The Genesis of Quanta (1890-1910)

Manoj K. Harbola

December 23, 2017 ABV IIITM Gwalior 1 Well established results in 1890 (1) Thermodynamics

First law: dQ dE  pdV

Second law: Entropy maximum at equilibrium. 2 S is negative E 2

December 23, 2017 ABV IIITM Gwalior 2 (2) Electromagnetic theory of radiation

Energy density of a plane electromagnetic wave : 1 u   E 2 2 0

Intensity of a plane electromagnetic wave:

I cu

Momentum carried by a plane electromagnetic wave: I p  u c

December 23, 2017 ABV IIITM Gwalior 3 Energy/second

1 2 = c   E  area 2 0

1 2 Energy content =  E  volume 2 0

1 2 Force =  0 E  area 2

December 23, 2017 ABV IIITM Gwalior 4 (3) Kinetic theory of gases

3 R Average energy per moleculein a gas T 2 N

(4) Statistical Mechanics developed by Boltzmann

Developed well but not widely accepted.

December 23, 2017 ABV IIITM Gwalior 5 Radiation from heated bodies

Emissive power = power radiated per unit area per unit wavelength normally in unit solid angle I d e d    dt d dA

Absorptive power = fraction of radiation falling on a body that is absorbed

radiation absorbed a dQ

December 23, 2017 ABV IIITM Gwalior 6 Black bodies

All the radiation of any wavelength falling on a black body gets completely absorbed i.e. none of it is either transmitted or reflected

a 1

Note: Lamp black absorbs about 96% of visible light. It transmits long wavelength radiation. A new material made Carbon nanotubes absorbs about 99.955%.

Of all objects at a given temperature, a black body has

the maximum emissive power. Call it E.

December 23, 2017 ABV IIITM Gwalior 7 How to design a black body ?

Theoretical arguments:

(1)In a hollow cavity of emissive power E' put a small

body of emissive power e and absorptive power a e (2) Energy balance at equilibrium gives E  a

(3) Take the small body to be a perfect black body  a 1

(4) Then the energy balance gives E E (5) Radiation inside a closed cavity is the same as that of a black body.

e (6) Kirchhoff’s law: E a December 23, 2017 ABV IIITM Gwalior 8 Diffused radiation in a closed cavity

I

Radiation is isotropic with energy density u u Pressure on the walls p  3 1 Intensity of radiation coming out I  cu 4 u and I have the same information December 23, 2017 ABV IIITM Gwalior 9 Black Body Radiation

• Spectrum of Black body : Plot of spectral density i.e. intensity per unit wavelength range versus wavelength

December 23, 2017 ABV IIITM Gwalior 10 December 23, 2017 ABV IIITM Gwalior 11 Power radiated from a black body

Stefan’s law (experimental, 1879):

Total power radiated T 4

Intensityof radiation  T 4

December 23, 2017 ABV IIITM Gwalior 12 Boltzmann’s derivation (1884):  S   U  TdS dU  pdV  T     p  V T  V T u p  3

 S   p  1  u   U   (uV )             u V V  V T  T V 3  T V  T  T

1  u  4 T   u   4 3  T V 3 u T

December 23, 2017 ABV IIITM Gwalior 13 Shape of the black-body spectrum curve

Wilhelm Wien (Nobel Prize 1911)

December 23, 2017 ABV IIITM Gwalior 14 Wien’s analysis (1893, 1894):

(1) A cavity expanding (contracting) adiabatically would cool down (heat up).

(2) The radiation inside would change to the black-body spectrum at the new temperature.

(3) Thus if we know the spectrum at one temperature, we should be able to connect it to the spectrum at another temperature by adiabatically expanding or contracting a cavity.

December 23, 2017 ABV IIITM Gwalior 15 Relation between size and temperature of a spherical cavity during an adiabatic expansion: u dQ dU  pdV dQ 0 U uV p  3 u du 4 dV 0 Vdu  udV  dV or  3 u 3 V 4  uV 3 constant 4 u aT 4 and V  r 3  Tr constant 3  1 December 23, 2017 T ABV IIITMr Gwalior 16 Change in wavelength during adiabatic expansion:

Why should a given wavelength change? It changes because of reflection from moving walls of the cavity (Doppler shift).

