Mystery of Planck's Constant Revealed

This is a clarification of this post on the physical meaning of Planck's constant $h$ and so of Quantum Mechanics QM as a whole. The basic message is that the numerical value of $h=6.62607015\times 10^{-34}$ Jouleseconds is chosen to make Planck's Law fit with observation and that this value is then inserted into Schrödinger's equation to preserve the linear relation between energy and frequency established in Planck's Law. 

Quantum Mechanics is based on a mysterious smallest quantum of energy/action $h$ named Planck's constant, which was introduced by Planck in 1900 as a "mathematical trick" to make Planck's Law of blackbody radiation fit with observations of radiation energy from glowing bodies of different temperatures. 

The mysterious Planck's constant  $h$ appears in Planck's Law in the combination $\frac{h\nu}{kT}$ where $\nu$ is frequency, $k$ is Boltzmann's constant and $T$ temperature with $kT$ a measure of energy (per degree of freedom) from thermodynamics. In particular  

•  $\nu_{max}=2.821\frac{kT}{h}$                   (*)

shows the frequency of maximal radiation intensity referred to as Wien's Displacement Law, which also serves as a cut-off frequency with quick decay of radiation intensity for frequencies $\nu >\nu_{max}$.  

If we translate (*) to wave length we get a corresponding smallest wave length 

• $\lambda_{min}= 0.2015\frac{hc}{kT}=\frac{0.0029}{T}$ meter
• $\lambda_{min} \approx 10^{-5}$m for $T=300$ K 
• $\lambda_{min} \approx 5\times 10^{-7}$m for $T=5778$ K (Sun)

We see that smallest wave length is orders of magnitude bigger that atomic size of $10^{-10}$ m, which tells that blackbody radiation is a collective wave phenomenon involving many atoms per radiated wave length.

Summary: 

• Planck's constant $h$ serves the role of setting a peak frequency scaling with temperature $T$ with corresponding smallest wave length scaling with $\frac{1}{T}$.
• The smallest wave length is many orders of magnitude bigger than atomic size showing blackbody radiation to be a collective wave phenomenon involving coordinated motion of many atoms. 
• Planck's constant $h$ thus has a physical meaning of setting a smallest spatial resolution size scaling with $\frac{1}{T}$ required for coordinated collective wave motion supporting radiation. 
• Higher temperature means more active atomic motion allowing smaller coordination length. 
• The standard interpretation of $h$ as smallest quanta of energy lacks physical representation.
• Connecting $h$ to coordination length is natural and gives $h$ a physical meaning without mystery. 
• Formally h = energy x time = momentum x length representing Heisenbergs Uncertainty Relation with h connecting to spatial resolution. Formally $E=h\nu=pc$ and so $h=p\lambda$.   

PS Recall that Schrödinger's equation for atoms and Maxwell's equations for light covers a very wide range of phenomena in what is referred to as a semi-classical model as half-quantum + half classical. In this model light is not quantised and there are no photons to worry about. The above meaning of $h$ from Planck's Law is understandable. The mystery is restored in Quantum Electro Dynamics QED where Maxwell's equations are replaced by the relativistic Dirac's equations and particles/photons appear as quantised excitations of fields. QED is way too complicated to be used for the wide range covered by QM and so is reserved for very special geometrically simplified situations. 

The Deep Secret of $E=h\nu$ Uncovered = 0

The value of Planck's constant $h$ is supposed to carry a deep secret of the atomic physics captured in the Schrödinger Equation SE of Quantum Mechanics QM as the foundation of modern physics. A deep secret of a microscopic world which is fundamentally different from the macroscopic world we can fathom by direct experience. A strange world of the modern physics emerging in the beginning of the 20th century, which "nobody understands" including the physical meaning of Planck's constant $h$. 

In the new 2019 SI standard of units, the value of $h$ is specified to be exactly $h=6.62607015\times 10^{−34}$ Joule-seconds, which is a very small number viewed to hide a deep secret, while appearing as an arbitrary unit conversion factor. 

Let us seek to untangle the secret in detail. We recall the message of modern physics of the existence of a smallest quantum of energy $h\nu$ associated to a wave of frequency of $\nu$ showing that the microscopic world is discrete and not continuous like the macroscopic world so well described by continuum mechanics. More precisely, light as a wave phenomenon is viewed to consist of a stream of light particles named photons each one carrying exactly the energy $h\nu$. Mind boggling, suggesting some deep secret.

Let us now trace the connection to SE for the Hydrogen atom taking the form: 

• $ih\frac{\partial\psi}{\partial t} + H\psi =0$                (SE)

where $\psi (x.t)$ is a complex-valued wave function depending on a 3d spatial coordinate $x$ and a time variable $t$ and $H$ is a (Hermitian) operator acting on $\psi$ with a discrete spectrum of real eigenvalues $E$ representing energies of normalised eigenfunctions $\Psi (x)$ satisfying $H\Psi =E\Psi$, which give wave solutions to (SE) of the form 

• $\psi (x,t)=\exp(i\frac{E}{h}t)\Psi (x)=\exp(i\nu t)\Psi (x)$ with
• $\nu =\frac{E}{h}$ or $E=h\nu$.  

We thus see a direct connection between the smallest quantum of energy $h\nu$ and energies $E=h\nu$ of eigenstates/functions of a Hydrogen atom, as a direct reflection of the form of (SE) including a first time derivative: Energy $E$ scales linearly with frequency $\nu$. 

The other way around, one can see (SE) as being formed by Schrödinger to include the connection $E=h\nu$ between energy $E$ and frequency $\nu$ (as a linear dispersion relation), because that fits with observed spectrum of the Hydrogen atom. Mathematical modeling to fit observation.   

More precisely, the spectrum of a Hydrogen atom comes out from differences of eigenvalues/energies $\Delta E$ translated to frequencies by $\Delta E =h\nu$. 

The basic heuristic idea of Einstein in 1905 was that  the energy of the electron of a Hydrogen atom can "jump" from one energy level to another by receiving/delivering exactly one photon of energy $\Delta E =h\nu$ in radiative equilibrium with light of frequency $\nu$: 

• Transition from one energy level to another with an energy jump $\Delta E$ of the electron of a Hydrogen atom involves receiving/delivering exactly the energy $\Delta E=h\nu$ of one photon of frequency $\nu =\frac{E}{h}$. 

This idea is supposed to convince us that the world of a Hydrogen atom is discrete operating with discrete chunks of energy $h\nu$ carried by discrete light particles/photons.

But this is an invented discreteness: SE is a continuum model of classical form in a wave function $\psi$ with $\vert\psi (x,t)\vert^2$ representing charge density, which has a discrete set of eigenvalues just like a vibrating string. The association of energy to frequency by $E=h\nu$ is simply a scaling of between energy and frequency with a scaling factor of $h$ with a value depending on choice of units.

From (SE) it follows that size of a Hydrogen atom scales with $h^2$ which connects to the discreteness of a Hydrogen atom with its only electron, which is described by the continuous model (SE) of classical continuum form. 

We thus find nothing fundamentally different from classical continuum mechanics point of view in the (SE) model of a Hydrogen atom in terms of a charge density. The association of an energy jump $\Delta E =h\nu $ to exactly one photon of frequency $\nu$ lacks real physical meaning and is just a convention which appeared as a heuristic idea in Einstein's mind in 1905. Planck's constant $h$ does not say that the microscopic world is discrete making it fundamentally different from a continuous macroscopic world. Planck's constant has a meaning as setting the physical scale of a Hydrogen atom, but not as a deep secret about the world. Of course atoms have spatial size just as specific macroscopic material objects with specific spatial extension. A Hydrogen atom is a like a continuous string of a violin of certain length and tension. No quantum.

In short, the quantum world of a Hydrogen atom can be understood in terms of classical continuum mechanics. 

The split appears when generalising (SE) to atoms with $N>1$ electrons following the route of Standard QM by Born-Bohr-Heisenberg into a linear wave equation in $3N$ spatial dimensions, with the wave function given a probabilistic unphysical meaning which makes StdQM "not understandable".

RealQM offers a fundamentally different generalisation without split away from classical continuum mechanics, which is understandable.  

