- #1
jjustinn
- 164
- 3
Note: I am NOT talking about the classical limit of quantum mechanics, where in the limit of numbers that are large compared to h the average values approach the classical values, nor am I talking about Lagrangin/Hamiltonian mechanics in phase space; I am talking about using vectors with classical-scale values in a Hilbert space INSTEAD of classical mechanics.
For example, take a bead constrained to move on a wire (so its position and momentum can be take values on the x axis), and say for this experiment we are interested if it is to the left of the origin,or at/to the right of the origin, and similarly for the momenta; therefore, our Hilbert Space contains four possible state vectors: (x<0, p<0), (x≥0, p<0), (x<0, p≥0), (x≥0, p≥0), and since it is a classical system, it will ALWAYS be in an eigenstate of the position/momentum operators, with two eigenvalues each.
It seems like a formulation like this would have great pedagogical value, since the results obtained can be compared to everyday experience (or to those calculated via the more traditional classical formalists-eg F=ma)...that is, supposing it works. I see no reason it shouldn't, but I'm having trouble wrapping my mind around it enough to work out a sample problem to test it out.
So -- does anyone know of any treatments of this, either online or in the literature? Or perhaps a simple reason why there wouldn't be any such treatments (eg “it won't work because...”)?
Thanks -
Justin
For example, take a bead constrained to move on a wire (so its position and momentum can be take values on the x axis), and say for this experiment we are interested if it is to the left of the origin,or at/to the right of the origin, and similarly for the momenta; therefore, our Hilbert Space contains four possible state vectors: (x<0, p<0), (x≥0, p<0), (x<0, p≥0), (x≥0, p≥0), and since it is a classical system, it will ALWAYS be in an eigenstate of the position/momentum operators, with two eigenvalues each.
It seems like a formulation like this would have great pedagogical value, since the results obtained can be compared to everyday experience (or to those calculated via the more traditional classical formalists-eg F=ma)...that is, supposing it works. I see no reason it shouldn't, but I'm having trouble wrapping my mind around it enough to work out a sample problem to test it out.
So -- does anyone know of any treatments of this, either online or in the literature? Or perhaps a simple reason why there wouldn't be any such treatments (eg “it won't work because...”)?
Thanks -
Justin