Gauge Institute Journal Volume 15, No 3, August 2019 H. Vic Dannon Zero Point Energy Does Not Imply Planck's Radiation Law, And Does Not Replace the Quantum Assumption H. Vic Dannon [email protected] March 1999 Revised July 2019 Abstract Zero point energy was introduced by Planck in 1912 in his second radiation law. In the derivation of his Radiation Law, Planck assumed that the radiation energy of his oscillator includes Zero Point Energy 1 w , 2 even at absolute temperature zero. Planck was unaware of his assumption. He believed that quantized energy implied the Zero Point Energy. Actually, Planck's Zero Point Energy is an assumption independent of the well-established fact that energy is quantized. In 1913, Einstein-Stern proposed that Zero Point Energy implies Planck's radiation law with no need for assuming quantized energy. 1 Gauge Institute Journal Volume 15, No 3, August 2019 H. Vic Dannon We show here that Einstein-Stern Dipole Model that assumes Zero Point Energy fails to produce Planck's Radiation Law, and is inconsistent with it. In 1916, Einstein gave up quietly on that claim. His final treatment of Radiation does not mention Zero Point Energy. But in 1969 that claim was revived in a paper by Boyer. Boyer never supplied any derivation of the radiation law, with or without quantum assumptions. We show here that his proposed equation leads to a Ricatti type differential equation, with infinitely many solutions, that are inconsistent with the radiation law. Keywords: Planck’s radiation law, Zero Point energy, Quantum hypothesis, Bose statistics, Entropy radiation law. Electromagnetic Radiation, Heat radiation, Blackbody Radiation, Radiation Quantum, PACS 12.90.+b. 2 Gauge Institute Journal Volume 15, No 3, August 2019 H. Vic Dannon Contents 1. Planck's Zero Point Energy 2. Einstein-Hopf Radiator 3. Einstein-Stern Zero Point Energy 4. Zero Point Energy of the Solution to Einstein-Stern Radiator 5. Einstein-Stern Dipole Model has Infinitely Many Solutions 6. Removing Zero Point Energy from the Einstein-Stern Radiator is Inconsistent with Planck's Radiation Law 7. Adding Zero Point Energy to the Einstein Model is Inconsistent with Planck's Radiation Law 8. Einstein's Giving Up on Zero Point Energy 9. Boyer's Revival of Einstein's Dead Radiator 10. Boyer Radiator has infinitely many solutions 11. Removing the Zero Point Energy from Boyer's Model is Inconsistent with Planck's radiation law 12. Adding Zero Point Energy to Boyer's Model is Inconsistent with Planck's Radiation Law 3 Gauge Institute Journal Volume 15, No 3, August 2019 H. Vic Dannon 1. Planck’s Zero Point Energy In 1912, Planck derived his second radiation law that includes Zero Point Energy by assuming that Zero Point Energy. Unaware of his assumption, he believed that his quantum hypothesis implied the zero point energy. Had he utilized his 1901 derivation to obtain his second radiation law, as we do here, he might have noticed that assumption. His 1912 derivation masked the assumption of his Zero Pint Energy. Thus, he was unable to refute Einstein’s claim that zero point energy alone without quantum hypothesis can imply the first radiation law. Planck was not aware of the necessity of both the ZPE and the quantum hypothesis in his models for the derivation of his second radiation law that includes Zero Point Energy. 1.1 Planck's Assumption of Zero Point Energy Planck’s assumptions in his 1912 paper define a harmonic oscillator with evenly spaced energy levels differing by w . The probability that the oscillator will radiate its energy is p The probability that it will retain it is 4 Gauge Institute Journal Volume 15, No 3, August 2019 H. Vic Dannon q= 1 - p The probability that the oscillator’s energy is between 0 and w is . p1 The probability that the energy is between w and 2w is = ... p2 p 1 q The probability that the energy is between2w and 3w is = 2 p3 p 1 q ............................................ We have = + + + + + 1p1 p 2 p 3 .... pn .... = + + 2+ + n- 1 + p1(1 q q .... q ....) 1 = p . 