Summary: 

1. Planck's constant $h$ serves as a formal conversion factor between energy $\Delta E$ and frequency $\nu$ with $\Delta E=h\nu$ in the setting of a radiating  Hydrogen atom. The size of a Hydrogen atom scales with $h^2$ which gives the specific value of Planck's constant $h$ a physical meaning, which is not some deep secreted of smallest quantum of energy. 
2. The generalisation to any atom by StdQM leaves classical continuum mechanics into a probabilistic quantum world "nobody can understand" where Planck's constant appears as a deep secret.
3. RealQM offers a generalisation staying within the form of classical continuum mechanics which "everybody can understand" where Planck's constant remains the simple conversion factor of 1. = No Secret = 0.
4. RealQM appears as "Quantum Mechanics without Quantum" which opens to unification with electromagnetics-Newtonian gravitation into a Unifies Field Theory as unfinished dream of Einstein. Let's get to work! 

Demystifying the New SI Base Units.

In the previous post we observed that Planck's constant $h$ appears as a conversion factor connecting light of frequency $\nu$ with attributed energy $h\nu$ (in eV or Joule) through the photoelectric effect with the release of an electron from a surface exposed to light (of sufficient high frequency). The inner mechanics of the atoms delivering the electrons upon excitation by exposure to light does not enter into the discussion and so Planck's constant can be given a meaning in macroscopic physics, thus without quantum mechanics, as a trade between light and electron energy and then further to mechanical energy. Its role in quantum mechanics then appears as an after construction.

Let us now turn to Boltzmann's constant $k$ to see its connection to Planck's constant and macroscopic physics. Boltzmann's constant appears in Planck's universal law of blackbody radiation law of the form

• $E(\nu ,T) = W(a)\, kT\nu^2$,
• $W(a) = \frac{a}{\exp(a )-1}$ with $a = \frac{h\nu}{kT}$,

where $E(\nu ,T)$ is the (suitably normalised) intensity of radiation of frequency $\nu$ from a blackbody of temperature $T$ and $W(a)$ is a cut-off factor with $W(a)=1 $ for small $a$ and
$W(a)$ small for medium to large $a$, expressing Wien's displacement law stating cut-off of high frequencies. We see that Planck’s constant only appears in the cut-off factor.

Experimental observation of $E(\nu ,T)$ makes it possible to determine $W(a)$ and thus $kT$ in terms of $h\nu$, from which Boltzmann's constant $k$ can be determined with respect to a chosen scale for temperatur $T$, or the other way around as in the new SI units by specifying by definition 

• $k=1.380650\times 10^{-23}\, J/K$,  

thus setting a new standard for Kelvin $K$ as measure of temperature. The connection between the energy measures $h\nu$ and $kT$ then shows to be

• $h\nu_{max} \approx 2.8214391\times kT$,

where $\nu = \nu_{max}$ gives maximum of the spectrum $E(\nu ,T)$.

Again, this can be done without having to invoke quantum mechanics in its standard form with $h$ as a "smallest quantum of action" as exposed in detail on Computational Blackbody Radiation, which presents a derivation of Planck's law using deterministic continuum physics instead of as usual statistics of discrete quanta. In particular, the new derivation captures the universality of blackbody radiation beyond specific inner atomic mechanics.

The universality of Planck's law is expressed by the fact that an ideal blackbody can take the form of a set of oscillators without very specific inner structure. In particular different blackbodies with different inner structure can share the same temperature scale.

To sum up, both Planck's constant and Boltzmann's constant are specified by definition in the new SI units, from which the new units kilogram and Kelvin can be determined by macroscopic experiments without resort to quantum mechanics in its standard form.



Hopefully this helps to demystify both Planck's and Boltzmann's constant, and the new SI units.
 

New Perspective on New Unit of Mass in terms of Planck's Constant

In the 2019 redefinition of the SI base units the kilogram as unit of mass is defined in terms of Planck's constant

• $h$ set by definition to exactly $6.62607015×10^{−34}$ Joule-second ($J\cdot s$), 

where

• $Joule = Newton\times m = M\times\frac{m}{s^2}\times m=M\frac{m^2}{s^2}=Mc^2$

with $m$ meter, $s$ second, $M$ mass in kilogram and $c$ the speed of light.

This defines kilogram in terms of Planck's constant $h$, second $s$ and speed of light $c$ with meter $m$ defined in terms of $c$. The relation $E=Mc^2$, viewed as a profound discovery attributed to Einstein's relativity theory, then appears simply as a definition (of mass).

The connection to quantum mechanics comes by attributing a certain energy $h\nu$ to light of frequency $\nu$ through the law of the photoelectric effect

• $h\nu = eV_0 + \phi = eV_0 + h\nu_0$, 

where $eV_0$ in electronVolts is the energy of a released electron with charge $e$ and $V_0$ a stopping potential in Volt, and $h\nu_0$ is the work to release an electron with $\nu_0$ a threshold frequency. This relation determines $h\approx 4.1357\times 10^{-15}$ in $eV\cdot s$, which fits with the new definition of $h$ in terms of $J\cdot s$ with the conversion $eV= 1.602176634×10^{−19} J$.

The photoelectric effect connects the macroscopic phenomena of light of different frequencies and stopping potential to the microscopic phenomenon of electron charge. In this connection there is nothing that says light of frequency $\nu$ is to be viewed as a stream of discrete photon particles of energy $h\nu$ and that Planck's constant $h$ has the physical meaning of a discrete smallest quantum of action.  Instead Planck's constant has the role of connecting light energy to electron potential energy ultimately to mechanical energy.

For a new continuum physics approach to blackbody radiation and the photoelectric effect with discrete quantum replaced by a threshold condition (as in the photoelectric effect), see Computational BlackBody Radiation.

The new definition of kilogram gives perspective on the very small size of Planck's constant $\sim 6.6\times 10^{-34}\, J\cdot s$ misleading to an idea of an absurdly small Planck length $\sim 1.6\times 10^{-35}\, m$ believed to have a physical meaning, in string theory in particular.  On the other hand, the length scale of atoms (and X-ray light of frequency about $3\times 10^{18}$) is about $10^{-10}\, m$ and that of a proton $10^{-15}\, m$, with the Planck scale 20 orders of magnitude smaller, way beyond any thinkable experimental exploration and thus meaning. Planck time  $\sim 5.3\times 10^{-44}\, s$ is even more absurd. No wonder that modern physics playing with Planck length and time is in a state of deep crisis, with the scientific madness come to full expression in the  Chronology of the Universe starting with the Planck Epoch before $10^{-43}\, s$ after Big Bang.

In Schrödingers equation $h$ multiplies the time derivative of the wave function, which means that
the atomic energy (potential + "kinetic" energy) of an eigenfunction of frequency $\nu$ is equal to $h\nu$, which comes to expression in the photoelectric effect.  Schrödinger's equation is a continuum model without any smallest quantum of action, only discrete eigenvalues representing different energies.

A reformulation of quantum mechanics in the form of a Schrödinger equation as a continuum model in real 3d space plus time without statistics, can be inspected at Real Quantum Mechanics.

To get an idea of the absurdly small Planck length scale $1.6\times 10^{-34}\, m$, one may compare with the estimated size of the observable Universe which is about $10^{27}\, m$ or $10^{33}\, \mu m$ with $\mu m$ mikrometer.



In short, Planck's constant $h$ converts light energy to mechanical energy through electron potential energy, and as such does not ask for a meaning as a "smallest quantum of action" in a mist of "quantisation" into absurdly small "quanta".

The new kilogram standard is specified to high precision using a Kibble balance, where gravitational force is balanced by an electromagnetic force (between two coils), which depends on Planck's constant. Mass is then derived by measuring the local gravitational constant.  

Comparing Blackbody Radiation Spectrum to Atomic Emission Spectrum

Planck's constant $h$ appears with different roles in a blackbody radiation spectrum and an atomic emission spectrum.  Blackbody radiation can be described as a near-resonance phenomenon in a forced harmonic oscillator with small damping in a mathematical model of the form

• $\ddot u (t) +\omega^2u(t) -\gamma\dddot u = f(t)\approx \sin(\omega t)$, 

where $u(t)$ is displacement as function of time $t$, $\omega$ is angular velocity, $\gamma$ is a small damping constant, $f(t)$ is forcing in near-resonance with $\omega$ and the dot signifies time differentiation. Here the oscillator described by $\ddot u (t) +\omega^2u(t)$ carries energy as background temperature and the dissipative term $-\gamma\dddot u$ gives off radiation balancing forcing $f(t)$.

