Is there a law which prohibits measuring En of a non-stationary state

[cooev769]

Hey.

Trying to wrap my head around this maths. And given that the wave function is a superposition of a bunch of stationary states, each with a different coefficient. The coefficients squared added add to one. And the probability of finding the particle in a given state is cn^2. I know all of this and I know that if you observe the particle you will find it in one of the stationary states. But is there anything in the maths which ensures you cannot measure the En to be say the E=E1+E2, why can't you measure energies in between, is there any mathematical rule which prohibits this. This still just rattles my brain.

Thanks.

 

[king vitamin]

The law you're looking for is known as the Born rule. It's not mathematical, it is one of the fundamental postulates of quantum mechanics - you will only get one of the energy eigenstates when you measure energy.

 

[cooev769]

Okay so it's just been observed and become a postulate of quantum mechanics?

 

[king vitamin]

Yup. It's an experimental fact that bound states have discrete energies, and a quantum theory needs to reproduce the discrete measurements as well as the probability distribution for the spectrum for a given state.

 

[bhobba]

Just to elaborate further its in the basic postulates of QM.

To fully understand it you need to see a proper axiomatic treatment.

I STRONGLY recommend getting a hold of Ballentine - Quantum mechanics - A Modern Development:
https://www.amazon.com/dp/9810241054/?tag=pfamazon01-20

Read the first 3 chapters and all will be clear - QM is basically the working out of just two axioms - the second one being the Born Rule mentioned previously in this thread.

That text is mathematically advanced, but don't worry about that, just skip the derivations that are a bit hairy and you will get the gist.

Interestingly Born's rule is not entirely independent of the first axiom being to some extent implied from the first via Gleason's Theorem.

QM from just one axiom. Obviously not - but that it can be reduced to reasonable assumptions is very interesting - the key one of which is non contextuality - but that is just by the by.

Thanks
Bill

 

[cooev769]

Yeah that is rather baffling. It would make sense to think that applying the energy operator which gives you the expected energy would be what is measured. Is there a reason for this, or is that just one of the mysteries of QM

 

[WannabeNewton]

But is there anything in the maths which ensures you cannot measure the En to be say the E=E1+E2, why can't you measure energies in between, is there any mathematical rule which prohibits this.

When one solves the Schrödinger equation for the energy spectrum of a given system one must impose boundary conditions on the solutions to the Schrödinger equation-it is these boundary conditions that restrict the energy spectrum to discrete values for bound systems like an infinite potential box or the hydrogen atom.