Theoretical Predictions For Cooling A Solid To Its Ground State

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1 Theoretical Predictions For Cooling A Solid To Its Ground State W. C. Troy Abstract The development of quantum computers is a central focus of research. A major goal is to drain all quanta from a solid at a positive temperature T 0 > 0, leaving the object in its ground state [10]. In 2010 the first complete success was reported when a quantum drum was cooled to its ground state at T 0 = 20mK [6]. However, as the number of quanta (q decreases to zero, the theoretical behavior of temperature (T is poorly understood. Current theory, based on the Bose- Einstein equation for a single vibrating atom, predicts that (i thermal energy is present at every positive temperature, and (ii T 0 as q 0 [14]. Here we prove that these predictions are false. Our proof is based on analyzing the Einstein and Debye specific heat models, which form natural extensions from a single vibrating atom to solids with large numbers of atoms. We reexamine the derivations of each model, and prove that mathematical errors arise in the resultant Bose-Einstein equations when T 0. Our resolution of these errors proves that T T 0 > 0 when q 0. Moreover, using experimental data in [6], we predict that T 0 = 9.1mK, in close agreement with the 20mK experimental claim. In 2009 and 2010, four groups, Park and Wang [7] Schliesser et al [12], Groblacher et al [5], and Rocheleau et [8] used used laser cooling techniques to the temperature of solids to a value where the number of remaining quanta are 63, 37, 30 and 4 respectively. Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, U.S.A. 1

2 In 2010 O Connell et al [6] achieved complete success by reducing the number q of qunta in their quantum drum to zero. At T = T 0 quantum effects are expected, e.g. superposition of states [6, 14]. The key to their breakthrough was to use a ceramic wafer whose natural frequency of vibration, 6 GHz., is much higher than frequencies in solids in previous studies. This advance allowed them to reach the ground state at T 0 = 20 mk. with 93% probability. In 2011 Teufel et al [14, 15] reduced the number of quanta in a drumlike membrane to less than one. Their device remains in the ground state for up to 100 microseconds, which is significantly longer than the 6 nano second result of O Connell et al [6]. Theory. The results described above apply to solids consisting of trillions of atoms. A Schrodinger equation modeling approach, which describes the behavior of each atom, is analytically intractable. Teufel et al [15] avoid this difficulty by assming that a single resonating oscillator represents the behavior of the entire solid. They assume that n, the mean number of quanta in the solid satisfies the Bose-Einstein equation n = 1 exp (, (0.1 hν kt 1 where T is temperature, k is Boltzman s constant, h is Planck s constant and ν is the resonant frequency. Analysis of (0.1 predicts that (i thermal energy is present at every positive temperature, and (ii T 0 as q 0. decreases to zero as the number of quanta decreases to zero. However, Teufel et al [15] claim that, to prepare an oscillator in its ground state, its temperature need only be reduced to a value satisfying 0 < kt < hν, so that 0 < n < 1 and less than one quantum of energy remains. O Connell et al [6] also point out that most approaches focus on measuring the behavior of a single mechanical resonance. They observe that cooling a mechanical resonance to its ground state is typically an enormous challenge as it requires temperatures such that T << hν k. (0.2 For their ceramic solid, this citerion is 0 < T << 30mK. In contrast to the predictions of (0.1, they claim that all quanta have been removed from their solid, with 93% probability, at T = 20mK. Their theoretical results have the following drawbacks: 2

3 (i They do not give rigororous arguments which either prove or disprove the claim that a solid can reach its ground state at a positive temperature. (ii Given that the ground state is reached at a critical positive temerature T 0 > 0, criterion (0.2 does not give an accurate prediction for the value of T 0. A sharp estimate for T 0 needs to be developed to give theoretical backing for their experimental claims. Our primary goal is to resolve the issues in (i-(ii, and give a firm connection between theory and the experimental results described above. Let q denote the number of quanta of energy, i.e. energy in excess of ground state energy, in the solid, and define T 0 = lim q 0 + T We focus on answering the following fundamental questions: Question 1. Is T 0 > 0? Question 2. If T 0 > 0, does the solid reach its ground state at T = T 0? To answer Questions 1 and 2 we investigate Einstein s 1907 model for specific heat of a solid [2]. The Einstein model is based on the following assumptions: (A1 Each of the N atoms in the lattice of the solid is a 3D quantum harmonic oscillator, and q > 0 quanta of energy have been added to the lattice. (A2 Each atom is attached to a preferred position by a spring. (A3 The oscillators have the same frequency, ν > 0. O Connell et al [6] and Teufel et al [14, 15] model the behavior of their solids with a single oscillator whose resonant frequency is representative of the entire material. The Einstein model, which also assumes monochromatic vibrations of the atoms, is the natural extension of the single oscillator model since it allows us to stduy how the collective behavior of the entire population of atoms affects the solid. Thus, the Einstein model is an ideal starting point to investigate Questions 1 and 2. Appendix A constains a discussion of the 1913 Debeye model which allows for a range of frequencies. Einstein showed that specific heat, at constant volume, of a solid is given by ( ǫ C V = 3N A k kt 2 exp( ǫ kt ( exp( ǫ kt 1 2, 0 < T <, (0.3 3

