Understanding the Deterministic Evolution of Wave Functions in Quantum Mechanics

[cragar]

One of the postulates of QM is that if the system is isolated from external interaction
that its wave function will evolve deterministically. So just the measurements of observables are not deterministic. What is our reason for assuming that the wave function will evolve deterministically?

 

[bhobba]

It's from the assumption that the time operator that transforms the vector space to another vector space is continuous. With that assumption, and Wigners Theorem, it turns out the time operator must be a linear unitary operator hence transforms Linear Observables to other linear Observables. Because of this and Stones Theorem the time operator has a generator that uniquely determines it and by definition is the energy of the system. Thus knowledge of energy operator uniquely determines the time operator so is deterministic. And it can be proven from the assumption of Galilaen invariance the Energy operator has the form standard to classical mechanics - where it is called the Hamiltonian -you will find a proof of this in Ballentine - QM A Modern Development Chapter 3. So basically determinism follows from the very reasonable assumption of continuity.

In applying it, it is assumed, again quite reasonably considering the theorem proved in Ballentine and the expectation of an operator goes over to the classical system equation, the Hamiltonian of the system you are quantising, is the same as the classical version.

Thanks
Bill

 

[cragar]

ok thanks for your response. You said it follows from continuity. When they first thought of this, was it viewed as more of a math fact or did they have intuition and they thought this is how nature worked.

 

[bhobba]

When they first thought of this the foundations were a mish mash and Schrodengers equation was assumed which implies determinism. Since then much work has been done on the foundations by people such as Wigner, Stone, Von Neumann and others and the view now is what I basically posted.

The reason it is not usually presented that way is it involves what mathematicians call decidedly non trivial mathematics (euphemism for difficult for guys like you and me to unserstand). However if you want to work through the detail the following is a good place to start:
https://www.amazon.com/dp/0387493859/?tag=pfamazon01-20

Thanks
Bill