In the case of the angle/angular momentum pair, since angles are dimensionless, angular momentum itself has same the units as an action, and one can swap measuring in units of $\hbar $ with measuring in units of an angular momentum characteristic of the system, as Carlo notes.įor any dimensionful quantity, one has to provide a scale. So, our inquisitive nature leads us to organize our physical quantities in conjugate pairs, and in each instance, the product of the pair has dimensions of an action - hence, action takes on such a central role. These operations form groups - the described structure is very natural from that point of view.
Fourier transform, the phase in your integral is $e^ L/\hbar )$. Now what is the impact? If you think of e.g. In some sense, this makes all the difference: in your favorite unit system, $\hbar$ has the numerical value $1$. The first observation is that Planck's constant has units, it is not a numerical constant but carries physical dimension. But if I understand correctly, you want to gain intuition ) Of course, the precise mathematical meaning is perhaps absent, so the answers are sort of heuristic. Related MO questions: Does quantum mechanics ever really quantize classical mechanics? Another purpose is that the Planck constant plays almost no role (and, in fact, is hardly mentioned) in the literature on quantum computation and quantum information, and I am curious about it. One purpose of this question is for me to try to get better early intuition towards a seminar we are running in the fall. What is the mathematical and physical meaning of letting the Planck constant tend to zero? (Or to infinity, if this ever happens.) Motivation: How does the Planck constant relate to the uncertainty principle and to mathematical formulations of the uncertainty principle? What is the role of the Planck constant in mathematical quantization? How does the Planck constant appear in mathematics of quantum mechanics? In particular, quantization is an important notion in mathematical physics and there are various forms of quantization for classical Hamiltonian systems. What are simple ways to think mathematically about the physical meanings of the Planck constant?
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