4 where N A is Avogadro s number, ǫ = hν is the quantum of energy, and ν is the characteristic frequency. Formula (0.3 has the following basic property: Let ǫ k be fixed. Then (see Figure 1, C V is well defined for all T > 0 and lim C T 0 + V = 0. (0.4 Property (0.4 is widely quoted in textbooks and reference books throughout the statistical mechanics and physics literature (e.g. see Prathia [11], p. 175 or Schroeder [12], p It follows from (0.4 that temperature can attain any positive value. An important implication of our investigation of Questions 1 and 2 is a proof that the lowest possible temperature is positive, and can go no lower. Thus, both parts of (0.4 are false. We emphasize that our main focus is not on studying specific heat properties per se. Our central aim is to understand the theoretical behavior of temperature and energy as the quanta are drained from a solid. Towards this end, our analytic technique is to reexamine each detail of the derivation of (0.3. There are two standard methods to derive (0.3. We follow the first, the micro canonical ensemble approach [11, 13], since it is simpler and based on standard statistical mechanics principles which allow us to precisely predict how temperature and energy change as the quanta are drained from the solid. The second method, the macro canonical approach, uses partition functions [3, 11]. Partition functions by themselves do not show the effect of decreasing the number of quanta. However, the details of their derivation do show such effects. Our first step is to show how expressions for internal energy, entropy and temperature lead to specific heat formula (0.3. We show how a previously unobserved mathematical error arises in one of the expressions as q 0 +. We then present our approach to resolve the error. Theorem 0.1 describes our new results. The possible energy states of the atoms are E k = ǫ ( n k + 1 2, 1 k N. Thus, internal energy is given by U = N k=1 ( ǫ n k + 1 = ǫq + U 0, (0.5 2 where q = N k=1 n k is the number of quanta and U 0 = N ǫ 2 is ground state energy. The Boltzman formula for entropy is S = k ln(w, where W is the number of ways that q quanta can be distributed over N = 3N degrees of freedom. Thus, W = (q+n 1! q!(n 1! 4

5 and S = k ln ( (q+n 1! q!(n 1!. Thus, S = k (ln((q + N 1! ln(q! ln((n 1!. (0.6 To proceed further, the standard statistical mechanics approach [11, 13] is to use Stirling s approximation ln(m! = M ln(m M, M >> 1, to simplify (0.6. Applying Stirling s approximation to (0.6, when q >> 1 and N >> 1, gives S = k ((q + N 1 ln((q + N 1 q ln(q (N 1 ln(n 1, (0.7 Temperature, T, and specific heat, C v, satisfy 1 T = S U and C V = 1 U n T, (0.8 where n = N N A is the number of moles. It follows from (0.5, (0.7 and (0.8 that 1 T = ds dq dq du = k ( q + N ǫ ln 1. (0.9 q An inversion of (0.9 gives the Bose-Einstein equation for the entire solid; q = N 1 e ǫ kt 1. (0.10 Ignore -1 in the numerator since N >> 1, substitute (0.10 into (0.5, and get U = ǫn e ǫ kt 1 + N ǫ 2. (0.11 Finally, set N = 3N and n = N N A, and combine (0.11 with C V = 1 n C V = 3N Aǫ 2 kt 2 du. This gives dt exp( ǫ kt ( exp( ǫ kt 1 2. (0.12 The Mathematical Error. We now focus on the Bose-Einstein equation (0.10. It follows from (0.10 that lim T = 0. (0.13 q 0 + Property (0.13 predicts that temperature must tend to absolute zero as the quanta are drained from the solid. This conclusion contradicts the 2010 experimental results of O Connell et al [6], whose solid is drained of all quanta at T = 20mK. However, 5

