition of matter waves with different wavelengths, amplitudes and phases must be chosen to interfere constructively over a limited region of space. The resultant wave packet or group can then be shown to travel with the same speed as the classical particle. 19. For a 11 understanding of light, we have to regard it as a wavicle (wave + particle ), a wave-particle object t that sometimes behaves like a classical wave and classical particle, and at other times like a mixture of the two! 20. Classical mechanics becomes invalid when the particle's de Broglie wavelength is comparable to or smaller than the smallest dimensions of the experiment.