Understanding the Deterministic Evolution of Wave Functions in Quantum Mechanics

In summary, the postulate of quantum mechanics states that an isolated system's wave function will evolve deterministically, but only the measurements of observables are not deterministic. This is based on the assumption of continuity and Wigner's Theorem, which leads to the conclusion that the time operator must be a linear unitary operator and the energy operator uniquely determines the time operator, making it deterministic. This assumption is also supported by the expectation of an operator going over to the classical system equation and the classical version of the Hamiltonian. Although this may seem like a purely mathematical concept, it has been proven through further research on the foundations of quantum mechanics. However, it is not often presented in this way due to the complexity of the mathematical concepts involved.
  • #1
cragar
2,552
3
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?
 
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  • #2
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
 
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  • #3
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.
 
  • #4
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
 
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1. What are observables in science?

Observables in science are physical quantities or properties that can be measured or observed. They are used to describe and understand the behavior of a system or phenomenon.

2. How are observables different from variables?

Observables and variables are closely related, but they have some key differences. Variables are typically used in mathematical equations to represent a quantity, while observables are directly measured or observed in an experiment. In other words, variables are often used to predict or explain observables.

3. What is the importance of observables in scientific research?

Observables are crucial in scientific research as they allow us to gather data and make objective conclusions about the natural world. They provide us with quantifiable information that can be used to test hypotheses and theories, and ultimately advance our understanding of the universe.

4. How are observables used in quantum mechanics?

In quantum mechanics, observables are represented by operators that act on the wave function of a system. These operators correspond to physical properties, such as position, momentum, and energy, and their eigenvalues represent the possible outcomes of measurements.

5. Can observables change over time?

Yes, observables can change over time. This is often seen in dynamic systems or processes, where the value of an observable may vary depending on the state of the system at a given time. Observables can also change due to external factors, such as environmental conditions or experimental manipulations.

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