Consider light being reflected from a moving mirror

2u cos θ u    c

December 23, 2017 ABV IIITM Gwalior 17 Now look at light being reflected inside an expanding spherical cavity: 2u cos u  (per reflection)   r c

r θ Number of reflections in time t u c u  t 2r cos

u Total change in wavelength in time t ut r      r r   r December 23, 2017 ABV IIITM Gwalior 18 Wien’s displacement law:  1  1 T  r and   r    T T constant

Experimental verification (Lummer and Pringsheim, 1899)

December 23, 2017 ABV IIITM Gwalior 19 Wien’s displacement law (experimental verification):

Temp. K max (micron) maxT(-K)

621 4.53 2814 723.0 4.08 2950 908.5 2.96 2980 998.5 2.96 2956 1094.5 2.71 2966 1259.0 2.35 2959 1460.4 2.04 2979 1646.0 1.78 2928

December 23, 2017 ABV IIITM Gwalior 20 Form of the spectrum curve:

T1 As the temperature changes from T1 to T2

4 u(1) u(1 )d1  T1  T2   u( )   2 u(2 )d2  T2 

d1 d2 T u( ) u( ) d  2 d 1 2 1 2  5  5 constant T1 T1 T2

5 5 1 u(1 ) 2 u(2 ) constant

December 23, 2017 ABV IIITM Gwalior 21 Experimental verification (Lummer and Pringsheim, 1899)

-5 Temp. K E at max ET (ergs sec-1cm-3) (ergs sec-1cm-3K-5) 621 2.026 2190 723.0 4.28 2166 908.5 13.66 2208 998.5 21.5 2166 1094.5 34.0 2164 1259.0 68.8 2176 1460.4 145.0 2184 1646.0 270.6 2246

December 23, 2017 ABV IIITM Gwalior 22 Finally, the general form of the energy density u(T):

When we go from one curve to the other such that T remain unchanged, the energy density satisfies 5 u() constant

Thus the constant should be a function of (T) f (T ) Therefore in general u (T )   5

 c  c 3   Since d d   d u (T)       2  T  December 23, 2017 ABV IIITM Gwalior 23 Wien’s formula for the Black-body spectrum (1896): 1 Frequency of radiation proportional to mv2 a kinetic energy of molecular dimension 2

Intensity of radiation proportional to number of resonators 1  mv2 / kT 2  a / kT E (T ) e  ( ) e  ( )

() a function of molecular velocity (and therefore of frequency or wavelength) only A E  e b / T  5 December 23, 2017 ABV IIITM Gwalior 24 Excerpts from the Nobel Prize presentation speech

“In 1893 Wien published a theoretical paper which was destined to acquire the utmost importance in the development of radiation theory”.

“Wien's displacement law provides half the answer to the problem”.

“The importance of Wien's displacement law extends in various directions”.

“The method has successfully been applied to the determination of the temperature of our light sources, of the sun and of some of the fixed stars, and has yielded extremely interesting results”.

“In 1894 he deduced a black body radiation law……at short wavelengths, it agrees with the …….. investigations by Lummer and Pringsheim”.

December 23, 2017 ABV IIITM Gwalior 25 Stage is now set for

to enter the scene

December 23, 2017 ABV IIITM Gwalior 26 Max Planck

Studied under Helmholtz and Kirchhoff at Berlin.

Read (self-studied) Clausius’ papers.

Fascinated by the second law of thermodynamics. Investigations into how it could be applied to different physics problems.

In 1894, turns his attention to the problem of black-body radiation (recall that the analysis of Wien had come out in that year).

Nobel Prize 1918

December 23, 2017 ABV IIITM Gwalior 27 Planck starts by analyzing the energy equilibrium between a radiating dipole and a radiation field (1899). u() = spectral density of radiation E() = average energy of the dipole 8 2 u( )  E( ) c 3 Dipoles (on the walls of the cavity) of frequency  have average energy E() Spectral energy density of radiation in the cavity is u()

If we calculate the average energy per oscillator, weDecember know 23, 2017 the spectralABV density. IIITM Gwalior 28 What did Planck do that was different?

“……at the very outset I hit upon the idea of correlating not the temperature of the oscillator but its entropy with its energy……While a host of outstanding physicists worked on the problem of spectral energy distribution……….every one of them directed his efforts solely towards exhibiting the dependence of intensity of radiation on temperature……….Nobody paid any attention to the method I adopted and I could work out my calculations completely at my leisure, with absolute thoroughness, without fear of interference or competition”.

