Part 3 Bonus Material

No-Nonsense Quantum Mechanics

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 : $\infty$.

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?

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.

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.

Which formulation of Quantum Mechanics is the best one?

Objectively they are all equivalent and therefore equally good. However, of course, you are free to pick your personal favorite.

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Edi

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.