The dynamics of near-resonance is quite subtle as explained in detail on Computational Blackbody Radiation showing that Planck's constant enters as a parameter in a high-frequency cut-off reflecting Wien's displacement law.   

Atomic emission can be described as an eigenvalue problem for Schrödinger's equation of the form

• $ih\dot\psi = E\psi$,

where $E$ is a real eigenvalue of an atomic Hamiltonian, with solution

• $\psi (t) =\exp(i\omega t) =\cos(\omega t)+i\sin(\omega t)$, 

which can be seen as a periodic exchange of two forms of energy represented by the real part $\cos(\omega t)$ and the complex part $\sin(\omega t)$ reflecting incoming-outgoing radiation. Atomic emission is thus a direct resonance phenomenon without background temperature. Planck's constant serves to convert angular velocity (angular momentum) $\omega$ to atomic energy $E$ as $\bar h\omega$ with $\bar h=\frac{h}{2\pi}$ with $E$ the sum of kinetic and potential energy. 

We conclude:

1. Blackbody radiation is a near-resonance phenomenon of molecules or collections of atoms modeled as a forced harmonic oscillator with small damping. Collections of atoms vibrate without electron configurations changing energy.   
2. Atomic radiation is a direct resonance phenomenon which can be modeled by a harmonic oscillator. Electrons oscillate between two energy levels representing eigenstates of an atom.

In both cases $h$ enters combined with frequency $\nu$ in the form $h\nu =\bar h\omega$ as quantity of energy serving in a threshold condition in blackbody radiation, and as an energy eigenvalue in atomic emission.

The value of $h$ as setting a conversion scale between light energy and electronic energy can be determined by the photoelectric effect and can then be used by definition in blackbody radiation and Schrödinger's equation. 

  

Water Dam Analog of Photoelectric Effect

                               Open sluice gates in the Three Gorges Dam in the Yangtze River.

Einstein was awarded the 1921 Nobel Prize in Physics for his "discovery of the law of the photoelectric effect", connecting frequency $\nu$ of light shining on a metallic surface with measured potential $U$:

• $h\nu = h\nu_0 + e\, U$ or $h(\nu -\nu_0) = e\, U$,

where $h$ is Planck's constant with dimension $eVs = electronvolt\,\times second$,  $\nu_0$ is the smallest frequency releasing electrons and $U$ in Volts $V$ is the stopping potential bringing the current to zero for $\nu >\nu_0$ and $e$ is the charge of an electron. Observing $U$ for different $\nu$ in a macroscopic experiment shows a linear relationship between $\nu -\nu_0$ and $U$ with $h$ as scale factor with reference value 

• $h = 4.135667516(91)\times 10^{-15}\, eVs$,

with Millikan's value from 1916 within $0.5\%$.

Determining $h$ this way makes Einstein's law of photoelectricity into an energy conversion standard attributing $h\nu$ electronvolts to the frequency $\nu$, without any implication concerning the microscopic nature of the photoelectric effect.

The award motivation "discovery of the law of the photoelectric effect" reflected that Einstein's derivation did not convince the committee as expressed by member Gullstrand: 

• When it was formulated it was only a tentatively poorly developed hunch, based on qualitative and partially correct observations. It would look peculiar if a prize was awarded to this particular work. 

To give perspective let us as an analog of the law of the photoelectric effect consider a water dam with sluice gates which automatically open when the level of water is $\nu_0$.  The sluice gates will then remain locked as long as the water level $\nu <\nu_0$.  Lock the sluice gates and let the dam fill to some water level $\nu >\nu_0$ and then unlock the sluices. The sluices will then open and water will flow through under transformation of potential energy into kinetic energy. Assuming the work to open the sluices corresponds to a level loss of $\nu_0$, a net level of $\nu -\nu_0$ potential energy will then be transformed into kinetic energy by the water flow through the sluices. 

The dam can be seen as an illustration of the photoelectric effect with the water level corresponding to frequency $\nu$ and the gravitational constant corresponding to $h$ and the width of the dam corresponding to the amplitude of incoming light. If $\nu <\nu_0$ then nothing will happen, if $\nu >\nu_0$ then the kinetic energy will scale with $h\nu$ and the total flow will scale with the width of the dam.

Notice that noting in this model requires the water to flow in discrete lumps or quanta. The only discrete effect is the threshold $\nu_0$ for opening the sluices.

Universal Quantum of Action: Standard Without Universality

In recent posts on we have seen that Plank's constant $h$ in physics text books being presented as a universal quantum of action as a smallest "packet of action" as a fundamental constant of fundamental significance in the "quantized" world we happen to be part of, in fact is nothing but a conversion standard between two measures of energy, in terms of frequency $\nu$ in periods per second and electronvolt (eV), determined by Einstein's law of photoelectricity

• $h(\nu - \nu_0) = e\, U$,

where $\nu_0$ is the smallest frequency releasing electrons from a metallic surface upon exposure of light, $U$ in Volts $V$ is the stopping potential bringing the current to zero for $\nu >\nu_0$ and $e$ is the charge of an electron. Observing $U$ for different $\nu$ shows a linear relationship between $\nu -\nu_0$ and $U$ with $h$ as the scale factor measured in $eVs$ $electronvolts\times second$ as $energy \times time$ as action. The reference value obtained this way is 

• $h = 4.135667516(91)\times 10^{-15}\, eVs$,

with Millikan's value from 1916 within $0.5\%$. Determining $h$ this way makes Einstein's law of photoelectricity simply into a conversion standard (that is, a definition) of energy attributing $h\nu$ electronvolts to the frequency $\nu$. Another way of finding the conversion from frequency to electronvolt is using a Josephson junction.

We now turn to Schrödinger's equation

• $i\bar h\frac{\partial\psi}{\partial t}+H\psi=0$,

where $\bar h=\frac{h}{2\pi}$ is Planck's reduced constant as conversion from periods $\nu$ per second to angular velocity $\omega$ with $h\nu =\bar h\omega$, and $H$ is a Hamiltonian of space dependence. An eigenvalue $E$ of the Hamiltonian represents energy with $\psi_E$ a corresponding space dependent eigenfunction satisfying $H\psi_E =E\psi_E$ and $\exp(i\omega t)\psi_E$ a corresponding solution of Schrödinger's equation with 

• $h\nu = \bar h\omega =  E$, 

expressing energy in terms of frequency. We see that the appearance of $\bar h$ with the time derivative in Schrödinger's equation accounts for the energy conversion and is completely normal and without mystery. 

Next, we consider the space dependent Hamiltonian in the basic case of the Hydrogen atom:

• $H\psi =  \frac{\bar h^2}{2m}\Delta\psi +  \frac{e^2}{r}\psi$  

where $\psi =\psi (x)$ with $x$ a space coordinate, $r =\vert x\vert$, and $m$ is the mass of the electron. Normalising by changing scale in space $x=a\bar x$ and time $t=b\bar t$, we obtain the Hamiltonian in normalized atomic units in the form

• $\bar H = \bar\Delta + \frac{2}{\bar r}$ with smallest eigenvalue $1$, 
• $a=\frac{\bar h^2}{me^2}$ as $Bohr\, radius$,
• $b=\frac{\bar h2a}{e^2}$ as $Bohr\, time$ with $\omega =\frac{1}{b}$ angular velocity
• $E =\frac{e^2}{2a}$ as $Rydberg\, energy$.

We now observe that

• $E\, b = \bar h$,

• $E = \bar h\, \omega$, 

which shows that the also the space dependent part of Schrödinger's equation is calibrated to the energy conversion standard. 

Finally, Planck's constant also appears in Planck's radiation law and then in the high-frequency cut-off factor

• $\frac{\alpha}{\exp(\alpha )-1}$
• $\alpha = \frac{h\nu}{kT}$,

where $k$ is Boltzmann's constant and $T$ temperature. We see that again $h\nu$ appears as an atomic energy measure with a value that is not very precisely determined in its role in the cut-off factor.

