3 ms·

Like the speed of light in vacuum provides a natural unit for velocity, i.e. for the ratio between length and time, Planck's constant (computed by him in 1899,

Like the speed of light in vacuum provides a natural unit for velocity, i.e. for the ratio between length and time, Planck's constant (computed by him in 1899, before any quantum theory, because it is a consequence of the laws of blackbody radiation) provides a natural unit for angular momentum, i.e. for the ratio between kinetic energy and frequency (a.k.a. angular velocity). (Many books claim that Planck's constant is a quantum of "action". This is extremely wrong, because action is not a quantized physical quantity, so it cannot have quanta, and moreover the ratio between energy and frequency is not an action, but an angular momentum. This ridiculous mistake is caused by the failure to understand that the unit of plane angle is a base unit that cannot be derived from any of the units of the other physical quantities and forgetting to write the plane angle in the dimensional relationships between physical quantities leads to very serious errors. Angular momentum is the ratio between action and plane angle, while energy is the ratio between action and time and the linear momentum is the ratio between action and length. Frequency is not the inverse of time, but the ratio between plane angle and time, as the old name of "Hertz", i.e. "cycles per second" was making this obvious.) Dividing or multiplying by combinations of Planck's constant with the speed of light and the elementary charge only changes the system of units, between the traditional units and natural units. There are several variants of "natural" systems of units and the difference between using them and using SI is that when using natural systems of units there are much less "universal" constants in the relationships between physical quantities. Like electric charge, angular momentum is one of the quantities that are discrete, not continuous (though angular momentum is not necessarily discrete like electric charge; depending on the physical system it may be either discrete or continuous). Like the elementary charge is the quantum of electric charge (when all quarks are bound in hadrons), a half of Planck's constant is the quantum of angular momentum (there are multiple possible definitions of the angular momentum, depending on the unit chosen for plane angle, which lead to multiple possible numeric values for Planck's constant). The importance of the c, e and h constants is due to them being respectively the limit value for velocity and the quanta of two fundamental discrete quantities (electric charge and angular momentum), which makes them appear in many relationships between physical quantities, unless the system of units is changed to a "natural" system of units, when the "universal" constants become "1", so they disappear from the formulae. When a natural system of units is not used in quantum mechanics, then all the equations that contain both energy or momentum and time or length, like the equation of Schroedinger, will contain Planck's constant, possibly combined with other universal constants. This will have as a consequence the appearance of those constants also in the expressions that give the pairs of quantities related by Fourier transforms.

wyager 2y ago
I don't think your explanation hits the bottom of the ontic causal hierarchy. The fact that hbar is the quantum of angular momentum can be derived from the boundary conditions of a wavefunction with a closed dimension (such as an angle), combined with the fact that one h of action phases a wavefunction by 360º. In particular, for a wavefunction with only angular dependence: boson: \psi(2pi) = \psi(0) fermion: \psi(4pi) = \psi(0) Because the value has to be the same when you get back to where you started. Therefore, (for a boson) psi must decompose into eigenfunctions of the form \psi(\theta) = exp(i * n * \theta) where n is an integer. These eigenfunctions satisfy the boundary condition stated above and are a complete basis We can take the fourier transform of this (closed) function and we get a dirac delta at n. I.e. the fourier transforms of this basis are just dirac deltas at integers in the fourier domain. But to convert this to physical units, we need to use the embedding from integers to reals, which is done by multiplication by planck's constant. So, some related questions that I think are further down the causal hierarchy: * Why does one h of action phase a wavefunction by 360º * Why does h necessarily show up in the fourier transform

wyager 2y ago

I don't think your explanation hits the bottom of the ontic causal hierarchy. The fact that hbar is the quantum of angular momentum can be derived from the boundary conditions of a wavefunction with a closed dimension (such as an angle), combined with the fact that one h of action phases a wavefunction by 360º. In particular, for a wavefunction with only angular dependence: boson: \psi(2pi) = \psi(0) fermion: \psi(4pi) = \psi(0) Because the value has to be the same when you get back to where you started. Therefore, (for a boson) psi must decompose into eigenfunctions of the form \psi(\theta) = exp(i * n * \theta) where n is an integer. These eigenfunctions satisfy the boundary condition stated above and are a complete basis We can take the fourier transform of this (closed) function and we get a dirac delta at n. I.e. the fourier transforms of this basis are just dirac deltas at integers in the fourier domain. But to convert this to physical units, we need to use the embedding from integers to reals, which is done by multiplication by planck's constant. So, some related questions that I think are further down the causal hierarchy: * Why does one h of action phase a wavefunction by 360º * Why does h necessarily show up in the fourier transform