Part 3 Bonus Material

No-Nonsense Quantum Mechanics

- A short but illuminating discussion of the
**different ways to describe Quantum Mechanics**is Nine formulations of quantum mechanics by D. Styer et. al. - Good starting points for some more information and further references on how
**walking droplets can mimic quantum behavior**are the Wikipedia article on hydrodynamic quantum analogs and the dotwave website. A good summary is the article titled “Walking Droplets – A form of wave-particle duality at macroscopic scale?” by Y. Couder et. al. - For another gentle introduction to
**path integrals**, see Path Integrals in Quantum Theories: A Pedagogic First Step by R. Klauber. - For an extremely interesting essay about the question how the concept “
**phase space**” was invented, see The tangled tale of phase space by D. Nolte - Lots of additional examples of product spaces can be found here.

Exercises

How many dimensions do the following mathematical arenas for a system consisting of N free particles have:

- everyday space
- configuration space
- phase space
- Hilbert space?

- everyday space : 3.
- configuration space : 3N . The configuration space of a free particle is 3-dimensional. Thus, for N particles we glue N differnt 3-dimensional configuration spaces together and the resulting space is 3N-dimensional.
- phase space : 6N. The phase space of a free particle is 6-dimensional: 3 to specify the location and 3 to specify the momentum. Thus for N particles we get a 6N-dimensional phase space.
- Hilbert space : $\infty$.

What's the difference between the Schrödinger picture and the Heisenberg picture?

Both pictures are formulation in Hilbert space but:

- In the Schrödinger picture, the states evolve in time while the operators do not change.
- In the Heisenberg pictures, the operators evolve in the and the states do not change.

In what sense is the pilot wave formulation in physical space? The functions rho, S and Q in (14.2) and (14.3) are functions of configuration space, if I understand correctly. Of course, if there is only one particle, configuration space turns into physical space. But in general, if there is more than one particle, the pilot wave formulation appears to be in configuration space.