The value of $h$ from photoelectricity can then serve also in Planck's law.

We conclude that Planck's constant $h$ is a conversion standard between two energy measures and as such has no meaning as a universal quantum of action or as integral multiples $nh\nu$ with $n=1,2,3,..$ of special significance other than by connection to eigenfunctions and eigenvalues.   

Ultimately, what is measured are atomic emission spectra in terms of frequencies and wave lengths which through Planck's constant can be translated to energies expressed in electronvolts (or Joule). Nothing of the internal atomic structure (in terms of $e$ and $m$) enters into this discussion.

Planck introduced $h$ in a statistical argument in 1900 long before atoms were known, Einstein picked up $h\nu$ in his 1905 article on photoelectricity, before atoms were known, and Schrödinger put $h$ into his equation in 1926 to describe atoms. This line of events supports the idea that Planck's constant $h$ is a convention without any universal significance.

Understanding the real role of Planck's constant may help to give Schrödinger's equation a physical interpretation which is free from mysteries of "quantization" and statistics. Versions of Schrödinger's equation based on an idea of smoothed particle mechanics then naturally present themselves, with $h$ acting as a smoothing parameter.

PS Notice that the fine structure constant $\alpha = \frac{e^2}{\bar hc}=\frac{1}{137}$ can be expressed as $\alpha =\frac{2}{c}\frac{a}{b}$ which shows that $\alpha$ relates $Bohr\, speed\, =\frac{a}{b}$ to the speed of light $c$. This relation is viewed to be fundamental, but why is hidden in mystery.   

Planck's Constant = Human Convention Standard Frequency vs Electronvolt

The recent posts on the photoelectric effect exhibits Planck's constant $h$ as a conversion standard between the units of light frequency $\nu$ in $Hz\, = 1/s$ as periods per second and electronvolt ($eV$), expressed in Einstein's law of photoelectricity:

• $h\times (\nu -\nu_0) = eU$, 

where $\nu_0$ is smallest frequency producing a photoelectric current, $e$ is the charge of an electron and $U$ the stopping potential in Volts $V$ for which the current is brought to zero for $\nu > \nu_0$. Einstein obtained, referring to Lenard's 1902 experiment with $\nu -\nu_0 = 1.03\times 10^{15}\, Hz$ corresponding to the ultraviolet limit of the solar spectrum and $U = 4.3\, V$ 

• $h = 4.17\times 10^{-15} eVs$

to be compared with the reference value $4.135667516(91)\times 10^{-15}\, eV$ used in Planck's radiation law. We see that here $h$ occurs as a conversion standard between Hertz $Hz$ and electronvolt $eV$  with 

• $1\, Hz  = 4.17\times 10^{-15}\, eV$ 

To connect to quantum mechanics, we recall that Schrödinger's equation is normalized with $h$ so that the first ionization energy of Hydrogen at frequency $\nu = 3.3\times 10^{15}\, Hz$ equals $13.6\, eV$, to be compared with $3.3\times 4.17 = 13.76\, eV$ corresponding to Lenard's photoelectric experiment. 

We understand that Planck's constant $h$ can be seen as a conversion standard between light energy measured by frequency and electron energy measured in electronvolts. The value of $h$ can then be determined by photoelectricity and thereafter calibrated into Schrödinger's equation to fit with ionization energies as well as into Planck's law as a parameter in the high-frequency cut-off (without a very precise value).  The universal character of $h$ as a smallest unit of action is then revealed to simply be a human convention standard without physical meaning. What a disappointment!

This fits with the article Quantum Theory without Planck's Constant ny John P. Ralston:

• Planck's constant was introduced as a fundamental scale in the early history of quantum mechanics. We find a modern approach where Planck's constant is absent: it is unobservable except as a constant of human convention.

Finally: It is natural to view frequency $\nu$ as a measure of energy per wavelength, since radiance as energy per unit of time scales with $\nu\times\nu$ in accordance with Planck's law, which can be viewed as $\nu$ wavelengths each of energy $\nu$ passing a specific location per unit of time. We thus expect to find a linear relation between frequency and electronvolt as two energy scales: If 1 € (Euro) is equal to 9 Skr (Swedish  Crowns), then 10 € is equal to 90 Skr.

Quantum Physics as Digital Continuum Physics

Quantum mechanics was born in 1900 in Planck's theoretical derivation of a modification of Rayleigh-Jeans law of blackbody radiation based on statistics of discrete "quanta of energy" of size $h\nu$, where $\nu$ is frequency and $h =6.626\times 10^{-34}\, Js$ is Planck's constant.

This was the result of a long fruitless struggle to explain the observed spectrum of radiating bodies using deterministic eletromagnetic wave theory, which ended in Planck's complete surrender to statistics as the only way he could see to avoid the "ultraviolet catastrophe" of infinite radiation energies, in a return to the safe haven of his dissertation work in 1889-90 based on Boltzmann's statistical theory of heat.

Planck described the critical step in his analysis of a radiating blackbody as a discrete collection of resonators as follows:

• We must now give the distribution of the energy over the separate resonators of each frequency, first of all the distribution of the energy $E$ over the $N$ resonators of frequency . If E is considered to be a continuously divisible quantity, this distribution is possible in infinitely many ways. 
• We consider,  however this is the most essential point of the whole calculation $E$ to be composed of a well-defined number of equal parts and use thereto the constant of nature $h = 6.55\times 10^{-27}\, erg sec$. This constant multiplied by the common frequency of the resonators gives us the energy element in $erg$, and dividing $E$ by we get the number $P$ of energy elements which must be divided over the $N$ resonators. 
• If the ratio thus calculated is not an integer, we take for $P$ an integer in the neighbourhood. It is clear that the distribution of P energy elements over $N$ resonators can only take place in a finite, well-defined number of ways.

We here see Planck introducing a constant of nature $h$, later referred to as Planck's constant, with a corresponding smallest quanta of energy $h\nu$ for radiation (light) of frequency $\nu$. 

Then Einstein entered in 1905 with a law of photoelectricity with $h\nu$ viewed as the energy of a light quanta of frequency $\nu$ later named photon and crowned as an elementary particle.

Finally, in 1926 Schrödinger formulated a wave equation for involving a formal momentum operator  $-ih\nabla$ including Planck's constant $h$, as the birth of quantum mechanics, as the incarnation of modern physics based on postulating that microscopic physics is

1. "quantized" with smallest quanta of energy $h\nu$,
2. indeterministic with discrete quantum jumps obeying laws of statistics.

However, microscopics based on statistics is contradictory, since it requires microscopics of microscopics in an endeless regression, which has led modern physics into an impasse of ever increasing irrationality into manyworlds and string theory as expressions of scientific regression to microscopics of microscopics. The idea of "quantization" of the microscopic world goes back to the atomism of Democritus, a primitive scientific idea rejected already by Aristotle arguing for the continuum, which however combined with modern statistics has ruined physics.  

But there is another way of avoiding the ultraviolet catastrophe without statistics, which is presented on Computational Blackbody Radiation with physics viewed as analog finite precision computation which can be modeled as digital computational simulation. 

This is physics governed by deterministic wave equations with solutions evolving in analog computational processes, which can be simulated digitally. This is physics without microscopic games of roulette as rational deterministic classical physics subject only to natural limitations of finite precision computation.

This opens to a view of quantum physics as digital continuum physics which can bring rationality back to physics. It opens to explore an analog physical atomistic world as a digital simulated world where the digital simulation reconnects to analog microelectronics. It opens to explore physics by exploring the digital model, readily available for inspection and analysis in contrast to analog physics hidden to inspection.

The microprocessor world is "quantized" into discrete processing units but it is a deterministic world with digital output:

Why the Same Universal Quantum of Action $h$ in Radiation, Photoelectricity and Quantum Mechanics?

