- 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
Exercise 1
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?
Solution to Exercise 1
- 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 : $\infty$.
Exercise 2
Let's assume the configuration space of one object is a line and the configuration space of a second object is a circle. How does the total configuration space look like?
Solution to Exercise 2
We have to glue a copy of the circle above each point of the line. What we end up with this way is a cylinder.
Exercise 3
What's the difference between the Schrödinger picture and the Heisenberg picture?
Solution to Exercise 3
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.
Exercise 4
Which formulation of Quantum Mechanics is the best one?
Solution to Exercise 4
Objectively they are all equivalent and therefore equally good. However, of course, you are free to pick your personal favorite.
From the book No-Nonsense Quantum Mechanics.