December 23,(M. 2017 Planck, 1950, ScientificABV IIITMautobiography Gwalior and Other papers) 29 Entropy of an Oscillator (June 1900) 8 2 Recall two formulae u( )  E( ) c 3 A A 3 u  e b/T  u( )  e b / cT  5 c4

  / T

This leads to E( )  e (  0 and   0)  S  1 and through    to  E V T E  E  2 S 1 1 S  ln     2    e  E  E December 23, 2017 ABV IIITM Gwalior 30 (October 1900) Rubens and Kurlbaum show inadequacy of Wien’s law at low frequencies and high temperatures. Since radiation law depends on the ratio (/T), Wien’s law does not hold for low values of this parameter.

They show that for low values of (/T), the intensity is proportional to the temperature T and tell Planck about their results.December 23, 2017 ABV IIITM Gwalior 31 The result

Planck gets back to work and a produces a formula. The paper is read at the meeting of the German Physical Society on October 19 1900.

December 23, 2017 ABV IIITM Gwalior 32 “On an Improvement of the Wien’s radiation law”, M. Planck, Verhandl der Deutschen Physikal. Gessellsch, 2, 202 (1900)

For low values of (/T) u T  E T

2 dS 1 dS 1 d S 1    which gives  dE T dE E dE 2 E 2 d 2S 1 For high values of (/T) Wien formula is good so  dE 2 E Combine the two formulae to get d 2 S a  (a  0, b  0) dE 2 E(b  E) December 23, 2017 ABV IIITM Gwalior 33 d 2S a 1 dS a  E   gives   ln  dE 2 E(b  E) T dE b  b  E 

b Invert this to get E  eb / aT  1

2 8 Combine this with to obtain u( )  3 E( ) c 8 2 b u( )  c 3 eb / aT  1

December 23, 2017 ABV IIITM Gwalior 34 8 2 b u( )  c 3 eb / aT  1

3   General form of the energy density u (T)     T 

8 3 1 u( )  b  and 3  / aT This gives c e  1

 E   E  E E  Entropy of the oscillator S a1  ln1   ln          

December 23, 2017 ABV IIITM Gwalior 35 On October 25, 1900 Rubens and Kurlbaum compare their new measurements with Planck’s formula and find it to give precise agreement with experiments. “However, even if the radiation formula should prove itself to be absolutely accurate, it would still only have, within the significance of a happily chosen interpolation formula, a strictly limited value”. (M. Planck, 1918, Nobel Lecture) “For this reason, I busied myself, from then on, that is, from the day of its establishment, with the task of elucidating a true physical character for the formula, and this problem led me automatically to a consideration of the connection between entropy and probability, that is, Boltzmann's trend of ideas; until after some weeks of the most strenuous work of my life, light came into the darkness, and a new undreamed- of perspective opened up before me”. (M. Planck, 1918, Nobel Lecture) December 23, 2017 ABV IIITM Gwalior 36 “On the theory of the Energy Distribution Law in the Normal Spectrum”, M. Planck, Verhandl der Deutschen Physikal. Gessellsch, 2, 237 (1900)

December 14, 1900 (Birthday of Quantum Theory)

Distribute the total energy EN among N oscillators with average energy E E N  N E

For the purpose of counting let the energy EN be made up of small elements of energy  so that there are P such elements E P  N  December 23, 2017 ABV IIITM Gwalior 37 Distribute P elements ( ) among N oscillators (separated by |)

P elements ( ) are equivalent; (N-1) separators (|) are equivalent (N  P  1)! (N  P)! (N  P)(N P) Number of ways W    (N  1)!P! N!P! N N P P  E   E  E E  Entropy per oscillator S k lnW k1  ln1   ln          

It is tempting to compare it with  E   E  E E  S k1  ln1   ln          

and conclude   December 23, 2017 ABV IIITM Gwalior 38 Planck’s argument :  8 2 3   and u (T )    u( )  E( )  T  c 3   E( )    T    E  1 1  E   f   or  f   Invert this to get T    T     1 dS 1  E   E    f    S F  T dE       

