is the minimum stopping potential for which the current between the plates reduces to zero.

Transcription

1 Module 1 :Quantum Mechanics Chapter 2 : Introduction to Quantum ideas Introduction to Quantum ideas We will now consider some experiments and their implications, which introduce us to quantum ideas. The choice of these will be influenced more by the logical need though some attention will be given to their historical importance. 2.1 Photoelectric effect Hertz found in 1887, that when the ultraviolet radiation is incident on the surface of an alkali metal (Na, K, Rb, Cs) plate, electrons are emitted. The number of these electrons and their energies can be studied by subjecting them to an electric field between the emitting plate and the receiving plate. Suppose an electron is emitted with kinetic energy. Then the maximum kinetic energy is given by where is the minimum stopping potential for which the current between the plates reduces to zero. Some important observations are: (1) The maximum kinetic energy and the stopping potential, (2.1) are independent of the intensity of the incident radiation. (2) They are proportional to, (2.2) where is the frequency of the incident radiation and is the critical frequency below which no electrons are emitted. (3) The electrons are emitted almost instantaneously without any noticeable time delay. These observations are not consistent with wave description of e.m. radiation. The results suggest (2.3) where the observed proportionality constant is Planck's constant. Einstein's interpretation (1905) was that radiation comes in energy quanta, photons, and their energy goes into overcoming the minimum binding energy called the work function, and into the kinetic energy of the emitted electrons. Some important points to be noted are: (1) Ultraviolet radiation is needed since and is of the order of 3-5 ev for alkali metals. The value of for visible radiation is about 2.5 ev. (2) Alkali metal plates are needed since they have the lowest work functions, 3-5 ev. (3) Only a small fraction of incident radiation, 5 percent is responsible for ejecting electrons, the rest is absorbed by the plate as a whole. (4) Photoelectric effect in atoms in the form of gases, is described as photo-ionization. Here since there is no collective absorption, number of photons absorbed is equal to the number of electrons ejected. (5)Energy for emission can come from heating, thermionic emission. In this case the rate of emission is determined by statistical properties in terms of and temperature T. (6) With the development of very high intensity radiation in the form of lasers, now we can have multiphoton absorption for which one has the relation

2 (2.4) for energy conservation, where is the number of photons absorbed by a single electron. It may be noted that momentum conservation relation is not used in the analysis of photoelectric effect since substantial amount of momentum is taken by the bulk plate which however is not directly measurable. The situation is similar to a wall and a ball case. For including momentum conservation relation, we need a small target. This is incorporated in Compton scattering where the target is a free electron or an atom. We note that for photons with zero mass, one has from special theory of relativity, (2.5) for photons as particles. One uses both energy and momentum conservation in the description of Compton scattering. 2.2 Compton Scattering (1922) When monochromatic X-ray radiation of wavelength is scattered by an electron in a thin film of metals, the scattered radiation has two major components in the intensity as a function of wavelength, one with unchanged wavelength, and one with shifted wavelength, (2.6) This is not expected for the scattering of a wave. However, it is explained quite accurately by regarding it as scattering of photon particles, each with energy and momentum by an electron or an atom. Consider the scattering of a photon with energy and momentum, scattered by a particle of mass at rest, which may be an electron or an atom. Let the scattered photon have energy and momentum making an angle with its initial momentum. Let the particle of mass, originally at rest have momentum after scattering, making an angle with the initial momentum of the photon. Then the conservation of momentum components parallel and perpendicular to the initial momentum of the photon, and the conservation of energy, lead to (2.7) where the last term is the relativistic energy of the scattered particle. Eliminating from the first two equations, one gets (2.8) The last equation in Eq.(2.7) leads to (2.9) Equating the two expressions for leads to (2.10)

