de Broglie inserted the Planck relation, Equation 1.3, for the energy and arrived at a relation between the wavelength and momentum of a photon: h (1.47) He then postulated that this relation also applied to material particles and that the wavelength of the matter wave is h (1.48) What is it that is doing the waving in the case of matter waves is, at this point, still not clear, but, for now let us simply note that this was indeed a brilliant step Experiments of the type described in Section 1. 1. 4 verified the validity of equate 1.48. It is interesting that the Bohr energy can be easily derived by simply requirin that a half-integral number of de Broglie wavelengths fit in each allowed Bohr or- bital circumference(see Problem 8). Why a half-integral number of wavelengths? Because that is what is required to establish a standing wave which, in this context, may be correlated with Bohr's stationary states. This is important because the de Broglie hypothesis is consistent with the Bohr model of the atom. 1.2.3 The Uncertainty Principle Among his many scientific accomplishments, one of Heisenbergs most important is his formulation of the uncertainty principle. For this, and other contributions, Werner Karl Heisenberg was awarded the Nobel Prize in Physics in 1932"for the creation of quantum mechanics, the application of which has, inter alia(among other hings), led to the discovery of the allotropic forms of hydrogen In his 1927 paper Heisenberg introduced the concept that position and momen tum could not be measured with unlimited precision. The principle may be stated Ax△px≥h/2 where Ax and A pr are the uncertainties in the measurements of position and mo- mentum,respectively. For simplicity we confine the discussion to one-dimension and specify the x-component of the momentum to emphasize that the principle holds only when the coordinate and the component of the linear momentum are the same It is important to understand that the uncertainties, Ar and Apx, are not the result of any flaw in asurement technique or apparatus. They are consequences of the wave-particle duality that, as we have seen above, is inherent in nature
1.2 Early Theory 19 de Broglie inserted the Planck relation, Equation 1.3, for the energy and arrived at a relation between the wavelength and momentum of a photon: p = h λ (1.47) He then postulated that this relation also applied to material particles and that the wavelength of the matter wave is λ = h p (1.48) What is it that is doing the waving in the case of matter waves is, at this point, still not clear, but, for now let us simply note that this was indeed a brilliant step. Experiments of the type described in Section 1.1.4 verified the validity of Equation 1.48. It is interesting that the Bohr energy can be easily derived by simply requiring that a half-integral number of de Broglie wavelengths fit in each allowed Bohr orbital circumference (see Problem 8). Why a half-integral number of wavelengths? Because that is what is required to establish a standing wave which, in this context, may be correlated with Bohr’s stationary states. This is important because the de Broglie hypothesis is consistent with the Bohr model of the atom. 1.2.3 The Uncertainty Principle Among his many scientific accomplishments, one of Heisenberg’s most important is his formulation of the uncertainty principle. For this, and other contributions, Werner Karl Heisenberg was awarded the Nobel Prize in Physics in 1932 “for the creation of quantum mechanics, the application of which has, inter alia (among other things), led to the discovery of the allotropic forms of hydrogen.” In his 1927 paper Heisenberg introduced the concept that position and momentum could not be measured with unlimited precision. The principle may be stated mathematically as xpx ≥ /2 (1.49) where x and px are the uncertainties in the measurements of position and momentum, respectively. For simplicity we confine the discussion to one-dimension and specify the x-component of the momentum to emphasize that the principle holds only when the coordinate and the component of the linear momentum are the same. It is important to understand that the uncertainties, x and px , are not the result of any flaw in our measurement technique or apparatus. They are consequences of the wave–particle duality that, as we have seen above, is inherent in nature.
