Multiple bound states for the Schrödinger-Poisson problem

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Mathematically, one has several ways to incorporate "dimensional" constants. One nice way is to look at graded algebras where the grading refers to the power of the dimension you are looking at. Then the grading helps to keep track of the correct dimensions. In particular in quantization theory this turned out to be a very useful tool. Equivalently, it could represent an angular momentum, I will get back to that at the end. To properly define this path integral is the central problem of the mathematics of quantization. There is no unique prescription for this. In that limit one will find that paths with very large velocity dominate.

[PDF] Ground and bound states for a static Schrodinger-Poisson-Slater problem - Semantic Scholar

This is one manifestation of the uncertainty principle. Finally, the OP remarks on the absence of Planck's constant in the literature of quantum computation and quantum information. To supplement the excellent answers previously given, some more remarks situated in the canonical quantization formalism, which is equivalent at physicist level to the path integral formalism referred to by Carlo Beenakker:. Soon afterwards, as inquisitive beings, we try to figure out how we can move around in our playground. We define conjugate objects, which, quantum mechanically, are the generators of these movements.

These operations form groups - the described structure is very natural from that point of view.

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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. For any dimensionful quantity, one has to provide a scale. It doesn't really make sense to declare a dimensionful quantity "large" or "small" without stating what one is comparing to. A fly will disagree with me about the size of a penny.

One place where this becomes very apparent is, indeed, in uncertainty relations. If the typical positions and momenta in the system are very large, as in a classical system, the uncertainties forced by this inequality become negligible by comparison, and quantum effects become invisible.

INFINITELY MANY BOUND STATE SOLUTIONS OF SCHRÖDINGER-POISSON EQUATIONS IN R 3 ∗

Note that we don't argue in quite the same simple fashion for the other conjugate pairs mentioned above. Until we consider relativity. The case of angles is tricky because of the compact nature of an angle - so, also there, we set up the algebra differently. So in some sense it might be reasonable to say that the quantitative measure of the departure from commutativity—that is, Planck's constant—provides at the same time a definite lower bound indicating the limits imposed by physical reality itself and not technological limitations of our probabilistic knowledge, as this is described in the frame of QM.

Planck considered a multiple-oscillator model with discrete energy spectrum borrowing an older "trick" of Ludwig Boltzmann , he then obtained an expression for the entropy per oscillator and demanded that this expression should be consistent with Wien's law of radiation. On the mathematics side, things are not so straightforward, mainly because, as mentioned by the OP,. I think the simplest viewpoint is the one provided by formal deformation quantization : QM is considered as a deformation of classical mechanics, with the Planck constant being the deformation parameter.

This, combined with the Wigner—Weyl transform , provides the opportunity to consider QM as a "smooth" deformation of the algebra of classical observables rather than a "sharp" or "discontinuous" change of our view of the mathematical nature of the observables functions on a manifold which suddenly become operators on a Hilbert space.

Regarding Q4 , i understand that the question is intimately related to the Correspondence principle or the so-called classical limit of QM introduced by Niels Bohr. In my understanding this is not a rigorous mathematical postulate but rather a heuristic argument quite common in the development of physical theories.


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Interesting details on the origin and the motivations of this description can be found in: The classical limit of quantum mechanical correlation functions , K. Hepp, Comm.

In the classical limit, QM probability distributions are expected to be identified with the probability distributions of suitable ensembles of classical trajectories. Brown, 40, , where this last point is analyzed in a quite rigorous and technical manner. Maybe the article: The classical limit of quantum theory, R. So the numerical value of Planck's constant has no significance.

More generally, it comes about because the position and momentum operators don't commute. This has implications for statistical mechanics. The third law of thermodynamics holds because of this discreteness of phase space. This is not the only way of making the correspondence principle pop out, nor does it always work. Another common way in which this limit occurs is the limit of large numbers of particles.

Another purpose is that the Planck constant plays almost no role and, in fact, is hardly mentioned in the literature of quantum computation and quantum information and I am curious about it. If you look at axiomatic formulations of quantum mechanics, they basically are descriptions of a kind of information theory. They don't have anything to do with space or motion. Examples below. The first, called the Kibble balance, works by offsetting the downward force of gravity on a chunk of metal with an upward magnetic force on a coil held in a magnetic field.

The second experiment, conceived by the International Avogadro Project, involves fabrication of a near-perfect sphere of silicon. As a result, the kilogram inherits the uncertainty that previously appeared in the Planck measurement. A better physical definition of the Planck permits normalization. A simple explanation without too much overlap is on the Wikipedia webpage " Natural Units ":. In physics, natural units are physical units of measurement based only on universal physical constants.

You need to be careful not to mix dimensioned with dimensionless quantities when using Planck base units and Hartree atomic units. You also can not normalize all your constants and need to choose your set carefully.


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In Wikipedia's explanation of derived units it mentions:. With Planck units, the units are defined by properties of quantum mechanics and gravity. Not coincidentally, the Planck unit of length is approximately the distance at which quantum gravity effects become important. Likewise, atomic units are based on the mass and charge of an electron, and not coincidentally the atomic unit of length is the Bohr radius describing the "orbit" of the electron in a hydrogen atom. In the Hartree system the numerical values of the following four fundamental physical constants are all unity by definition:.

In Hartree atomic units, the speed of light is approximately atomic units of velocity.

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See the webpage: " 8. Cosmology Main article: Chronology of the Universe. There is no currently available physical theory to describe such short times, and it is not clear in what sense the concept of time is meaningful for values smaller than the Planck time. It is generally assumed that quantum effects of gravity dominate physical interactions at this time scale. At this scale, the unified force of the Standard Model is assumed to be unified with gravitation. How does the Planck constant appears in mathematics of quantum mechanics. What is the role of the Planck constant in mathematical quantization.

See: " Planck's constant in geometric quantization ".

4. Solutions to Schrödinger Equation, Energy Quantization

How does the Planck constant relate to the uncertainty principle and to mathematical formulations of the uncertainty principle. Any calculation of the position and momentum of an object at the quantum level involves some uncertainty and Heisenberg's uncertainty principle states that complementary properties cannot be observed or measured simultaneously.

What is the mathematical and physical meaning of letting the Planck constant tending to zero. When and why does it fail? Waves electromagnetic, acoustic, etc carry energy.