3 In terms of, one gets (2.11) which leads us to identify the Compton wavelength in Eq.(2.6) as (2.12) which depends only on the mass of the scattering particle, and has a value of for an electron. The observed intensity of scattered X-rays has two peaks, one at given by in Eq.(2.11) with for an electron, and the other with unshifted wavelength. The unshifted wavelength is due to scattering by the atom as a whole which has a mass greater than the electron mass by a factor of about. For this is smaller by a factor of about compared with the electron Compton wavelength of, and one can take for this case. This is known as the Thomson component. Some significant points are: (1) That the shift in the wavelength is due to scattering by an electron is confirmed by the observation of the scattered electron for which Eq.(2.7) leads to (2.13) (2) The observed intensity has a spread around the peak values. The main reason for this is that the initial electron is in general not at rest but has some momentum with some spread. The correction due to this momentum leads to a spread in the observed wavelength in Eq.(2.11). (3) Though is independent of, the intensity of scattering increases with. Therefore it is easier to observe the Compton effect for higher frequency radiation, e.g. X-ray radiation. 2.3 Diffraction of matter particles The experiments described demonstrate that electromagnetic waves have particle-like properties in the form of photons. One could reverse the arguments and state that particles called photons have wavelike properties. Can one generalise this to other particles? For doing this we note that e.m. waves are associated with wave functions: (2.14) This is in a form easy to generalise to particles other than photons. It means that one can associate (2.15) as a wavelength for all particles with momentum. One can then expect interference and diffraction patterns for particles with characteristic wavelength called the de Broglie wavelength (1923). This was observed for electron beams scattered by crystal lattices. The experiments were done in two situations. In the earlier Davisson-Germer (1927) experiment, the incoming beam of electrons was incident perpendicular to the surface of monocrystals (Ni). The intensity

4 of the scattered beam was observed at angle Bright fringes were observed at angle, with respect to the line perpendicular to the surface. (2.16) n an integer, where is the separation between two successive scattering rows on the surface, and is the path difference in the wave components scattered by two successive rows. It is interesting to observe that similar fringes were observed for scattering of X-rays with wavelength and of electron beams with energy for which the de Broglie wavelength is approximately (2.17) In the slightly later experiments by Thomson and Tartakovsky (1927), the scattering was by polycrystalline materials. In this case, bright fringes were observed for (2.18) n is an integer, where is now the separation between two successive layers of the crystal, is the angle the incoming beam makes with the planar surface, the beam is scattered at an angle equal to the angle of incidence, and is the path difference for beam components scattered by two successive layers. Following important points should be noted. 1. The diffraction pattern of scattered particles is not because of interaction between the particles. The pattern remains the same when the intensity is varied, with electrons coming almost one at a time for low intensity. This suggests the pattern may be related to the probability for individual particles. 2. Since the pattern is independent of the interaction, we have interference for neutral particle beams, neutrons and molecules. One point is that for neutrons with, the particle energy is (2.19) Therefore we essentially need thermal neutrons, at. 3. X-ray, electron and neutron diffraction have become indispensible tools for the study of crystal surfaces and lattice structures. 4. Electrons at high energies, kev have very small de Broglie wavelength and hence are used in electron microscopes for studying very small specimens. 5. Neutrons interact mainly with the nuclei, and hence are important in the study of isotopes. 6. Since neutrons interact strongly with protons, neutron beams are important in the study of organic compounds. 7. Neutron beams are important in the study of magnetic materials. 8. The main constraint of neutron diffraction is that it is difficult to get high intensity neutron beams with nearly thermal energies. 2.4 Black-body radiation Historically, Planck's explanation (1900) of black-body radiation was the first quantum description of a physical phenomenon. Black-body is a perfect absorber of incident radiation. Because of thermal equilibrium, it is also the best emitter of radiation. It may be simulated by a hole grilled in a cavity. Experimentally, the radiation emitted by a black-body has a well-defined intensity distribution as a function of wavelength, which has a maximum at some wavelength. As the temperature of the black-body is raised, the radiation intensity increases at all wavelengths and shifts to a smaller value such that (2.20) known as Wien's displacement law (1993). To start with we have e.m. radiation described by the wave