6 (0.13 is mathematically illegitimate since the derivations of both (0.10 and (0.13 are dependent on the Stirling approximation ln(q! = q ln(q q, which is accurate when q >> 1, but rapidly losses accuracy as q 0 + : the relative error is 13% at q = 10, and 188% at q = 2. To corect this unacceptable loss of accuracy, our approach is to return to formula (0.6 for S, and replace each term of the form ln(m! with the exact value ln(m! = ln(γ(m + 1, where Γ is the Gamma function. This transforms (0.6 into S = k (ln(γ(q + N ln (Γ(q + 1 ln (Γ(N q 0. (0.14 In Theorem 0.1 we use (0.14 to derive new expressions for T, U and C V which, for the first time, are completely accurate over the entire range 0 < q <. Theorem 0.1 Let ǫ k > 0 be fixed. Temperature is given by T = ǫ k ( Γ (q + N 1 Γ(q + N Γ (q + 1 > 0 q 0. (0.15 Γ(q + 1 T is an increasing function of q. Its lowest possible value is T 0 = lim q 0 + T = ǫ k where γ.577 is Euler s constant. Energy satisfies Specific heat is given by C V = k n ( Γ (N 1 Γ(N + γ > 0, (0.16 lim U = N ǫ = Ground State Energy. (0.17 q ( Γ (q + N 2 [ ( Γ(q + N Γ (q + 1 d Γ (q + N ] 1 Γ(q + 1 dq Γ(q + N Γ (q + 1 > 0 q 0. Γ(q + 1 (0.18 The lowest possible value of C V is positive, and is given by C 0 = lim q 0 + C V = k n ( Γ (N 2 [ ( d Γ Γ(N + γ (q + N 1 dq Γ(q + N Γ (q + 1 > 0. Γ(q + 1 q=0] (0.19 The proof of Theorem 0.1, and further generalizations, are given in Appendix A. We make the following conclusions: 6

7 6 C v 5 4 Diamond New C v C v 0.9 x Blowup New C v C T(Kelvin Einstein C 0.3 v T 0 T Figure 1: Left panel: Einstein s C v function (0.3 vs. T for diamond: data from [3], frequency ν = Superimposed is the new C v formula (0.18 Right panel: blowup showing both functions vs T when 23.1 < T < 23.3 The red dot corresponds to T 0 = , the temerature where ths solid reaches its ground state. (I Comparison with experiment. It follows from (0.15 and (0.16 that temperature decreases to a positive value T 0 > 0 as q 0 +, and cannot go lower. This result overturns prediction (0.13, the long standing theoretical belief derived from the Bose-Einstein equation (0.10, that lim q 0 + T = 0. Also, (0.17 shows that energy decreases to its ground state as q 0 +, i.e. as T T 0 +. To test the predictive ability of formula (0.16, we use data from O Connell et al [6], whose device reaches its ground state at T = 20mK. To compare their result with the prediction of (0.16, we use their values for frequency, ω = 6GHz, and N = 10 trillion atoms. Thus, we substitute ǫ = νh = ω h = and N = 3N = into (0.16 and obtain T 0 = 9.1mK. (0.20 This theoretical prediction for T 0 is in close agreement with the 20mK experimental result of O Connell et al [6].. (II Diamond. When n = 1 mole, (0.16 predicts that T 0 depends only on frequency: T 0 = ν, 0 < ν <. (0.21 Formula (0.21 shows that T 0 can be large, potentially reaching room temperature, 7

8 when ν is large. For example, when ν = , the value given by Einstein [3] for diamond, (0.21 gives (see Figure 1, right panel T 0 = 23.2K This high value is expected since diamond is an extremely hard substance, hence its natural frequency is high. This prediction is also in agreement with the experimental results of O Connell et al [6], who showed that a hard, high frequency solid could more easily reach its ground state at a positive temperature. (III Specific heat. We conclude from (0.18-(0.19 that C v decreases to a positive value C 0 > 0, which is independent of frequency, as q 0 +, and cannot go lower (Figure 1, right panel. This result overturns (0.4, the widely acclaimed, long standing belief that C v is well defined for all T > 0, and lim T 0 + C v = 0. References [1] Debye, P. Zur Theorie der spezifischen Warme. Annalen der Physik (Leipzig 39(4, 789 (1912 [2] Einstein, A. Die Plancksche Theorie der Strahlung und die Theorie der spezifischen Warme., Annalen der Physik, 22, (1907 [3] Eyring, H, Henerson, D., Stover, B. J. & Eyring, E. M. Statistical Mechanics and Dynamics. Wiley, New York, New York (1964, second Edition (1982 [4] Griffiths, D. Introduction To Quantum Mechanics. Pearson Prentice Hall, (1995, second Edition (2005 [5] Groblacher, S., Hertzberg, B. Vanner, M., Cole, G., Gigan, Schwab, S. K. & Aspelmeyer, M. Demonstration of an ultracold micro-optomechanical oscillator in a cryogenic cavity. Nature Physics, 5, (2009 [6] O Connell, A. D., Hofheinz, M., Ansmann, M., Bialczak, C., Lenander, M., Lucero, E., Neeley, E. M., Sank, D., Wang, H., Weides, M., 8

