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.

## Discussion