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The Derivation of the Planck Formula

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The Derivation of the Planck Formula Chapter 10 The Derivation of the Planck Formula Topics The Planck formula for black-body radiation. Revision of waves in a box. Radiation in thermal equilibrium. The equipartition theorem and the ultraviolet catastrophe. The photoelectric e®ect. The wave-particle duality. Quantisation of radiation and the derivation of the Planck spectrum. The Stefan-Boltzmann law. 10.1 Introduction In the ¯rst lecture, we stated that the energy den- sity of radiation per unit frequency interval u(º) Figure 10.1. Spectrum of black body radiation for black-body radiation is described by the Planck x3 f(x) dx / dx formula (Figure 10.1), ex ¡ 1 in terms of the dimensionless frequency x. 8¼hº3 1 u(º) dº = dº (10.1) 3 hº=kT 1.5 c (e ¡ 1) ........................ ...... ........ .... ....... ... ...... ... ..... ..... ..... ... .... ¡34 ... .... ... .... where Planck's constant, h = 6:626 £ 10 J s. ... .... ... .... ... .... ... .... ... .... In this lecture, we demonstrate why quantum con- .. .... hº ... .... .. .... 1.0 .. .... x = .. .... cepts are necessary to account for this formula. The .. .... .. .... kT .. .... .. .... .. .... .. .... programme to derive this formula is as follows. .. .... f(x) .. .... .. .... .. .... .. .... .. .... .. ..... .. ..... .. ..... .. ..... 0.5 .. ..... ² First, we consider the properties of waves in .. ..... .. ..... .. ..... .. ..... .. ..... a box and work out an expression for the ra- .. ..... .. ..... .. ...... .. ....... .. ......... .. ......... diation spectrum in thermal equilibrium at ... .......... .. ............ ... ................ .... ......... temperature T . 0.0 ...... 0 1 2 3 4x 5 6 7 8 9 10 1 The Derivation of the Planck Formula 2 ² Application of the law of equipartition of en- ergy leads to the ultraviolet catastrophe, which shows that something is seriously wrong with the classical argument. ² Then, we introduce Einstein's deduction that light has to be quantised in order to account for the observed features of the photoelectric e®ect. ² Finally, we work out the mean energy per mode of the radiation in the box assuming the radiation is quantised. This leads to Planck's radiation formula. This calculation indicates clearly the necessity of introducing the concepts of quantisation and quanta into physics. 10.2 Waves in a Box - Revision Let us revise the expression for an electromagnetic (or light) wave travelling at the speed of light in some ar- bitary direction, say, in the direction of the vector r. If the wave has wavelength ¸, at some instant the ampli- tude of the wave in the r-direction is 2¼r A(r) = A sin 0 ¸ (see Figure 10.2). In terms of the wave vector k, we can write Figure 10.2. The properties of sine and A(r) = A0 sin(k ¢ r) = A0 sin kr; cosine waves. where jkj = 2¼=¸. Wave vectors will prove to be very important quantities in what follows. The wave travels at the speed of light c in the r-direction and so, after time t, the whole wave pattern is shifted a distance ct in the positive r-direction and the pattern is 0 A0 sin kr , where we have shifted the origin to the point ct along the r-axis such that r = r0 + ct. Thus, the expression for the wave after time t is 0 A(r; t) = A0 sin kr = A0 sin(kr ¡ kct): But, if we observe the wave at a ¯xed value of r, we ob- serve the amplitude to oscillate at frequency º. There- fore, the time dependence of the wave amplitude is sin(2¼t=T ) where T = º¡1 is the period of oscillation of the wave. Therefore, the time dependence of the wave at any point The Derivation of the Planck Formula 3 is sin !t, where ! = 2¼º is the angular frequency of the wave. Therefore, the expression for the wave is A(r; t) = A0 sin(kr ¡ !t); and the speed of the wave is c = !