Please pay! Planck’s quantum of action – from the hotfix to quantum physics

With “World Quantum Day” on April 14th, quantum scientists want to enlighten the general public about quantum research and technology. Whether quantum physics or quantum computers, the linchpin is and remains “Planck’s quantum of action”.

h = 6,626 × 10-34 Js (joules × second) or – as particle physicists like to write in electron volts: h = 4.135 669 2(12) × 10-15 eV Hz-1 – belongs to the universal natural constants. The chosen date for today’s World Quantum Day is derived from the latter notation in rounded form of the American date format 4/14.

Max Planck “discovered” h when he tried to theoretically describe the thermal radiation of black bodies in 1899. Physicists call a black body a system that absorbs all radiation. Because such a system can be excellently described by a cavity with a tiny measurement opening, it is also referred to as “cavity radiation”.

At high frequencies, the measured values ​​could be described well with Wien’s radiation law, which was determined empirically, but which could not be derived from classical thermodynamics. It describes the spectrum using a negative exponential factor that depends on the quotient of frequency and temperature.

In this section, we present amazing, impressive, informative and funny figures from the fields of IT, science, art, business, politics and of course mathematics every Tuesday.

For his attempt at a theoretical derivation, Planck considered cavity radiation as a multitude of harmonic oscillators, taking entropy into account. From this, Planck theoretically derived a formula in the form of Wien’s radiation law, one of whose parameters is 6.62607015 × 10−34 Js was only slightly above the constant h, which was later called the constant of action. The meaning of the parameter was still completely in the dark.

Berlin-Brandenburg Academy of Sciences

Berlin-Brandenburg Academy of Sciences

In the meeting reports of the Royal Prussian Academy of Sciences in Berlin, Planck gave the value of h for the first time (still simply referred to as parameter b there), which, converted into Js, is only a few percent above the actual value of h.

(Image: Berlin-Brandenburg Academy of Sciences / emphasis heise online)

However, more detailed measurements of blackbody thermal radiation revealed that the formula was wrong for low frequencies (or long wavelengths). Instead, the spectrum could be described here using the Rayleigh-Jeans law derived from classical thermodynamics. This would in turn lead to an “ultraviolet catastrophe” at high frequencies, so it cannot be entirely correct. So the truth had to lie somewhere in between. To a certain extent, Planck created an interpolating formula as a “hotfix”, which became the Rayleigh-Jeans law for low frequencies and Wien’s radiation law for high frequencies.

Subsequently, Planck also laid the theoretical foundations by deriving his radiation law from the hotfix. However, this presupposes that the oscillators can only exchange energy in discrete (quantized) energy levels ΔE = h · f, where h is a constant and f is the frequency of the oscillator. The letter h simply stood for “auxiliary quantity”, but later established itself as an abbreviation for the quantum of action. What is special about Planck’s simple formula: it combines particle properties (energy) with wave properties (frequency). Although Planck introduced the quantization of the energy states, he did not understand this as a property of the light waves, but attributed them to the cavity radiators themselves.

The light quantum hypothesis was first introduced by Albert Einstein. He attributed the energy E = h · f to the light quanta (or photons) and was thus able to explain the photoelectric effect, which earned him the Nobel Prize in 1921 (was only awarded in 1922). Einstein recognized that electromagnetic radiation sometimes exhibited wave properties and sometimes particle properties. This wave-particle duality brought … and still brings … some physicists to despair 😉

As if that were not enough, Werner Heisenberg (the physicist, not the code name from “Breaking Bad”) derived his famous uncertainty principle in 1927, in which the quantum of action also plays an elementary role:

Δp × Δx ≥ h/4π

Δx is the inaccuracy of the measurement of the location of an object, Δp that of the impulse. One often encounters the uncertainty principle in the following notation: Δp × Δx ≥ ħ/2. ħ (“h across”) is an abbreviation for ħ = h/(2π) – physicists love it.

The fuzziness or indeterminacy principle thus states that when measuring the location of an object, its momentum is necessarily influenced. The more precisely you want to determine the location, the more imprecise the momentum measurement and vice versa. Position and momentum can therefore never be measured with any degree of accuracy at the same time. The relation can also be extended to other conjugate (related) measured quantities. Because of the tiny nature of h, the uncertainty principle usually plays no role in everyday macroscopic practice; it only becomes relevant at the microscopic level. However, it also played an important role in redefining the kilogram. In 2019 it was defined as a derived quantity from Planck’s constant, the speed of light and the cesium frequency:

1 kg = 1,475521 × 1040 (h·fCs/c²)

Some of you may remember a somewhat simplified form of the uncertainty principle from school:

Δp × Δx ≈ h

This is typically derived when explaining the wave-particle duality with a diffraction experiment at a slit:

Diffraction of an electron beam with momentum p at a slit.  The spatial blur is given by half the slit width:

Diffraction of an electron beam with momentum p at a slit.  The spatial blur is given by half the slit width:

Slit experiment for the simple derivation of Heisenberg’s uncertainty principle in typical whiteboard notation

The spatial unsharpness Δx is determined by the slit width, the momentum unsharpness by the first minimum of the diffraction pattern. The path difference of the wave at the first minimum is at least one wavelength, i.e. λ = Δx × sin α, the momentum uncertainty is Δp ≈ p × sin α. In order to establish a relationship between wavelength and momentum, one also needs the de Broglie wavelength: λ = h/p, with the help of which the equation Δp × Δx ≈ h follows. The missing factor 4π does not play a significant role, since the uncertainty principle is primarily about the orders of magnitude.

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