Planck's constant $h$ as The Universal Quantum of Action was introduced by Planck in 1900 as a mathematical statistical trick to supply the classical Rayleigh-Jeans radiation law $I(\nu ,T)=\gamma T\nu^2$ with a high-frequency cut-off factor $\theta (\nu ,T)$ to make it fit with observations including Wien's displacement law, where

• $\theta (\nu ,T) =\frac{\alpha}{\exp(\alpha )-1}$,
• $\alpha =\frac{h\nu}{kT}$, 

$\nu$ is frequency, $T$ temperature in Kelvin $K$ and $k =1.38066\times 10^{-23}\, J/K$ is Boltzmann's constant and $\gamma =\frac{2k}{c}$ with $c\, m/s$ the speed of light in vaccum. Planck then determined $h$ from experimental radiation spectra to have a value of $6.55\times 10^{-34} Js$, as well as Boltzmann's constant to be $1.346\times 10^{-23}\, J/K$ with $\frac{h}{k}= 4.87\times 10^{-11}\, Ks$ as the effective parameter in the cut-off.  

Planck viewed $h$ as a fictional mathematical quantity without real physical meaning, with $h\nu$ a fictional mathematical quantity as a smallest packet of energy of a wave of frequency $\nu$, but in 1905 the young ambitious Einstein suggested an energy balance for photoelectricity of the form 

• $h\nu = W + E$,

with $W$ the energy required to release one electron from a metallic surface and E the energy of a released electron with $h\nu$ interpreted as the energy of a light photon of frequency $\nu$ as a discrete lump of energy. Since the left hand side $h\nu$ in this law of photoelectricity was determined by the value of $h$ in Planck's radiation law, a new energy measure for electrons of electronvolt was defined by the relation $W + E =h\nu$. As if by magic the same Universal Quantum of Action $h$ then appeared to serve a fundamental role in both radiation and photoelectricity.

What a wonderful magical coincidence that the energy of a light photon of frequency $\nu$ showed to be exactly $h\nu \, Joule$! In one shot Planck's fictional smallest quanta of energy $h\nu$ in the hands of the young ambitious Einstein had been turned into a reality as the energy of a light photon of frequency $h\nu$, and of course because a photon carries a definite packet of energy a photon must be real. Voila!

In 1926 Planck's constant $h$ showed up again in a new context, now in Schrödinger's equation

• $-\frac{\bar h^2}{2m}\Delta\psi = E\psi$

 with the formal connection   

• $p = -i\bar h \nabla$ with $\bar h =\frac{h}{2\pi}$,
• $\frac{\vert p\vert^2}{2m} = E$, 

as a formal analog of the classical expression of kinetic energy $\frac{\vert p\vert ^2}{2m}$ with $p=mv$ momentum, $m$ mass and $v$ velocity.

Planck's constant $h$ originally determined to make theory fit with observations of radiation spectra and then by Planck in 1900 canonized as The Universal Quantum of Action, thus in 1905 served to attribute the energy $h\nu$ to the new fictional formal quantity of a photon of frequency $\nu$ . In 1926 a similar formal connection was made in the formulation of Schrödinger's wave equation.  

The result is that the same Universal Quantum of Action $h$ by all modern physicists is claimed to play a fundamental role in both (i) radiation, (ii) photolelectricity and (iii) quantum mechanics of the atom. This is taken as an expression of a deep mystical one-ness of physics which only physicists can grasp,  while it in fact it is a play with definitions without mystery, where $h$ appears as a parameter in a high-frequency cut-off factor in Planck's Law, or rather in the combination $\hat h =\frac{h}{k}$,  and then is transferred into (ii) and (iii) by definition.  Universality can this way be created by human hands by definition. The power of thinking has no limitations, or cut-off.

No wonder that Schrödinger had lifelong interest in the Vedanta philosophy of Hinduism "played out on one universal consciousness".

But Einstein's invention of the photon as light quanta in 1905 haunted him through life and approaching the end in 1954, he confessed:

• All these fifty years of conscious brooding have brought me no nearer to the answer to the question, "What are light quanta?" Nowadays every Tom, Dick and Harry thinks he knows it, but he is mistaken. 

Real physics always shows up to be more interesting than fictional physics, cf. Dr Faustus ofd Modern Physics.

PS Planck's constant $h$ is usually measured by (ii) and is then transferred to (i) and (iii) by ad hoc definition.

The True Meaning of Planck's Constant as Measure of Wavelength of Maximal Radiance and Small-Wavelength Cut-off.

The modern physics of quantum mechanics was born in 1900 when Max Planck after many unsuccessful attempts in an "act of despair" introduced a universal smallest quantum of action  $h= 6.626\times 10^{-34}\, Js = 4.12\times 10^{-15}\, eVs$ named Planck's constant in a theoretical justification of the spectrum of radiating bodies observed in experiments, based on statistics of packets of energy of size $h\nu$ with $\nu$ frequency.

Planck describes this monumental moment in the history of science in his 1918 Nobel Lecture as follows:

• For many years, such an aim for me was to find the solution to the problem of the distribution of energy in the normal spectrum of radiating heat.
• Nevertheless, the result meant no more than a preparatory step towards the initial onslaught on the particular problem which now towered with all its fearsome height even steeper before me. The first attempt upon it went wrong…
• So there was nothing left for me but to tackle the problem from the opposite side, that of thermodynamics, in which field I felt, moreover, more confident. 
• Since the whole problem concerned a universal law of Nature, and since at that time, as still today, I held the unshakeable opinion that the simpler the presentation of a particular law of Nature, the more general it is… 
• 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.

Planck thus finally succeeded to prove Planck's radiation law as a modification of Rayleigh-Jeans law with a high-frequency cut-off factor eliminating "the ultraviolet catastrophe" which had paralyzed physics shortly after the introduction of Maxwell's wave equations for electromagnetics as the culmination of classical physics.

Planck's constant $h$ enters Planck's law

• $I(\nu ,T)=\gamma \theta (\nu , T)\nu^2 T$, where $\gamma =\frac{2k}{c^2}$,

where $I(\nu ,T)$ is normalized radiance, as a parameter in the multiplicative factor

• $\theta (\nu ,T)=\frac{\alpha}{e^{\alpha} -1}$, 
• $\alpha=\frac{h\nu}{kT}$,

where $\nu$ is frequency, $T$ temperature in Kelvin $K$ and $k = 1.38\times 10^{-23}\, J/K = 8.62\times 10^{-5}\, eV/K$ is Boltzmann's constant and $c\, m/s$ the speed of light.

We see that $\theta (\nu ,T)\approx 1$ for small $\alpha$ and enforces a high-frequency small-wavelength cut-off for $\alpha > 10$, that is, for   

• $\nu > \nu_{max}\approx \frac{10T}{\hat h}$ where $\hat h =\frac{h}{k}=4.8\times 10^{-11}\, Ks$,
• $\lambda < \lambda_{min}\approx \frac{c}{10T}\hat h$ where $\nu\lambda =c$,

with maximal radiance occuring for $\alpha = 2.821$ in accordance with Wien's displacement law.  With $T = 1000\, K$ the cut-off is in the visible range for $\nu\approx 2\times 10^{14}$ and $\lambda\approx 10^{-6}\, m$. We see that the relation 

• $\frac{c}{10T}\hat h =\lambda_{min}$,

gives $\hat h$ a physical meaning as measure of wave-length of maximal radiance and small-wavelength cut-off of atomic size scaling with $\frac{c}{T}$. 

Modern physicsts are trained to believe that Planck's constant $h$ as the universal quantum of action represents a smallest unit of a "quantized" world with a corresponding Planck length $l_p= 1.62\times 10^{-35}$ as a smallest unit of length, about 20 orders of magnitude smaller than the proton diameter.

We have seen that Planck's constant enters in Planck's radiation law in the form $\hat h =\frac{h}{k}$, and not as $h$, and that $\hat h$ has the role of setting a small-wavelength cut-off scaling with $\frac{c}{T}$.

Small-wavelength cut-off in the radiation from a body is possible to envision in wave mechanics as an expression of finite precision analog computation. In this perspective Planck's universal quantum of action emerges as unnecessary fiction about exceedingly small quantities beyond reason and reality.

Lagrange's Biggest Mistake: Least Action Principle Not Physics!

The reader will find no figures in this work. The methods which I set forth do not require either constructions or geometrical or mechanical reasonings: but only algebraic operations, subject to a regular and uniform rule of procedure. (Preface to Mécanique Analytique)

The Principle of Least Action formulated by Lagrange in his monumental treatise Mecanique Analytique (1811) collecting 50 years work, is viewed to be the crown jewel of the Calculus of Newton and Leibniz as the mathematical basis of the scientific revolution:

• The equations of motion of a dynamical system are the same equations that express that the action as the integral over time of the difference of kinetic and potential energies, is stationary that is does not change under small variations.   