9 Wenner, J., Martinis, J. M. & Cleland, A. N. Quantum ground state and single-photon control of a mechanical resonator. Nature 464, (2010 [7] Park, Y & Wang, H. Resolved-sideband and cryogenic cooling of an optomechanical resonator. Nature Phyiscs 5, (2009 [8] Rocheleau, T., Ndukum1, T., Macklin, C., Hertzberg, J. B., Clerk, A. A. & Schwab, K. C. Preparation and detection of a mechanical resonator near the ground state of motion. Nature 463, (2010 [9] Planck, M. On the law of distribution of energy in the normal spectrum. Annalen der Phgysik, 4, 553 (1901 [10] Powell, D.Moved by Light Science News, 179, No. 10, (2011 [11] Prathia, D. K. Statistical Mechanics. Addison-Wesley (1999 [12] A. Schliesser, A., Arcizet1, O. R. Rivire1, O. R., Anetsberger, G. & Kippenberg, T. J. Resolved-sideband cooling and position measurement of a micromechanical oscillator close to the Heisenberg uncertainty limit. Nature Physics 5, (2009 [13] Schroeder, D. V. An Introduction To Thermal Physics. Addison-Wesley (1999 [14] Teufel, J. D., Li, D., Allman, M. S., Cicak, K., Sirois, A.J., Whittaker, J. D. & Simmonds, R. W. Circuit electromechanics cavity in the strong coupling regime. Nature 471, 204 (2011 [15] Teufel, J. D., Donner, T., LI, D., Harlow, W., Allman, M. S., Cicak, K., Sirois, A.J., Whittaker, J. D., Lehnert, K. W., & Simmonds, R. W. Sideband Cooling Micromechanical Motion to the Qunatum Ground State. Nature, to appear (2011 [16] Yariv, A. Quantum Electronics. Wiley (1975 9

10 1 Appendix A. This Appendix has two parts: Part I. The proof of Theorem 0.1 Part II. We analyze Debye s model of specific heat of a solid [1]. This model, which is an extension of the Einstein model, takes into account the effects of variable frequency in the lattice of atoms. Our goal is the same as for the Einstein model. We show that, as the quanta are drained from the solid, its temperature approaches a positive value T 0 > 0, and the solid reaches the ground state at T = T 0. Part I: Proof of Theorem 0.1. The first step is to prove crucial properties of entropy, S, and internal energy, U. We conclude from (0.14 and (0.5 ( ds Γ dq = k (q + N Γ(q + N Γ (q + 1 and du = ǫ q > 0. (1.1 Γ(q + 1 dq Temperature. It follows from the fact that 1 = S, and (1.1 that T U 1 T = ds dq dq du = k ( Γ (q + N ǫ Γ(q + N Γ (q + 1 q 0. (1.2 Γ(q + 1 This, together with the property ( d Γ (x = dx Γ(x implies that T = ǫ k This proves property ( te xt 1 e tdt > 0 x > 0, (1.3 ( Γ (q + N 1 Γ(q + N Γ (q + 1 > 0 q 0. (1.4 Γ(q + 1 Next, let ǫ, k and N be held fixed in (1.4. Then the only way that T can increase, or decrease, is to change q. We claim that T is an increasing function of q. The first step in proving this claim is to differentiate (1.4. This gives dt dq = ǫ ( Γ (q + N 2 ( k Γ(q + N Γ (q + 1 d Γ (q + N Γ(q + 1 dq Γ(q + N Γ (q + 1 Γ(q + 1 q 0. (1.5 To prove that T is an increasing function of q, we need to show that the right side of (1.5 is positive when q 0. The second term is negative since d 2 ( Γ (x t 2 e xt = dx 2 Γ(x 0 1 e tdt < 0 x > 0. (1.6 10

