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"For my PhD thesis I found out that you didn't need an orthogonal basis set to do quantum mechanical calculations but rather you could do it just fine with an o

by ianai 5y ago

"For my PhD thesis I found out that you didn't need an orthogonal basis set to do quantum mechanical calculations but rather you could do it just fine with an overdetermined basis set such as coherent states over p and q so long as you had resolution of unity over the basis set" Isn't this just a fact from the definition of R^n topologies? i.e. There are only n-basis vectors possible and any transformation of them which is still R^n will be linearly related to those original n-basis vectors. (It's been over a decade since LA, though, so I'm not 100% on my verbiage.) Put another way, any non-zero determinate/linearly independent set of vectors within R^n can have at most n-elements.

PaulHoule 5y ago
... but also the vector space can be infinite (like position q or momentum p) which are very familiar but you can take the trace integrating over both q and p to get the density of states. There is a community in chemical physics that does this all the time and here is a run-of-the-mill paper that uses the method. https://www.researchgate.net/publication/263170709_Wavepacket_dynamics_and_quantum_mechanical_energy_densities_in_the_quartet_N_2_O_2_system https://www.researchgate.net/publication/263170709_Wavepacke... Roughly a wavepacket which is localized in both q and p propagates on a path similar to a classical particle located at q, p and you can use trick to do quantum mechanical calculations based on the classical mechanics. It is not a method of investigating the "classical quantum correspondence" but rather a method of calculating QM quantities.
- ianai 5y ago
  Which is what makes simulating things within QM compute-able in finite time, too, as I recall. I wonder if unitarity could be tested with analogs to black holes like the sonic analog. Otherwise this seems testable in a weak sense. [edit] I just want to point out that, to my intuitive idea of how a black hole would work in string theory - it seems all the information would not be lost. It would be compressed to the event horizon along with everything else.

by ianai 5y ago

PaulHoule 5y ago

... but also the vector space can be infinite (like position q or momentum p) which are very familiar but you can take the trace integrating over both q and p to get the density of states. There is a community in chemical physics that does this all the time and here is a run-of-the-mill paper that uses the method. https://www.researchgate.net/publication/263170709_Wavepacket_dynamics_and_quantum_mechanical_energy_densities_in_the_quartet_N_2_O_2_system https://www.researchgate.net/publication/263170709_Wavepacke... Roughly a wavepacket which is localized in both q and p propagates on a path similar to a classical particle located at q, p and you can use trick to do quantum mechanical calculations based on the classical mechanics. It is not a method of investigating the "classical quantum correspondence" but rather a method of calculating QM quantities.

ianai 5y ago
Which is what makes simulating things within QM compute-able in finite time, too, as I recall. I wonder if unitarity could be tested with analogs to black holes like the sonic analog. Otherwise this seems testable in a weak sense. [edit] I just want to point out that, to my intuitive idea of how a black hole would work in string theory - it seems all the information would not be lost. It would be compressed to the event horizon along with everything else.