This implies that the energy element must be  h December 23, 2017 ABV IIITM Gwalior 39 h Work backwards to obtain E( )  eh / kT  1 8h 3 1 and u( )  c3 eh / kT  1

h and k are universal constant

From Stefan-Boltzmann constant  and Wien’s constant b get

h = 6.55  10−27 erg . sec

k = 1.346  10−16 erg/deg

December 23, 2017 ABV IIITM Gwalior 40 Planck on k: “constant is…..connected with the definition of temperature. If temperature were to be defined as the average kinetic energy of a molecule in an ideal gas, … the constant would have the value 2/3……At the time when I carried out the corresponding calculation from the radiation law, an exact proof of the number obtained was quite impossible” (M. Planck, 1918, Nobel Lecture) Calculations by Planck on the basis of k:

Avogadro’s number from gas constant R N  6.1751023 per mole A k

F  10 electronic charge from Faraday constant e  4.6910 esu N A

December 23, 2017 ABV IIITM Gwalior 41 Planck on h:

“The explanation of the second universal constant of the radiation law was not so easy. Because it represents the product of energy and time …… I described it as the elementary quantum of action. Whilst it was completely indispensable for obtaining the correct expression for entropy ……. it proved elusive and resistant to all efforts to fit it into the framework of classical theory…. Either the quantum of action was a fictional quantity, then the whole deduction of the radiation law was in the main illusory and represented nothing more than an empty non-significant play on formulae, or the derivation of the radiation law was based on a sound physical conception. In this case the quantum of action must play a fundamental role in physics. December 23, 2017 ABV IIITM Gwalior 42 Experiment has decided for the second alternative. That the decision could be made so soon and so definitely was due not to the proving of the energy distribution law of heat radiation, still less to the special derivation of that law devised by me, but rather it should be attributed to the restless forward thrusting work of those research workers who used the quantum of action to help them in their own investigations and experiments” (M. Planck, 1918, Nobel Lecture)

“My futile attempts to fit the elementary quantum of action somehow into the classical theory continued for a number of years and they cost me a great deal of effort. Many of my colleagues saw in this something bordering on tragedy. But I feel differently about it. For the thorough enlightenment I thus received was all the more valuable”. (M. Planck, 1950, Scientific autobiography and Other papers) December 23, 2017 ABV IIITM Gwalior 43 Excerpt from the Nobel Prize presentation speech

“Planck's radiation theory is, in truth, the most significant lodestar for modern physical research, and it seems that it will be a long time before the treasures will be exhausted which have been unearthed as a result of Planck's genius”.

December 23, 2017 ABV IIITM Gwalior 44 Do oscillators really have energy elements  = h ?

Albert Einstein (Nobel Prize 1921)

December 23, 2017 ABV IIITM Gwalior 45 Criticism of Planck’s theory

(1) Planck did not follow Boltzmann’s procedure for finding the equilibrium distribution. Einstein’s objection “The manner in which Mr. Planck uses Boltzmann’s equation is rather strange to me in that a probability of a state W is introduced without a physical definition of this quantity” (A. Einstein, The theory of radiation and quanta, Transactions of the first Solvay conference, 1912).

(2) Logical inconsistency : Oscillators are assumed to have quantized energies and yet a classical result is used to work out the rate of radiation.

December 23, 2017 ABV IIITM Gwalior 46 Boltzmann’s procedure applied to oscillators , A. Einstein, Ann. der Physik 22, 180 (1907) For a system at temperature T probability P() that the system has energy  P(E)  e   / kT

For a quantized oscillator of frequency  the average energy

nhe nh / kT  h E( )  n   e nh / kT eh / kT  1 n

8 2 h

This gives u( )  3 h / kT December 23, 2017 ABV IIITM Gwaliorc e  1 47 Einstein’s views on logical inconsistency: “It should be kept in mind that optical observations refer to values averaged over time and not to instantaneous values. Despite the complete experimental verification of the theory…., it is conceivable that a theory of light operating with continuous three-dimensional functions will lead to conflicts with experience if it is applied to the phenomena of light generation and conversion”.