11 Therefore, the right side of (1.5 is positive, hence dt dq > 0 q 0. (1.7 Thus, T is an increasing function of q. From this and (1.4 we conclude that the lowest possible value of temperature is positive, and is given by lim q 0 + T = T 0 = ǫ k ( Γ (N Γ(N Γ (1 Γ(1 1 = ǫ k since Γ (N > 0, Γ(N > 0, Γ(1 = 1 and Γ (1 = γ. ( Γ (N Γ(N + γ Energy. It follows from (0.5 that internal energy, U, is given by 1 > 0, (1.8 U = ǫq + U 0, (1.9 where U 0 = N ǫ 2 is ground state energy, and q = N k=1 ( nk is the total number of quanta of thermal energy. We conlude from (1.9 that This proves property (0.17. lim U = N ǫ q Specific Heat. We conclude from (0.8 and (1.1 that It follows from (1.5 and (1.7 that C V = 1 U n T = 1 U n q = Ground State Energy. (1.10 q T = ǫ q n T. (1.11 ( q Γ T = k (q + N 2 [ ( ǫ Γ(q + N Γ (q + 1 d Γ (q + N ] 1 Γ(q + 1 dq Γ(q + N Γ (q + 1 > 0 q 0. Γ(q + 1 (1.12 Combining (1.11 and (1.12 gives C V = k n ( Γ (q + N 2 [ ( Γ(q + N Γ (q + 1 d Γ (q + N ] 1 Γ(q + 1 dq Γ(q + N Γ (q + 1 > 0 q 0. Γ(q + 1 (1.13 From (1.13 it follows that C V is an increasing function of q. Thus, the lowest possible value of specific heat is found by letting q 0 +. This gives lim q 0 + C V = C 0 = k n ( Γ (N Γ(N + γ 2 [ d dq 11 ( Γ (q + N 1 Γ(q + N Γ (q + 1 > 0. Γ(q + 1 q=0] (1.14

12 This proves property (0.19. The proof of Theorem 0.1 is now complete. Part II: The Debye Model Here we present the results of our study of the Debye theory of a solid. Our primary is goal is the same as in our analysis of the Einstein model. That is, we show that as the solid is drained of its quanta, its temperature approaches a positive value, and its energy approaches the ground state. Towards this end we again let q denote the number of quanta in the solid, and define T 0 = lim q 0 + T. We focus on the following fundamental questions: Question 1. Is T 0 > 0? Question 2. If T 0 > 0, does the solid reach its ground state at T = T 0? The first step is to state the key features of the Debye theory. In 1913 Debye [1] derived the following formula for specific heat, at constant volume, of a solid: ( T 3 TD /T C v = 9Nk TD 0 x 4 e x (e x 1 2dx, 0 < T <, (1.15 where N >> 1 is the number of atoms, k is Boltzman s constant, T is temperature, and T D is a characteristic temperature. A widely acclaimed property is that C v is well defined for all T > 0 and lim C T 0 + v = 0. (1.16 Remark. We show that T 0 > 0, hence T cannot approach zero and both claims in (1.16 are false. Debye theory assumes that the atoms are three dimensional quantum oscillators, and that corresponding to the oscillators there is a finite number, D > 0, of energy states 0 < E 1 < E 2 < < E D. (1.17 Associate with each energy E i is a degeneracy d i, which is assumed to satisfy d i >> 1, and a frequnecy ω i satisfying E i = ω i h, 1 i D, (

13 A key assumption is that sound waves propagate through the crystal, and that along the frequency set {ω i }, the energy density is given by ρ(ω i, T i = hω i 3 ( ( π 2 c 3 exp hωi, 1 i D, 0 < T i <, (1.19 k B T 1 i where c denotes the speed of sound, and T i is the temperature of the i th mode. As in our study of the Einstein model, we again emphasize that our focus is not on deriving the specific heat formula. Instead, our central aim is to determine the theoretical behavior of temperature and energy as the quanta are drained from the solid. For this, we reexamine the statistical mechanics based details behind the derivation of (1.19. We show that, corresponding to the frequency set {ω i }, there is an associated set of Bose-Einstein equations, and we prove that a mathematical error arises in each of these equations. To understand how the error arises, and how we rectify these errors, it is necessary to give complete detials of the drivation of the Bose-Einstein equations. We follow the details in Griffiths [4] (pp Consider an arbitrary potential, for which the one-particle energies are 0 < E 1 < E 2 < with degeneracies d 1, d 2,.. Suppose that N particles, all with the same mass, are put into this potential, and consider a configuration (N 1, N 2, N 3,.., (1.20 where N i is the number of particles with energy E i. Let Q(N 1, N 2,... denote the number of states corresponding to (1.20. Assume that the particles are bosons. Then Q = i=1 (N i + d i 1!. (1.21 N i!(d i 1! In thermal equilibrium, each state with a given particle number N, and total energy E = N i E i, (1.22 i=1 is equally probable. The most probable configuration is the one for which Q is a maximum, subject to constraint (1.22. Since ln(q is an increasing function, it suffices to find the maximum of ln(q subject to (1.22. The standard approach is to use the method of Lagrange multipliers. Consider the function ( G = ln(q + β E N i E i, ( i=1