1 1 - q Hence, = - = p1 1 q p and = n-1 pn pq The oscillator’s entropy is = - - - - - s kp1log p 1 kp 2 log p 2 kp 3 log p 3 kp 4 log p 4 .... = - - - 2 2- 3 3 - kplog p kpq log pq kpq log pq kpq log pq .... logp+ log q log p+ 2log q log p+ 3log q = - + + 2+ - + + 2 + - kqq(1 ...)log ppk (1 2 qq 3 ...) pqq log ... 1 d =p( q+ q2+ q 3 + ...) 1-q dq 1 -1 1-q 1 = -klog p- k pq log q (1- q )2 5 Gauge Institute Journal Volume 15, No 3, August 2019 H. Vic Dannon 1 - p = -klog p- k log(1- p ) p 1 = k-log p- log(1- p )+ log(1- p ) p 1 1 = k- log(1- p )+ log - 1 pè p ø 1 1é ù 1 = k- log p- ë log(1- p )- log p û + log - 1 p p è p ø 1 1 1 1 = k log- - 1 log - 1. p pè pø è p ø To obtain the Radiation Law with Zero Point Energy, the oscillator must have at the first energy level, between 0, and w , an average energy of 1 w with probability , that is, 1 w 2 p1 2 p1 at the second energy level, between w , and 2w , an average energy of 3 w with probability , that is, 3 w 2 p2 2 p2 at the third energy level, between 2w , and 3w , an average energy of 5 w with probability , that is, 5 w 2 p3 2 p3 ........................................ the average energy of the oscillator must be = 1w + 3w + 5 w + u2 p1 2 p 2 2 p 3 ... = 1 w + + 2 + 2 p(1 3 q 5 q ...) 6 Gauge Institute Journal Volume 15, No 3, August 2019 H. Vic Dannon = 1wp(24+ q+ 6 q2+ ...)- 1 w p (1+ q+ q 2 + ...) 2 2 + + 2 + 2(1 2q 3 q ...) 1= 1 1-q p = wp(1+ 2 q+ 3 q2 + ...) - 1 w 2 d {q+ q2+ q 3 +...} dq d = wp{ q+ q2+ q 3 + ...} - 1 w dq 2 1 -1 1-q 1 = wp - 1 w - 2 2 (1q ) 1 p2 1 = w - 1 w p 2 1 1 = w - 1 + w è p ø 2 Here we note the assumption of 1 w Zero Point Energy, which 2 was assigned the nonzero probability p . Thus, 1u 1 - 1 = - , p w 2 1= u + 1 p w 2 1 1 1 1 s= k log- - 1 log - 1 p pè pø è p ø u1 u 1 u 1 u 1 = k + log + - - log - . èw 2ø èw 2ø èw 2ø èw 2 ø 7 Gauge Institute Journal Volume 15, No 3, August 2019 H. Vic Dannon ¶s 1 u 1 1 1 u 1 1 = k log + + - k log - + ¶u w èw 2ø w w èw 2 ø w u + 1 k w = log 2 w u - 1 w 2 ¶ Since s = 1 , ¶u T u + 1 k w 1 log 2 = w u- 1 T w 2 u 1 + w w 2 = ekT u - 1 w 2 w 1 1 u+ w = ekT u - w 2è 2 ø w w 1 kT+ = kT - we1 e 1 u 2 è ø è ø w 1ekT + 1 u = w 2 w ekT - 1 w 1ekT - 1+ 2 = w 2 w ekT - 1 1 w = w + . 2 w ekT - 1 8 Gauge Institute Journal Volume 15, No 3, August 2019 H. Vic Dannon This derivation’s assumptions are equivalent to the harmonic oscillator given by the Schrödinger equation. The assumptions that guarantee the radiation law with ZPE for the harmonic oscillator model include the evenly spaced energy levels, the ZPE, and the quantum hypothesis. We see that the quantum hypothesis in Planck’s derivation of his radiation law, does not imply the zero point energy. It is the hidden assumption of ZPE in the oscillator’s average energy that leads to ZPE in the radiation law. Thus, Planck’s derivation does not establish the existence of ZPE. It only recovers what was assumed at the start. 1.2 The Likelihood of Zero Point Energy From 1= u + 1 p w 2 = 1+ 1+ 1 2w 2 ekT - 1 w kT = e , w ekT - 1 we have w ekT - 1 p = w ekT 9 Gauge Institute Journal Volume 15, No 3, August 2019 H. Vic Dannon w - = 1 - e kT Zero Point Energy is assumed with probability. The ZPE appears in the radiation law for u()w with probability w - p= 1 - e kT . Computing with - h = (6.6260755)1034 Joul/Kelvin n ∼ (5)⋅ 1014 Cycles/sec, for visible light - k = (1.380658)⋅ 1023 Joul/Kelvin ° T = 300 Kelvin, for room Temperature hn - ∼ -79.98692773 kT w - p= 1 - e kT - ∼ 1 - e 80 » 1. That is, Planck assumed Zero Point Energy in his derivation with almost 100% certainty. Planck’s “discovery” of zero point energy was instrumental in prompting Einstein-Stern to look for an experimental confirmation in Eucken’s specific heat data. Their analysis precedes Mulliken’s observation of zero point energy in vibrations of isotopes. [Mulliken]. 10 Gauge Institute Journal Volume 15, No 3, August 2019 H. Vic Dannon 2. Einstein-Hopf Radiator In 1910, Einstein and Hopf utilized an oscillating dipole radiator that was used by Planck to obtain r()w , the radiation energy volume density per frequency in the frequency interval [,]w w + dw , in terms of u()w , the average energy per mode in [,]w w + dw .