Different classical limits of quantum mechanics

[spaghetti3451]

There is only one way to reduce the equations of special relativity (aka Lorentz Transformations) to the equations of Newtonian mechanics (aka Galilean Transformations).

In light of the above, why are there multiple ways to reduce quantum-mechanical equations of motion into classical equations of motion? For example, you could either take the Planck's constant to zero, or take the quantum numbers to infinity to reduce to the classical behaviour.

 

[Nugatory]

There is only one way to reduce the equations of special relativity (aka Lorentz Transformations) to the equations of Newtonian mechanics (aka Galilean Transformations).

In light of the above, why are there multiple ways to reduce quantum-mechanical equations of motion into classical equations of motion? For example, you could either take the Planck's constant to zero, or take the quantum numbers to infinity to reduce to the classical behaviour.

They two approaches are pretty much the same thing: we're considering the limit as the spacing between adjacent energy levels approaches zero.

I could as easily say that there are multiple ways of reducing the equations of SR to Newtonian mechanics: take $c$ to be infinite, take $v/c$ to be zero, take $\gamma$ to be one.

 

[spaghetti3451]

Is it always true in physics that a limiting theory can be derived from a general theory by taking the limit of one unique parameter?

 

[Vanadium 50]

Did you not read Nugatory's post? He provides a counterexample right there!

 

[jerromyjon]

take v/c to be zero

I had to think a minute about that one but the others are obvious.

 

[spaghetti3451]

Did you not read Nugatory's post? He provides a counterexample right there!

Well, the parameters $c$, $\frac{v}{c}$, and $\gamma$ are all related to each other, as these parameters are simply various arithmetic combinations of $v$ and $c$. That, I understand.

However, what I was actually wanting to ask is if there are two or more parameters, $\textit{which cannot be related to each other in any which way}$, which still produce the special limit of $\textit{some}$ general theory.

 

[Vanadium 50]

If you can reach the same limit by taking two different parameters to some limit, are not the two parameters related?

 

[spaghetti3451]

Yes, they must be! It's clear to me now!

I was wondering if there is an introduction to quantum mechanics that follows naturally from classical mechanics.

In classical mechanics, there are three equivalent formalisms: Newtonian, Lagrangian, and Hamiltonian.
In quantum mechanics, there are also three equivalent formalisms: Schrödinger, Heisenberg, and Feynman.
It turns out that Schrödinger and Heisenberg formalisms are differential and integral forms of the same underlying principle, and generalise Hamiltonian mechanics.
Furthermore, Feynman's path integral formalism generalise Lagrangian mechanics.

Is there no quantum- mechanical generalisation of Newton's laws of motion?

Also, is there some textbook or resource that brings out the relationship of the Schrödinger and Heisenberg formalisms to Hamiltonian mechanics elegantly and in comprehensive detail: for instance, I understand that variables generalise to operators and Poisson brackets generalise to Lie brackets, but I don't see how eigenvalues arise in quantum mechanics from classical mechanics.

 

[andresB]

You are seeing it backwards. Quantum mechanics don't arises from classical mechanics, at most is the other way around.

And yes, there is a quantum mechanics-like description of classical mechanics(perhaps its more a description of classical statistical mechanics), but it is not a generalization but just a way to describe it with a different mathematics .