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
- $\nu_{max}=2.821\frac{kT}{h}$ (*)
- $\lambda_{min}= 0.2015\frac{hc}{kT}=\frac{0.0029}{T}$ meter
- $\lambda_{min} \approx 10^{-5}$m for $T=300$ K
- $\lambda_{min} \approx 5\times 10^{-7}$m for $T=5778$ K (Sun)
- 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 $h=p\lambda$.
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