I Introduction The principle can be illustrated by examining the familiar single-slit diffraction experiment from physical optics as is illustrated in Fig. 1.6. If the monochromatic waves of wavelength A that are incident on the single slit of width a are light waves, the diffraction pattern observed on the screen will be proportional to sin- P/p where P=asin e/A From physical optics, the position of the first minimum is sin b= (1.50) which we take to be the angular spread We can also imagine the incident "waves" to be monoenergetic electrons is, electrons that all have the same kinetic energy p/2me. Therefore, all of the electrons will have the same de broglie wavelength A = h/p and the same pattern will be observed. (Recall we have not yet specified what it is that is doing the wav ing.) What causes the diffraction pattern in either case, photons or electrons, is uncertainty in the x-component of momentum. This uncertainty is given by px= psI 入 (1.51) Note that, from relativity and from the de broglie relation, Equation 1. 48, the mo- mentum is(h/A) for either photons or electrons. Now, the uncertainty in x, Ar, is the width of the slit a, so we have recovered the uncertainty principle. The impor tant point to be made here is that the wave properties of light and matter and th uncertainty principle are inextricably linked. There is another experiment that can be carried out using electrons and slits that demonstrates the wave nature of matter. It is analogous to the two-slit experiment first performed by Young in 1801. We will not go into detail about this experiment here other than to note that it also demonstrates the interference properties of matter waves n Equation 1.49 is not the only uncertainty relationship. Because h has units of gular momentum it is, at least dimensionally, correct(always a good start). There are, however, other combinations of variables, the products of which have the correct units, that could provide suitable uncertainty relations. An important such relation is Fig 1. 6 Schematic diagram experiment showing the ntensity of the diffraction pattern. The secondary maxIma e been exaggerated for clarity
20 1 Introduction The principle can be illustrated by examining the familiar single-slit diffraction experiment from physical optics as is illustrated in Fig. 1.6. If the monochromatic waves of wavelength λ that are incident on the single slit of width a are light waves, the diffraction pattern observed on the screen will be proportional to sin2 β/β2 where β = πa sin θ/λ. From physical optics, the position of the first minimum is sin θ = λ a (1.50) which we take to be the angular spread. We can also imagine the incident “waves” to be monoenergetic electrons, that is, electrons that all have the same kinetic energy p2/2me. Therefore, all of the electrons will have the same de Broglie wavelength λ = h/p and the same pattern will be observed. (Recall we have not yet specified what it is that is doing the waving.) What causes the diffraction pattern in either case, photons or electrons, is an uncertainty in the x-component of momentum. This uncertainty is given by px = p sin θ = h λ λ a (1.51) Note that, from relativity and from the de Broglie relation, Equation 1.48, the momentum is (h/λ) for either photons or electrons. Now, the uncertainty in x, x, is the width of the slit a, so we have recovered the uncertainty principle. The important point to be made here is that the wave properties of light and matter and the uncertainty principle are inextricably linked. There is another experiment that can be carried out using electrons and slits that demonstrates the wave nature of matter. It is analogous to the two-slit experiment first performed by Young in 1801. We will not go into detail about this experiment here other than to note that it also demonstrates the interference properties of matter waves. Equation 1.49 is not the only uncertainty relationship. Because has units of angular momentum it is, at least dimensionally, correct (always a good start). There are, however, other combinations of variables, the products of which have the correct units, that could provide suitable uncertainty relations. An important such relation is Fig. 1.6 Schematic diagram of a single-slit diffraction experiment showing the intensity of the diffraction pattern. The secondary maxima have been exaggerated for clarity
1. 2 Early Theory the energy-time uncertainty relation. We can rationalize the relationship by imagin ing a particle having kinetic energy E= p-/2m. The uncertainty in energy is then 4E=P4 Ax But Ax/u=At, the uncertainty in time which leads to the relation △EMt≥b/2 (1.53) This energy-time uncertainty relation has many consequences. For exampl even if the resolution of the photographic plates were perfect and there were no other mechanism for broadening the lines observed in atomic spectroscopy, some lines would be broader than others This is because the lifetimes of the initial states involved in the transitions can differ by orders of magnitudes. These finite lifetimes, which may be regarded as At in the uncertainty relation, are accompanied by an uncertainty in the energy. Thus, the photon energies are not truly monoenergetic and a broadened line is observed on the photographic plate. In practice there are other mechanisms that serve to broaden the lines, but modern spectroscopic techniques can eliminate these so the" natural"linewidth, that associated with the uncertainty in the energy of the state, can be observed 1. 