The basic idea goes back to Leibniz:

• In change of motion, the quantity of action takes on a Maximum or Minimum. 

And to Maupertis (1746):

• Whenever any action occurs in nature, the quantity of action employed in this change is the least possible.

In mathematical terms, the Principle of Least Action expresses that the trajectory $u(t)$ followed by a dynamical system over a given time interval $I$ with time coordinate $t$, is determined by the condition of stationarity of the action:

• $\frac{d}{d\epsilon}\int_I(T(u(t)+\epsilon v(t)) - V(u(t)+\epsilon v(t)))\, dt =0$,  

where $T(u(t))$ is kinetic energy and $V(u(t))$ is potential energy of $u(t)$ at time $t$, and $v(t)$ is an arbitrary perturbation of $u(t)$,  combined with an initial condition. In the basic case of a harmonic oscillator;

• $T(u(t))=\frac{1}{2}\dot u^2(t)$ with $\dot u=\frac{du}{dt}$,
• $V(u(t))=\frac{1}{2}u^2(t)$
• stationarity is expressed as force balance as Newton's 2nd law: $\ddot u (t) +u(t) = 0$.  

The Principle of Least Action is viewed as a constructive way of deriving the equations of motion expressing force balance according to Newton's 2nd law, in situations with specific choices of coordinates for which direct establishment of the equations is tricky. 

From the success in this respect the Principle of Least Action has been elevated from mathematical trick to physical principle asking Nature to arrange itself so as to keep the action stationary, as if Nature could compare the action integral for different trajectories and choose the trajectory with least action towards a teleological final cause, while in fact Nature can only respond to forces as expressed in equations of motion.

But if Nature does not have the capability of evaluating and comparing action integrals, it can be misleading to think this way. In the worst case it leads to invention of physics without real meaning, which is acknowledged by Lagrange in the Preface to Mecanique Analytique.

The ultimate example is the very foundation of quantum physics as the pillar of modern physics based on a concept of elementary (smallest) quantum of action  denoted by $h$ and named Planck's constant with dimension $force \times time$. Physicists are trained to view the elementary quantum of action to represent a "quantization" of reality expressed as follows on Wikipedia:

• In physics, a quantum (plural: quanta) is the minimum amount of any physical entity involved in an interaction. 
• Behind this, one finds the fundamental notion that a physical property may be "quantized," referred to as "the hypothesis of quantization".This means that the magnitude can take on only certain discrete values.
• A photon is a single quantum of light, and is referred to as a "light quantum".

In the quantum world light consists of a stream of discrete light quanta named photons. Although Einstein in his 1905 article on the photoelectric effect found it useful as a heuristic idea to speak about light quanta, he later changed mind:

• The quanta really are a hopeless mess. (to Pauli)
• All these fifty years of conscious brooding have brought me no nearer to the answer to the question, 'What are light quanta?' Nowadays every Tom, Dick and Harry thinks he knows it, but he is mistaken. (1954)

But nobody listened to Einstein and there we are today with an elementary quantum of action which is viewed as the basis of modern physics but has not physical reality. Schrödinger supported by Einstein said:

• There are no particles or quanta. All is waves.

Connecting to the previous post, note that to compute a solution according the Principle of Least Action typically an iterative method based on relaxation of the equations of motion is used, which has a physical meaning as response to imbalance of forces. This shows the strong connection between computational mathematics as iterative time-stepping and analog physics as motion in time subject to forces, which can be seen as a mindless evolution towards a hidden final cause, as if directed by an invisible hand of a mind understanding the final cause.   

Physics as Analog Computation instead of Physics as Observation

Bohr plotting the Copenhagen Interpretation of quantum mechanics together with Heisenberg and Pauli (left) and Bohr wondering what he did 30 years later (right).

To view physics as a form of analog computation which can be simulated by digital computation offers resolutions of the following main unsolved problems of modern microscopic and classical macroscopic  physics:

1. Interaction between subject (experimental apparatus) and object under observation.
2. Meaning of smallest quantum of action named Planck's constant $h$.
3. Contradiction between particle and wave qualities. Particle-wave duality.
4. Meaning of the 2nd law of thermodynamics and direction of time.
5. Meaning of Heisenberg's Uncertainity Principle.
6. Loss of cause-effect relation by resort of microscopic statistics.
7. Statistical interpretation of Schrödinger's multidimensional wave function.
8. Meaning of Bohr's Complementarity Principle. 
9. Meaning of Least Action Principle. 

This view is explored on The World as Computation and Computational Blackbody Radiation suggesting the following answers to these basic problems:

1. Observation by digital simulation is possible without interference with physical object.
2. Planck's constant $h$ can be viewed as a computational mesh size parameter.
3. All is wave. There are no particles. No particle-wave duality.
4. Dissipation as an effect of finite precision computation gives a 2nd law and direction of time.
5. Uncertainty Principle as effect of finite precision computation.
6. Statistics replaced by finite precision computation.
7. Schrödinger's wave equation as system in 3d without statistical interpretation.
8. No contradiction between complementary properties. No need of Complementarity Principle.
9. Least Action Principle as computational mathematical principle without physical reality. 

The textbook physics harboring the unsolved problems is well summarized by Bohr:

• There is no quantum world. There is only an abstract quantum physical description. It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature…
• Everything we call real is made of things that cannot be regarded as real. If quantum mechanics hasn't profoundly shocked you, you haven't understood it yet.
• We are all agreed that your theory is crazy. The question which divides us is whether it is crazy enough to have a chance of being correct. My own feeling is that it is not crazy enough. 
• We must be clear that when it comes to atoms, language can be used only as in poetry. The poet, too, is not nearly so concerned with describing facts as with creating images and establishing mental connections.

Quantum Mechanics from Blackbody Radiation as "Act of Despair"

Max Planck: The whole procedure was an act of despair because a theoretical interpretation (of black-body radiation) had to be found at any price, no matter how high that might be…I was ready to sacrifice any of my previous convictions about physics..For this reason, on the very first day when I formulated this law, I began to devote myself to the task of investing it with true physical meaning.

The textbook history of modern physics tells that quantum mechanics was born from Planck's proof of the universal law of blackbody radiation based on an statistics of discrete lumps of energy or energy quanta $h\nu$, where $h$ is Planck's constant and $\nu$ frequency. The textbook definition of a blackbody is a body which absorbs all, reflects none and re-emits all of incident radiation:

• A black body is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. (Wikipedia)
• "Blackbody radiation" or "cavity radiation" refers to an object or system which absorbs all radiation incident upon it and re-radiates energy which is characteristic of this radiating system only, not dependent upon the type of radiation which is incident upon it. (Hyperphysics)
• Theoretical surface that absorbs all radiant energy that falls on it, and radiates electromagnetic energy at all frequencies, from radio waves to gamma rays, with an intensity distribution dependent on its temperature. (Merriam-Webster)
• An ideal object that is a perfect absorber of light (hence the name since it would appear completely black if it were cold), and also a perfect emitter of light. (Astro Virginia)
• A black body is a theoretical object that absorbs 100% of the radiation that hits it. Therefore it reflects no radiation and appears perfectly black. (Egglescliff)
• A hypothetic body that completely absorbs all wavelengths of thermal radiation incident on it. (Eric Weisstein's World of Physics) 

But there is something more to a blackbody and that is a the high frequency cut-off, expressed in Wien's displacement law, of the principal form 

• $\nu < \frac{T}{\hat h}$,

where $\nu$ is frequency, $T$ temperature  and $\hat h$ a Planck constant, stating that only frequencies below the cut-off $\frac{T}{\hat h}$ are re-emitted. Absorbed frequencies above the cut-off will then be stored as internal energy in the body under increasing temperature,

Bodies which absorb all incident radiation made of different materials will have different high-frequency cut-off and an (ideal) blackbody should then be characterized as having maximal cut-off, that is smallest Planck constant $\hat h$, with the maximum taken over all real bodies. 