“In accordance with the assumption to be considered here, the energy of light ray spreading out from a point source is not continuously distributed over an increasing space but consists of a finite number of energy quanta which are localized at points in space, which move without dividing, and which can only be produced and absorbed as complete units”. (A. Einstein, Ann. der Physik, 17, 132 (1905). December 23, 2017 ABV IIITM Gwalior 48 How does he go about doing it? Calculate the entropy of low density radiation at equilibrium. Show that it is like that of an ideal gas. Write entropy as S ( )d with E u( )d dS 1  1 Then    dE T u T 8  3 Use Wien’s formula for low density radiation u( )  c 3 e  / T 1  1 8 3 to get   ln T u  c 3u u  uc3  and   ln  1   3  December 23, 2017 ABV IIITM Gwalior8 v  49 Consider the total radiation in the spectral range  to   in volume V with energy E=Vu E  Ec3  Then the entropy S V  ln  1  3    8 V 

Now suppose the volume is changed from V0 to V keeping the total energy the same E V Then the entropy change S  S0  ln  V0

Compare this with the entropy change in an ideal gas of N molecules

December 23, 2017 ABV IIITM Gwalior 50 E V For low density radiation S  S0  ln  V0 V For an ideal gas S  S0 kN ln V0 h E With   the two expressions are the same with N  k h

Einstein concludes that low density radiation behaves thermodynamically as if it has N independent quanta each of energy h.

December 23, 2017 ABV IIITM Gwalior 51 Consider phenomena which cannot be explained by classical physics and see if the quantum picture of light explains them.

Einstein explained three phenomena using his theory.

Stokes’ rule : Frequency of photoluminescent emission is less than that of incident light.

The photoelectric effect : Energies of photoelectrons independent of the intensity of incident light.

Photoionization of gases : A minimum frequency of light required to ionize a gas.

December 23, 2017 ABV IIITM Gwalior 52 Is Quantization of energy limited to radiating dipoles?

“For the following question forces itself upon us. If the elementary oscillators that are used in the theory of the energy exchange between radiation and matter cannot be interpreted in the sense of the present kinetic molecular theory, must we not also modify the theory for the other oscillators that are used in the molecular theory of heat?” [A. Einstein, Ann. der Physik 22, 180 (1907)]

Einstein is referring to the problem of specific heat of solids.

December 23, 2017 ABV IIITM Gwalior 53 Dulong-Pettit law : Specific heat of solids is 3R per mole

Experimental situation in 1900:

Some materials do not obey Dulong-Petit law. The light elements have much smaller heat capacities that 3R.

Specific heat of some elements vary rapidly with temperature and attain the value of 3R at high temperatures.

December 23, 2017 ABV IIITM Gwalior 54 Einstein’s solution: h Average energy per vibrating atom in a solid E  e h / kT  1

Energy for N atoms with three degrees of freedom h U 3N e h / kT  1

2 dU  h  e h / kT Specific heat 3R  2 dT  kT  e h / kT  1

Explains : smaller specific heat of lighter elements (their frequency is higher) Predicts : Predicts that specific heat goes to zero as T0. This is confirmed experimentally by Nernst. December 23, 2017 ABV IIITM Gwalior 55 Excerpts from the Nobel Prize presentation speech

“There is probably no physicist living today whose name has become so widely known as that of Albert Einstein”.

“A third group of studies, for which in particular Einstein has received the Nobel Prize, falls within the domain of the quantum theory founded by Planck in 1900. This theory asserts that radiant energy consists of individual particles, termed "quanta", approximately in the same way as matter is made up of particles, i.e. atoms. This remarkable theory, for which Planck received the Nobel Prize for Physics in 1918, suffered from a variety of drawbacks and about the middle of the first decade of this century it reached a kind of impasse. Then Einstein came forward with his work on specific heat and the photoelectric effect”. December 23, 2017 ABV IIITM Gwalior 56 Quoted from a Letter from M. Planck to R.W. Wood (October 7, 1931)

“Briefly summarized, what I did can be described as simply an act of desperation ………..I had been wrestling unsuccessfully for six years (since 1894) with the problem of equilibrium between radiation and matter and I knew this problem was of fundamental importance to physics; I also knew the formula that expresses the energy distribution in the normal spectrum. A theoretical interpretation therefore had to be found at any cost, no matter how high……………..This approach was opened to me by maintaining the two laws of thermodynamics. For the rest, I was ready to sacrifice every one of my previous conviction about physical laws.”

December 23, 2017 ABV IIITM Gwalior 57 T h a n k Y o u

December 23, 2017 ABV IIITM Gwalior 58