5 equation (2.21) which follows from the Maxwell's equations. The solutions to this equations are of the type (2.22) where is the wave number. For the radiation confined to a cubic box of length, the boundary conditions imply (2.23) with being a positive integer (negative integers give the same modes). For this wave function to satisfy Eq.(2.21), one has (2.24) It is to be noted that collective dynamics of particle leads to normal modes with well-defined frequencies. The number of normal modes is equal to the number of particles. Now in statistical mechanics, every degree of freedom is assigned an energy. However, for oscillatory motion one has two terms, kinetic energy and potential energy. Therefore, one can assign an energy of for each mode. For the number of modes in the present case, one has for the number of modes with an upper limit of, (2.25) for the number of modes per unit volume, per unit frequency. Here, is introduced because are only positive integers, and is introduced because there are two degrees of freedom in polarization for. This leads to (2.26) for the energy density per unit volume, per unit frequency. This is known as Rayleigh-Jeans law which is appropriate for large, small. However it is not acceptable in other domains since it implies that energy density increases indefinitely as increases and total energy per unit volume becomes infinite. This is known as ultra-violet catastrophe. Max Planck reanalysed (1900) the black-body radiation with the new assumption that each mode is associated with an oscillator in the wall with allowed quantum energies, where is an integer and is known as Planck's constant. Using Maxwell-Boltzmann distribution, it leads to an average energy for each mode,

6 (2.27) Finally we have for the energy density per unit frequency, (2.28) which is the celebrated Planck's formula for black-body radiation density per unit volume per unit frequency. It leads to several important results: (1) The energy/area/time/frequency emitted at the surface can be deduced by considering the contribution from different volume elements inside, (2.29) where is the area vector perpendicular to the surface, is a unit vector pointing from the volume element towards the surface area element. In terms of spherical coordinates, this leads (2.30) where the limits for are taken to be for. (2) Total energy emitted per unit area per unit time, is (2.31) where the sum of can be deduced from Fourier series for and. This is stated as Stefan- Boltzmann law, (2.32)

7 in MKS units. (3) We deduce Wien's displacement law by first considering energy density per unit volume per unit wavelength, (2.33) Using the expression for in Eq.(2.28), one gets (2.34) For finding for which is a maximum, one has (2.35) Neglecting the exponential term, we get (2.36) which demonstrates Wien's displacement law. (4) Stefan-Boltzmann law in Eq.(2.32) allows us to get a good estimation of sun's surface temperature by considering the earth and the sun to be black-bodies in equilibrium. Taking and to be the surface temperatures of the earth and the sun, the total energy emitted and received by the earth are, (2.37 where is the radius of the earth, is the radius of the sun, is the distance of the earth from the sun. Equating the two for the equilibrium condition, one has (2.38) where the angular size of the sun as observed from the earth, is about degree. Taking The earth's surface temperature to be, we get (2.39)

8 2.5 Atomic spectra de Broglie hypothesis and its experimental verification in terms of diffraction experiments suggests other possibilities. For example, radiation or some other waves when confined to a region develop standing waves with well-defined frequencies which implies that the corresponding photons have specific discrete energies. One may therefore expect that particles in general, confined to a region may develop standing waves with discrete energies. Indeed one had observed discrete frequencies in atomic spectra from glowing vapour flames and atomic gas discharges. Balmer (1885) had observed discrete lines with wavelengths (2.40) in hydrogen spectrum, (2.41) mainly in the visible region. Then Rydberg (1890) suggested the form (2.42) This was quickly generalised to (2.43) with (Lyman, 1906), (Balmer, 1885), (Paschen, 1908), (Bracket, 1922) series. For other atoms also one could write a similar expression, (2.44) However, in general is rather complicated, one of the very useful relations being due to Rydberg, (2.45) where is known as the quantum defect which is approximately independent of. Similar but dark lines are observed in absorption spectra corresponding to changes in standing wave frequency of electron. Helium was first observed in the absorption lines of solar spectra (helios). It is very suggestive that each term could be associated with the energy of the electron with a standing wave. One then has (2.46) In particular one has for the hydrogen atom,