14 where the constant β is the Lagrange multiplier. Equation (1.23 expands into ( G = (ln ((N i + d i 1! ln (N i! ln ((d i 1! + β E N i E i. (1.24 i=1 i=1 Stirling s approximation is ln(m! = M ln(m M, M >> 1. Applying Stirling s approximation to (1.24, under the assumprion that each N i >> 1 and d i >> 1, gives G = i=1 ((N i + d i 1 ln (N i + d i 1 N i ln(n i (d i 1 ln(d i 1 + β (E i=1 N i E i. (1.25 To maximize Q, subject to (1.22, we need to solve G N k = 0. This gives Solving (1.26 for N i, we obtain 0 = ln((n i + d i 1 ln (N i + βe i (1.26 N i = d i 1 exp (βe i 1. (1.27 Set β = 1 kt, E i = hω i and ignore the -1 in the numerator in (1.27 since d i >> 1. Then the number N i of quanta having energy E i satisfies the Bose-Einstein equation N i = d i exp ( hω i kt i 1, (1.28 An inversion of (1.28 shows that T = hω i k ln ( d i N i + 1. (1.29 The Mathematical Error. It follows from (1.29 that, for each i 1, lim T = 0. (1.30 N i 0 + Property (1.30 predicts that temperature tends to zero as the number N i of quanta tends to zero. However, it follows exactly as in our analysis of the Einstein model, that property (1.30 is mathematically illegitimate since the derivations of (1.28, (1.29 and (1.30 depend on applyting Stirling s approximation ln(n i! = N i ln(n i N i, (

15 to reduce (1.24 to (1.25. Approximation (1.31 is accurate when N i >> 1, but rapidly losses accuracy as N i 0 +. To corect this unacceptable loss of accuracy, our approach is exactly the same as in our analysis of the Einstein solid. That is, in (1.24 we replace every term of the form ln(m! with the exact value ln(m! = ln(γ(m + 1, where Γ is the Gamma function. Repetition of the steps leading to (1.29 gives, for each i 1, the following new formula for temperature of the i th mode which is completely accurate for all N i 0 : T i = ǫ k ( Γ 1 (N i + d i Γ(N i + d i Γ (N i + 1 > 0 N i 0. (1.32 Γ(N i + 1 Remark. Formula (1.32 is exactly the same as (0.15 in Theorem 0.1 We now focus our attention on the Debye model, which assumes that the energies of the atoms are restricted to the values E 1, E 2,.., E D which satisfy which satisfy (1.17 and (1.18. It follows from (1.32 that T i is an increasing function of N i. Its lowest value is T i 0 = lim N i 0 + T i = ǫ k ( Γ 1 (d i Γ(d i + γ > 0, 1 i D, (1.33 where γ.577 is Euler s constant. Next, let q = D i=1 N i denote the total number of quanta in the solid. We conclude from the definition of q, (1.32 and (1.33 that T 0 = lim q 0 + T = min { T i 0 1 i D} > 0. (1.34 Finally, we show that the solid reaches its ground state at T = T 0. First, the energy of the solid is given by U = i=d i=1 ( Ni E i + U i 0, (1.35 where U0 i = d E i i 2 = Ground State Energy of the ith mode. (1.36 Thus, (1.35 and (1.36 imply that, as the quanta are drained from the solid, energy satisfies i=d lim U = q 0 + i=1 d i E i 2 = Ground State Energy. (1.37 In summary, properties (1.34 and (1.37 demonstrate that as the qunata are drained from solid, its temperature approaches a positive value, T 0 > 0, and the solid reaches 15

16 ground state at T = T 0, as claimed. An important consequence of (1.34 and (1.37 is that the widely acclaimed specific heat properties stated in (1.16 must be false. That is, the limiting theoretical temperature T 0 is strictly postive, hence T cannot approach zero as (1.16 requires. 16