2. 4 The Compton Wavelength revisited It was noted in our discussion of the Bohr atom that, although not strictly correct, it provides a good model for visualization and it leads to correct orders of magnitudes and scaling for atomic parameters. In view of the uncertainty principle, we wish examine the question of whether it is reasonable to consider the electron(with its wavelike properties)to really be"pointlike"as compared with the size of an atom, To this end we ask the question: when are the nonrelativistic treatment offered by the Bohr theory and the(nonrelativistic) quantum mechanics in this book valid Evidently, relativity will become important when the kinetic energy T of any of the particles (including massless particles) is sufficient to cause creation of particle pairs, that is, when T N mec2. Let us imagine performing a Compton scattering experiment in which we try to confine an electron within a small distance 8(we ill use one-dimension for simplicity). The more precisely we try to confine the electron, that is, the smaller we wish to make 8(which is essentially the uncertainty in position), the greater is the uncertainty in momentum Ap of the photon. As Ap
1.2 Early Theory 21 the energy–time uncertainty relation. We can rationalize the relationship by imagining a particle having kinetic energy E = p2/2m. The uncertainty in energy is then E = pp m ≥ v /2 x = /2 x/v (1.52) But x/v = t, the uncertainty in time which leads to the relation Et ≥ /2 (1.53) This energy–time uncertainty relation has many consequences. For example, even if the resolution of the photographic plates were perfect and there were no other mechanism for broadening the lines observed in atomic spectroscopy, some lines would be broader than others. This is because the lifetimes of the initial states involved in the transitions can differ by orders of magnitudes. These finite lifetimes, which may be regarded as t in the uncertainty relation, are accompanied by an uncertainty in the energy. Thus, the photon energies are not truly monoenergetic and a broadened line is observed on the photographic plate. In practice there are other mechanisms that serve to broaden the lines, but modern spectroscopic techniques can eliminate these so the “natural” linewidth, that associated with the uncertainty in the energy of the state, can be observed. 1.2.4 The Compton Wavelength Revisited It was noted in our discussion of the Bohr atom that, although not strictly correct, it provides a good model for visualization and it leads to correct orders of magnitudes and scaling for atomic parameters. In view of the uncertainty principle, we wish to examine the question of whether it is reasonable to consider the electron (with its wavelike properties) to really be “pointlike” as compared with the size of an atom, ∼ a0. To this end we ask the question: when are the nonrelativistic treatment offered by the Bohr theory and the (nonrelativistic) quantum mechanics in this book valid? Evidently, relativity will become important when the kinetic energy T of any of the particles (including massless particles) is sufficient to cause creation of particle pairs, that is, when T ∼ mec2. Let us imagine performing a Compton scattering experiment in which we try to confine an electron within a small distance δ (we will use one-dimension for simplicity). The more precisely we try to confine the electron, that is, the smaller we wish to make δ (which is essentially the uncertainty in position), the greater is the uncertainty in momentum p of the photon. As p
I Introduction increases, so does the energy of the incident photon. In terms of the uncertainty principle this means that h 8=△p Multiplying numerator and denominator by c makes the denominator the energy of the(massless)photon in accord with the relativistic formula given in Equation 1.45 If we now require that this energy be less than the amount of energy required create an electron-positron pair, namely, w mec, we have 2cA hc Thus, the minimum dimension for 8 in which the electron may be localized before relativistic considerations are required is the order of the Compton wavelength. We may, therefore, regard the Compton wavelength as the intrinsic quantum mechani- cal"size"of an electron. The Compton wavelength is, very roughly, the minimum length in which a particle may be localized according to quantum mechanics. To localize it further would require such a high momentum that the energy would be sufficient for pair production Returning to the Bohr model, we may compare the size of the electron Ac with the size of the atom According to Equation 1.31 (4丌∈0) =/(4z6oa (1.56) where a is the fine structure constant, Equation 1.35. Therefo terms of the bohr radius the size of the electron, the Compton wavelength, is which means that the electron's intrinsic quantum mechanical size is roughly two orders of magnitude smaller than the diameter of the atom we conclude, therefore