A cavity with graphite walls is used as a reference blackbody defined by the following properties: 

1. absorption of all incident radiation
2. maximal cut-off - smallest Planck constant $\hat h\approx 4.8\times 10^{-11}\, Ks$,     

and $\hat h =\frac{h}{k}$ is Planck's constant $h$ scaled by Boltzmann's constant $k$. 

Planck viewed the high frequency cut-off defined by the Planck constant $\hat h$ to be inexplicable in Maxwell's classical electromagnetic wave theory. In an "act of despair" to save physics from collapse in an "ultraviolet catastrophe", a role which Planck had taken on,  Planck then resorted to statistics of discrete energy quanta $h\nu$, which in the 1920s resurfaced as a basic element of quantum mechanics. 

But a high frequency cut-off in wave mechanics is not inexplicable, but is a well known phenomenon in all forms of waves including elastic, acoustic and electromagnetic waves, and can be modeled as a disspiative loss effect, where high frequency wave motion is broken down into chaotic motion stored as internal heat energy. For details, see Computational Blackbody Radiation.

It is a mystery why this was not understood by Planck. Science created in an "act of despair" runs the risk of being irrational and flat wrong, and that is if anything the trademark of quantum mechanics based on discrete quanta. 

Quantum mechanics as deterministic wave mechanics may be rational and understandable. Quantum mechanics as statistics of quanta is irrational and confusing. All the troubles and mysteries of quantum mechanics emanate from the idea of discrete quanta. Schrödinger had the solution:

• I insist upon the view that all is waves.
• If all this damned quantum jumping were really here to stay, I should be sorry I ever got involved with quantum theory.

But Schrödinger was overpowered by Bohr and Heisenberg, who have twisted the brains of modern physicists with devastating consequences... 

Unphysical Combination of Complementary Experiments

Let us take a look at how Bohr in his famous 1927 Como Lecture  describes complementarity as a fundamental aspect of Bohr's Copenhagen Interpretation still dominating textbook presentations of  quantum mechanics:

• The quantum theory is characterised by the acknowledgment of a fundamental limitation in the classical physical ideas when applied to atomic phenomena. The situation thus created is of a peculiar nature, since our interpretation of the experimental material rests essentially upon the classical concepts.
• Notwithstanding the difficulties which hence are involved in the formulation of the quantum theory, it seems, as we shall see, that its essence may be expressed in the so-called quantum postulate, which attributes to any atomic process an essential discontinuity, or rather individuality, completely foreign to the classical theories and symbolised by Planck's quantum of action.

OK, we learn that quantum theory is based on a quantum postulate about an essential discontinuity symbolised as Planck's constant $h=6.626\times 10^{-34}\, Js$ as a quantum of action. Next we read about necessary interaction between the phenomena under observation and the observer:   

• Now the quantum postulate implies that any observation of atomic phenomena will involve an interaction with the agency of observation not to be neglected. 
• Accordingly, an independent reality in the ordinary physical sense can neither be ascribed to the phenomena nor to the agencies of observation.
• The circumstance, however, that in interpreting observations use has always to be made of theoretical notions, entails that for every particular case it is a question of convenience at what point the concept of observation involving the quantum postulate with its inherent 'irrationality' is brought in.

Next, Bohr emphasizes the contrast between the quantum of action and classical concepts:

• The fundamental contrast between the quantum of action and the classical concepts is immediately apparent from the simple formulas which form the common foundation of the theory of light quanta and of the wave theory of material particles. If Planck's constant be denoted by $h$, as is well known: $E\tau = I \lambda = h$, where $E$ and $I$ are energy and momentum respectively, $\tau$ and $\lambda$  the corresponding period of vibration and wave-length. 
• In these formulae the two notions of light and also of matter enter in sharp contrast. 
• While energy and momentum are associated with the concept of particles, and hence may be characterised according to the classical point of view by definite space-time co-ordinates, the period of vibration and wave-length refer to a plane harmonic wave train of unlimited extent in space and time.
• Just this situation brings out most strikingly the complementary character of the description of atomic phenomena which appears as an inevitable consequence of the contrast between the quantum postulate and the distinction between object and agency of measurement, inherent in our very idea of observation.

Bohr clearly brings out the unphysical aspects of the basic action formula

• $E\tau = I \lambda = h$,  

where energy $E$ and momentum $I$ related to particle are combined with period $\tau$ and wave-length $\lambda$ related to wave.

Bohr then seeks to resolve the contradiction by naming it complementarity as an effect of interaction between instrument and object:

• Consequently, evidence obtained under different experimental conditions cannot be comprehended within a single picture, but must be regarded as complementary in the sense that only the totality of the phenomena exhausts the possible information about the objects. 
• In quantum mechanics, however, evidence about atomic objects obtained by different experimental arrangements exhibits a novel kind of complementary relationship. 
• … the notion of complementarity simply characterizes the answers we can receive by such inquiry, whenever the interaction between the measuring instruments and the objects form an integral part of the phenomena. 

Bohr's complementarity principle has been questioned by many over the years:

• Bohr’s interpretation of quantum mechanics has been criticized as incoherent and opportunistic, and based on doubtful philosophical premises. (Simon Saunders)
• Despite the expenditure of much effort, I have been unable to obtain a clear understanding of Bohr’s principle of complementarity (Einstein).

Of course an object may have complementary qualities such as e.g. color and weight, which can be measured in different experiments, but it is meaningless to form a new concept as color times weight or colorweight and then desperately seek to give it a meaning. 

In the New View presented on Computational Blackbody Radiation the concept of action as e.g position times velocity has a meaning in a threshold condition for dissipation, but is not a measure of a quantity which is carried by a physical object such as mass and energy.

The ruling Copenhagen interpretation was developed by Bohr contributing a complementarity principle and Heisenberg contributing a related uncertainty principle based position times momentum (or velocity) as Bohr's unphysical complementary combination. The uncertainty principle is often expressed as a lower bound on the product of weighted norms of a function and its Fourier transform, and then interpreted as combat between localization in space and frequency or between particle and wave. In this form of the uncertainty principle the unphysical aspect of a product of position and frequency is hidden by mathematics.

The Copenhagen Interpretation was completed by Born's suggestion to view (the square of the modulus of) Schrödinger's wave function as a probability distribution for particle configuration, which in the absence of something better became the accepted way to handle the apparent wave-particle contradiction, by viewing it as a combination of probability wave with particle distribution.     

Mystery of Quantum of Photoelectricity Replaced by Non-Mystery of Threshold Value

The deepest mystery of quantum mechanics is the smallest quantum of action introduced by Planck in his proof of Planck's law:

• $h\nu$ 

where $h=6.626\times 10^{-34}\, Js$ is Planck's constant and $\nu$ is frequency. 

The smallest quantum of action $h\nu$ appears in Einstein's celebrated law of the photoelectic effect:

• $h\nu  = W + P$, 

where $W$ is the energy required to liberate an electron and $P$ and the energy of a liberated electron.

Einstein's formula models generation of an electric current corresponding to $P >0$ from input of light of frequency $h\nu$, scaling with the intensity of the light.

Einstein interpreted $h\nu$ as the "smallest packet of energy" of a wave of frequency $\nu$ later named photon with $h\nu - W>0$ required for the generation of an electric current with $P > 0$. We can view

• $h\nu > W$

as a cut-off condition for photo-electricity similar to the cut-off condition in Planck's law with $T$ temperature and $k$ Boltzmann's constant:

• $h\nu > kT$

which according to Computational Blackbody Radiation expresses internal heating from exposure to radiation of frequency $\nu$. 

We can thus view both photo-electricity and internal heating from exposure to radiation as being determined by a high-frequency cut-off condition expressing that no effect occurs for input frequencies below the cut-off, and that the effect scales with the intensity above the cut-off. 

The idea of smallest packet of energy, which is strange as physical concept, can thus be replaced by the concept of threshold value on frequency or wave length, which is not strange from physics point of view. For details, see the book Computational Blackbody Radiation.

Recall that Einstein was awarded the 1921 Nobel Prize in Pbysics for his above law of photoelectricity, but it was explicitly stated in the award motivation that he was not given the Prize because of his derivation of the formula based on photons, only for the "discovery" of the formula as if formulas are laying around waiting to be discovered.

