3 ms·

I have read a lot of documents on ncatlab and other places to try to pin down a coherent model of the physical role of planck's constant in fourier transforms o

by wyager 2y ago

I have read a lot of documents on ncatlab and other places to try to pin down a coherent model of the physical role of planck's constant in fourier transforms on physical systems. I understand that it often serves as the scale factor for embedding the integers into the reals, but it's not totally clear to me what its role is in physical pontryagin duality/fourier transforms. It's some kind of volume in phase space, but where does that volume come from? For a constant like c, we have the narrative "c is the ratio of unit lengths in time and space", but I have not yet found a good narrative about the meaning of h that works for fourier transforms. Would appreciate any articles on the matter.

CarpaDorada 2y ago
I hope this answers your question. Let me preface this by saying that there are (probably) no satisfying answers for these questions, and that I'm not an expert. There is a classical limit <https://en.wikipedia.org/wiki/Classical_limit https://en.wikipedia.org/wiki/Classical_limit> that will recover classical equations from quantum equations from the limit ℏ -> 0. Such a thing is a heuristic, which means that we just know some equations/models where it works, but have not discovered a general truth. There are also situations where you may take c -> +∞ for example, and that would be called the non-relativistic limit. Why do we take these limits? Because when we did, the answer was not complete nonsense. We don't know what to make of them, i.e. we don't have complete theories. Also, what these limits mean is not a simple matter of calculus, they are not point-wise limits. In one such instance I've been studying for years, the WKB approximation, I've realized two things: 1) the approximations are not well understood and 2) the mathematics are quite complicated, but these points notwithstanding the equations are used in experiments. You can read the few-page introduction in "Lectures on the Geometry of Quantization" by Bates & Weinstein <https://math.berkeley.edu/~alanw/GofQ.pdf https://math.berkeley.edu/~alanw/GofQ.pdf> to see some of this, in particular the subsection "Quantization and the classical limit". I'll just quote the relevant paragraph: > Although there remain some unsettled issues connected with the question, “How can ℏ become small?” the answer is essentially the following. For any particular mechanical system, there are usually characteristic distances, masses, velocities, . . . from which a unit of action appropriate to the system can be derived, and the classical limit is applicable when ℏ divided by this unit is much less than 1. But remember, this is just one approach to the subject. Another heuristic is this: h has dimensions energy x time, which means it converts frequency into energy, e.g. E = hf. In the Fourier transform, the character is exp(2πihx·ξ), where ξ is the frequency. The effect of h -> 0 would be to dampen high-energy waves. Irregularity comes from high frequencies (think of it like this: a sum of sines of large periods would not have many kinks.) When you "iron out" the irregularity of the quantum solution, you end up with a classical one. Again disclaimer: not a physicist, nor an expert.
adrian_b 2y ago
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.

by wyager 2y ago

CarpaDorada 2y ago

I hope this answers your question. Let me preface this by saying that there are (probably) no satisfying answers for these questions, and that I'm not an expert. There is a classical limit <https://en.wikipedia.org/wiki/Classical_limit https://en.wikipedia.org/wiki/Classical_limit> that will recover classical equations from quantum equations from the limit ℏ -> 0. Such a thing is a heuristic, which means that we just know some equations/models where it works, but have not discovered a general truth. There are also situations where you may take c -> +∞ for example, and that would be called the non-relativistic limit. Why do we take these limits? Because when we did, the answer was not complete nonsense. We don't know what to make of them, i.e. we don't have complete theories. Also, what these limits mean is not a simple matter of calculus, they are not point-wise limits. In one such instance I've been studying for years, the WKB approximation, I've realized two things: 1) the approximations are not well understood and 2) the mathematics are quite complicated, but these points notwithstanding the equations are used in experiments. You can read the few-page introduction in "Lectures on the Geometry of Quantization" by Bates & Weinstein <https://math.berkeley.edu/~alanw/GofQ.pdf https://math.berkeley.edu/~alanw/GofQ.pdf> to see some of this, in particular the subsection "Quantization and the classical limit". I'll just quote the relevant paragraph: > Although there remain some unsettled issues connected with the question, “How can ℏ become small?” the answer is essentially the following. For any particular mechanical system, there are usually characteristic distances, masses, velocities, . . . from which a unit of action appropriate to the system can be derived, and the classical limit is applicable when ℏ divided by this unit is much less than 1. But remember, this is just one approach to the subject. Another heuristic is this: h has dimensions energy x time, which means it converts frequency into energy, e.g. E = hf. In the Fourier transform, the character is exp(2πihx·ξ), where ξ is the frequency. The effect of h -> 0 would be to dampen high-energy waves. Irregularity comes from high frequencies (think of it like this: a sum of sines of large periods would not have many kinks.) When you "iron out" the irregularity of the quantum solution, you end up with a classical one. Again disclaimer: not a physicist, nor an expert.

adrian_b 2y ago

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.