22 1 Introduction increases, so does the energy of the incident photon. In terms of the uncertainty principle this means that δ = /2 p (1.54) Multiplying numerator and denominator by c makes the denominator the energy of the (massless) photon in accord with the relativistic formula given in Equation 1.45. If we now require that this energy be less than the amount of energy required to create an electron–positron pair, namely, ∼ mec2, we have δ = c 2cp = c 2mec2 ∼ λc (1.55) Thus, the minimum dimension for δ in which the electron may be localized before relativistic considerations are required is the order of the Compton wavelength. We may, therefore, regard the Compton wavelength as the intrinsic quantum mechanical “size” of an electron. The Compton wavelength is, very roughly, the minimum length in which a particle may be localized according to quantum mechanics. To localize it further would require such a high momentum that the energy would be sufficient for pair production. Returning to the Bohr model, we may compare the size of the electron λc with the size of the atom ∼ a0. According to Equation 1.31, a0 = (4π 0) 2 mee2 = (4π 0) c e2 mec = 1 α λc (1.56) where α is the fine structure constant, Equation 1.35. Therefore, in terms of the Bohr radius the size of the electron, the Compton wavelength, is λc = αa0 ∼ 1 137a0 (1.57) which means that the electron’s intrinsic quantum mechanical size is roughly two orders of magnitude smaller than the diameter of the atom. We conclude, therefore,
1. 2 Early Theory hat the bohr model of the atom is viable in the sense that it can, indeed be viewed as a point electron orbiting a stationary nucleus 1.2.5 The Classical Radius of the electron While the fact that the Compton wavelength is much smaller than the Bohr radius validates the assumption of nonrelativistic quantum mechanics, there is another quantity of interest, the classical radius of the electron. The assumptions that go into the calculation of the classical radius are of dubious validity, but comparison of it with the Compton wavelength of the electron is interesting. The calculation is simple. We imagine that the charge e of the electron is dis- ibuted uniformly over the surface of a sphere of radius Re, the classical radius of the electron. We now assume that the energy required to assemble this charge, W, is equal to the rest energy of the electron. There are several ways to calculate the energy of the charge distribution. One way is to use the fact that the energy to assemble the charges is the sum of charge multiplied by the electric potential Adapted to the current problem this means that we must integrate the product of the surface charge density and the potential at r= Re over the surface of the sphere and multiply by one-half. Thus, 2(4丌R 4r∈0/dS 2(4r∈oR equating W to the rest energy of the electron mec2leads 2(4e 4∈o which shows that the classical radius of the electron is roughly two orders of magni- ude smaller than the Compton wavelength of the electron and four orders of magni- tude smaller than an atom. Thus, although the concept upon which the calculation of the classical radius of the electron is suspect, it is consistent with the visualization of the electron as a " probability cloud, "the radius of which is the Compton wavelength. Moreover, physicists love it when quantities can be expressed in te known parameters multiplied by the fine structure constant
1.2 Early Theory 23 that the Bohr model of the atom is viable in the sense that it can, indeed, be viewed as a point electron orbiting a stationary nucleus. 1.2.5 The Classical Radius of the Electron While the fact that the Compton wavelength is much smaller than the Bohr radius validates the assumption of nonrelativistic quantum mechanics, there is another quantity of interest, the classical radius of the electron. The assumptions that go into the calculation of the classical radius are of dubious validity, but comparison of it with the Compton wavelength of the electron is interesting. The calculation is simple. We imagine that the charge e of the electron is distributed uniformly over the surface of a sphere of radius Re, the classical radius of the electron. We now assume that the energy required to assemble this charge, W, is equal to the rest energy of the electron. There are several ways to calculate the energy of the charge distribution. One way is to use the fact that the energy to assemble the charges is the sum of charge multiplied by the electric potential. Adapted to the current problem this means that we must integrate the product of the surface charge density and the potential at r = Re over the surface of the sphere and multiply by one-half. Thus, W = 1 2 e 4π R2 e e 4π 0Re S d S = 1 2 e2 4π 0Re (1.58) Equating W to the rest energy of the electron mec2 leads to Re = 1 2 e2 4π 0 1 mec2 = 1 2 e2 4π 0c mec ∼ αλc ∼ α2 a0 (1.59) which shows that the classical radius of the electron is roughly two orders of magnitude smaller than the Compton wavelength of the electron and four orders of magnitude smaller than an atom. Thus, although the concept upon which the calculation of the classical radius of the electron is suspect, it is consistent with the visualization of the electron as a “probability cloud,” the radius of which is the Compton wavelength. Moreover, physicists love it when quantities can be expressed in terms of previously known parameters multiplied by the fine structure constant.