=k. 10.3 Electromagnetic Modes in a Box { More Revision Consider a cubical box of side L and imagine waves bouncing back and forth inside it. The box has ¯xed, rigid, perfectly conducting walls. Therefore, the electric ¯eld of the electromagnetic wave must be zero at the walls of the box and so we can only ¯t waves into the box which are multiples of half a wavelength. The ¯rst few examples are shown in Figure 10.3. In the x-direction, the wavelengths of the waves which can be ¯tted into the box are those for which l¸ x = L 2 where l takes any positive integral value, 1, 2, 3, ::: . Figure 10.3. Waves which can be ¯tted into a Similarly, for the y and z directions, box with perfectly conducting walls. m¸ n¸ y = L and z = L; 2 2 where m and n are also positive integers. The expression for the waves which ¯t in the box in the x-direction is A(x) = A0 sin kxx Now, kx = 2¼=¸x where kx is the component of the wave-vector of the mode of oscillation in the x-direction. Hence, the values of kx which ¯t into the box are those for which ¸x = 2L=l and so 2¼l ¼l k = = ; x 2L L where l takes the values l = 1; 2; 3 ::: . Similar results are found in the y and z directions: ¼m ¼n k = ; k = : y L z L Let us now plot a three-dimensional diagram with axes kx, ky and kz showing the allowed values of kx, ky and kz. These form a regular cubical array of points, each of them de¯ned by the three integers, l; m; n (Figure 10.4). This is exactly the same as the velocity, or mo- mentum, space which we introduced for particles. The Derivation of the Planck Formula 4 The waves can oscillate in three dimensions but the com- ponents of their k-vectors, kx, ky and kz, must be such that they are associated with one of the points of the lattice in k-space. A wave oscillating in three dimen- sions with any of the allowed values of l; m; n satis¯es the boundary conditions and so every point in the lat- tice represents a possible mode of oscillation of the waves within the box, consistent with the boundary conditions. Thus, in three-dimensions, the modes of oscillation can be written A(x; y; z) = A0 sin(kxx) sin(kyy) sin(kzz) (10.2) To ¯nd the relation between kx; ky; kz and the angular frequency ! of the mode, we insert this trial solution into the three-dimensional wave equation Figure 10.4. Illustrating the values of l and @2A @2A @2A 1 @2A + + = : (10.3) m which result in components of wavevectors @x2 @y2 @z2 c2 @t2 which can ¯t in the box. The time dependence of the wave is also sinusoidal, A = A0 sin !t and so we can ¯nd the dispersion relation for the waves, that is, the relation between ! and kx; ky; kz, by substituting the trial solution into (10.3); !2 jkj2 = (k2 + k2 + k2) = : x y z c2 where k is the three-dimensional wave-vector. Now, ¼2 k2 = k2 + k2 + k2 = (l2 + m2 + n2); x y z L2 and so !2 ¼2 ¼2p2 = (l2 + m2 + n2) = ; (10.4) c2 L2 L2 where p2 = l2 + m2 + n2. 10.4 Counting the Modes We now return to statistical physics. We need the number of modes of oscillation in the frequency in- terval º to º + dº. This is now straightforward, since we need only count up the number of lat- tice points in the interval of k-space k to k + dk corresponding to º to º + dº. We carried out a similar calculation in converting from dvx dvy dvz to 4¼v2 dv. The Derivation of the Planck Formula 5 In Figure 10.4, the number density of lattice points is one per unit volume of (l; m; n) space. We are only interested in positive values of l, m and n and so we need only consider one-eighth of the sphere of radius p. The volume of a spherical shell of radius p and thickness dp is 4¼p2 dp and so the number of modes in the octant is µ ¶ 1 dN(p) = N(p) dp = 4¼p2 dp: 8 Since k = ¼p=L and dk = ¼ dp=L, we ¯nd Note on the Polarisation of Electro- magnetic Waves L3 dN(p) = k2 dk: 2¼2 But, L3 = V is the volume of the box and k = 2¼º=c. Therefore, we can rewrite the expression V V 8¼3º2 4¼º2V dN(p) = k2 dk = dº = dº 2¼2 2¼2 c3 c3 Finally, for electromagnetic waves, we are always Illustrating the electric and magnetic allowed two independent modes, or polarisations, ¯elds of an electromagnetic wave. The E per state and so we have to multiply the result by and B ¯elds of the wave are perpendicular two. Because of the nature of light waves, there are to each other and to the direction of prop- two independent states associated with each lattice agation of the wave. There is an indepen- point (l; m; n). The ¯nal result is that the number dent mode of propagation in which E and of modes of oscillation in the frequency interval º B are rotated through 90± with respect to to º + dº is the direction of propagation C. Any po- 8¼º2V dN = dº larisation of the wave can be formed by c3 the sum of these two independent modes Thus, per unit volume, the number of states is of propagation. 8¼º2 dN = dº (10.5) c3 This is a really important equation. 10.5 The Average Energy per Mode and the Ultraviolet Catastrophe We now introduce the idea that the waves are in thermodynamic equilibrium at some temperature T .
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