9 (2.47) Thomson tried to get these energies by considering the oscillatory motion of an electron embedded in a distribution of positive charge. However this is inconsistent with the observation of particle scattering from massive atoms for which a very good description of the observations is obtained by regarding the process as scattering of particles by a massive, point-like target with charge (Rutherford, 1911). Consider the scattering of particles by atoms in a thin film target. For each target particle, let be the cross-sectional area of the incoming beam for which the beam-particles are scattered into solid angle with respect to the target. Then the number of particles scattered into over time is (2.48) where is the particle flux in the incoming beam. Then (2.49) is the number of particles scattered per unit solid angle, per unit flux, per unit time, per target. This is described as the differential cross section. For the impact parameter which is the distance of approach for the incoming particle when it is far away, let the particle be scattered at angle, and for at angle. Then we have (2.50) where is the angular variation in the plane perpendicular to the incoming beam direction taken to be the axis. Now the magnitude of the incoming momentum is equal to the magnitude of the outgoing momentum, so that the change in the momentum vector is along the line from the target towards the nearest point of the projectile path, (2.51) where is the angle of scattering. Taking the angular position of the projectile with respect to the line towards its nearest position from the target to be, we have (2.52) In terms of the angular momentum, one gets (2.53) (2.54)

10 where is the kinetic energy of the incoming particle. Finally we get for the differential cross section, (2.55) with for the particle. Some important points to be noted are: (1) The dependence of on are verified experimentally. (2)The results are not valid for, for large, when the effect of other atoms will be significant, nor are they valid for when the nuclear interaction will be significant for small. (3) It is interesting to observe that quantum description also leads to the same expression for as in Eq.(2.55), which supports Rutherford's model of a point-like, massive nucleus with charge. (4) The model of an atom with electrons moving around point-like nucleus is unable to explain the stability of an atom, or the observation of discrete spectra of radiation emitted or absorbed by an atom. 2.6 Bohr's model (1913) Bohr's model of an atom is based on Rutherford's model of the atom, but it modifies the classical laws to be applicable to the motion of an electron around the nucleus. The modifications in the form of postulates are: 1. Electrons that are bound move around in discrete orbits and do not emit radiation when they remain in these discrete orbits. 2. For the allowed circular orbits, one has the quantum condition (2.56) 3. Emission or absorption of radiation takes place when the electron undergoes transition between these discrete orbits, and the frequency of the radiation is given by (2.57) where and refer to the initial and final states. For applying the model to the one electron atom or ion, we first separate out the centre of mass motion, with (2.58) where and refer to the electron and the nucleus. It then follows that the total kinetic energy is (2.59) where is the reduced mass. Leaving out the c.m. motion, we have for the total energy, (2.60)

11 With central force relation, we get for the circular orbits, (2.61) Using Bohr's quantization condition in Eq.(2.56), one obtains (2.62) for the quantised energies, and the Bohr's radius for the orbits. Some examples of the model are,,,, positronium, muonic atoms. The quantum nature of the energy levels was directly verified by the Franck-Hertz (1914) experiment in which atoms were excited by electrons with appropriate kinetic energies. It was also confirmed by the appearance of specific spectral lines in the radiation from the excited atoms. The ideas about the discrete orbits are a bit ad-hoc. Also, Bohr's model considers only circular orbits. It is essentially a one electron atom/ion model which does not allow simple generalization to many electron systems. However, the idea of an atom with discrete electron orbits has retained its utility for qualitative understanding. Sommerfeld generalised the model to include non-circular orbits in terms of other coordinates. He suggested quantization conditions for periodic orbits, Bohr-Sommerfeld quantization condition, (2.63) for each pair of coordinate and its canonical momentum, with the integration over the period. For the one electron atom/ion, one has with polar coordinates and, (2.64) for closed orbits, where and are integers. Carrying out the somewhat complicated integration, it leads to (2.65) which has a degeneracy of order, with the same energy. The Bohr- Sommerfeld quantization condition in Eq.(2.63) allows us to analyse the quantum properties in 1- dimension also. For example, in the case of simple harmonic oscillator it leads to

12 (2.66) Carrying out the integration, one obtains (2.67)