Wikileaks: Heisenberg's Uncertainty Principle Fundamentally Misleading!

Wikipedia gives the following information about Heisenberg's uncertainty principle:

• In quantum mechanics, the uncertainty principle is any of a variety of mathematical inequalities asserting a fundamental limit to the precision with which certain pairs of physical properties of a particle known as complementary variables, such as position x and momentum p, can be known simultaneously.
• Though widely repeated in textbooks, this physical argument is now known to be fundamentally misleading. While the act of measurement does lead to uncertainty, the loss of precision is less than that predicted by Heisenberg's argument; the formal mathematical result remains valid, however.
• Thus, the uncertainty principle actually states a fundamental property of quantum systems, and is not a statement about the observational success of current technology. 
• It must be emphasized that measurement does not mean only a process in which a physicist-observer takes part, but rather any interaction between classical and quantum objects regardless of any observer.
• Thus, actually states a fundamental property of quantum systems, and is not a statement about the observational success of current technology.

The uncertainty principle, which in current text books on quantum mechanics serves a fundamental role, is in fact fundamentally misleading. Wow! This must be a leak from a true Wikileaks whistleblower. Imagine what will happen if this message is understood by the scientific community.

The leak opens to a fresh look at the uncertainty principle as in Computational Blackbody Radiation suggesting the following fundamental property of atomistic quantum systems: Finite precision computation introduces a high frequency cut-off as expressed in Plank's law:

• $\nu < \frac{T}{\hat h}$ where $\hat h =\frac{h}{k}$, 

with $\nu$ frequency, $h$ Planck's constant, $k$ Bolzmann's constant and $\hat h =4.8\times 10^{-11}\, Ks$. Planck's constant $\hat h$ is then determined by the reference blackbody as the blackbody with maximal cut-off frequency (smallest $\hat h$) = peep hole of empty box with graphite walls.

The high frequency cut-off can alternatively be expressed as a restriction on wave length $\lambda$ of the form

• $\lambda > \frac{c}{T}\hat h$,

which can be seen as a smallest coordination length required for emission of radiation from atomistic oscillation subject to finite precision.

To see the connection to Heisenberg's uncertainty principle, consider a wave of frequency $\nu$ of amplitude $u_\nu$ with $\dot u_\nu \equiv\frac{du_\nu}{dt}=\nu u_\nu$ and $T =\dot u_\nu^2$ for which the high-frequency cut-off condition $\nu < \frac{T}{\hat h}$, can be expressed as

• $\dot u_\nu u_\nu  >  h$.

We see that high-frequency cut-off from finite precision computation can be seen as a substitute for an uncertainty principle which today is viewed as fundamentally misleading. 

We note that the idea of viewing the uncertainty principle as a relation between a function and its Fourier transform also seems to be fundamentally misleading. 

New Objective View of Heisenberg's Uncertainty Principle

Heisenberg's Uncertainty Principle stating a lower bound of accuracy in observation of position $x$ and momentum (velocity) $p$.

Computational Blackbody Radiation gives a new view on Planck's constant $h$ as effectively a high-frequency cut-off: Only frequencies $\nu$ such that

• $\nu < \frac{T}{\hat h}$,

will be radiated, where $T$ is temperature in Kelvin K, and $\hat h =\frac{h}{k} \approx 4.8\times 10^{-11}\, Ks$ where $k$ is Planck's constant. The cut-off condition can alternatively be expressed as 

1. $u_\nu\dot u_\nu > \hat h$

where $u_\nu$ is amplitude of wave frequency $\nu$ and $\dot u_\nu=\frac{du}{dt}=\nu u$ and $\dot u_\nu^2 =T$. 

This relation 1 is similar to Heisenberg's Uncertainty Principle as a lower bound on the product of position (amplitude) and velocity, but with a different physical meaning. Whereas Heisenberg's Uncertainty Principle concerns the product of errors in position $\Delta x$ and momentum/velocity $\Delta p$ vs Planck's constant $h$, the relation 1 concerns the product of amplitude and velocity vs the scaled Planck constant $\hat h$.  

The relation expresses that radiation of a certain frequency $\nu$ requires either a sufficiently large amplitude $u_\nu$ or velocity $\dot u_\nu$, as a requirement for coordinated oscillation under finite precision computation.

We thus replace uncertainty in observation by finite precision in actual physics, which reduces the subjective observer aspect of Heisenberg's Uncertainty Principle.  There is a connection to observation in finite precision computation cut-off of high frequencies, in the sense that only frequencies which effectively are emitted, can be observed. It is here not the observer who sets limits of observational accuracy by interacting with the observed object, but rather the object itself.

We recall that the central idea is to view physics as analog finite precision computation, which can be simulated by digitial computation allowing observation without interference and thus eliminates a basic difficulty in quantum mechanics.      

The Real Physical Meaning of Planck's Constant

The mystery of discrete lumps or quanta and the strange energy relation $E=h\nu$ with strange dimension of energy x time. 

Planck's constant $h = 6.626 \times 10^{-34} Js$ is supposed to represent a fundamental property of the Universe we happen to live in.

Prandtl introduced $h$ as a fictional mysterious quantity in his proof of Planck's radiation law in 1900 based on statistics with $h$ representing a smallest "quantum of action". Today 114 years later the fiction and mystery remains, and it is time to pass on to reality.  In the recent series of posts we have seen that $h$ enters into Planck's law in a high-frequency cut-off condition of the form

• $\nu > \frac{kT}{h}=\frac{T}{\hat h}$ where $\hat h =\frac{h}{k}$,

$\nu$ is frequency, $T$ is temperature in Kelvin $K$ and $k = 1.3806488\times 10^{−23}\, J/K$. Here  

• $\hat h =4.8\times 10^{-11}\, Ks$,

shows up as the real effective Planck constant in Planck's law. 

We see that $\frac{T}{\hat h}$ acts as a threshold value for frequency $\nu$, or equivalently $\hat h\frac{c}{T}$ acts as a threshold value for wave length $\lambda =\frac{c}{\nu}\, m$ with $c\, m/s$ the speed of light. The real effective Planck constant $\hat h$ thus has the form of a material parameter for a blackbody as a web of oscillators with a characteristic high-frequency cut-off  $\frac{T}{\hat h}$ or wave-length cut-off $\hat h\frac{c}{T}$, which expresses Wien's displacement law.

The mystery of $h$ as a "smallest quantum of action" thus can be deconstructed and the real meaning as $\hat h$ can be readily understood, all following Einstein's device to make everything as simple as possible, but not simpler. 

For the full story, see Computational Blackbody Radiation.

Newton's Laws vs Lagrange's Principle of Least Action

There is a firm belief among physicists that Nature somehow functions so as to find the least action, for example that light finds its way by choosing among all paths connecting two points the quickest path. There is good reason to question this idea on the ground that physics does not have any means to compute, store and compare action integrals, just because mathematicians can do that by writing symbols on a piece of paper.

There are two mathematically equivalent ways of describing a dynamical system of classical mechanics:

1. Newton's laws expressing equilibrium of forces.
2. Lagrange's principle of least action.

Here 1 expresses stationarity (or minimality) of the action integral

• $\int_0^T (K(t) -V(t))\, dt$, 

where $K(t)$ and $V(t)$ are kinetic and potential energies as functions of time $t$ over a given time interval $[0,T]$. We see that the dimension of the action is energy x time.

While equilibrium of forces has a direct physical reality in the sense that a dynamical system directly reacts to forces according to Newton's laws, this is less clear for Lagrange's principle of least action since it requires evaluating and comparing different action integrals and then choosing the one statisfying stationarity or minimality as the physical one. 

While the kinetic and potential energies have physical representations, the action as the integral of kinetic and potential energies does not seem to have a physical representation. We are thus led to the conclusion that Lagrange's principle of least action does not describe physics, only mathematics, and confusion arises if least action nevertheless is believed to have a physical reality.

This directly connects to the definition of Planck's constant $h$ which has the dimension of energy x time as an integral of energy over time.  The physical meaning of Planck's constant $h$ was a mystery to Planck and the mystery remains today, if you give it a careful thought and not just accept that since so many are speaking about $h$, it must have a definite meaning as an "elementary quantum of action" or something like that…or something according to your own favorite idea...