Why Eigenvalues in Quantum Measurements

Most recent answer: 07/31/2014

Q:
In quantum mechanics there is state of the system defined by the wavefunction. It contains information about all the physical quantities associated with the system. There is a way to extract this information and it is by operation on wavefunction. That is we operate on wavefunction an operator associated with the physical quantity and then we expect the result to be multiple of the wavefunction. And if this is so then we say that the "multiple" (eigenvalue) is the value of the physical quantity that the system can possess. Why this is so? Why eigenvalues are only allowed? And not the values calculated by some other way.
- Ashutosh (age 20)
Mumbai,India
A:

That's a very deep question I an give a partial answer.

What is the apparatus that measures the quantity X? By definition, it's something that acquires different macroscopic (large-scale) states depending on the value of X. Now let's say you saw an output of the measurement operation that was not in an eigenvalue of X. Then, by definition of the measurement apparatus, it would also not be in an eigenstate of the macroscopic properties. You would be seeing, for example, some superposition of a detector needle in visibly different positions.

Now here's where my answer gets fuzzy. There's a rule of nature that our experience has only definite values of macro phenomena. Cats are not found in states consisting of very healthy and very dead versions. How that rule arises is at the core of the "measurement" problem. Is it that the full quantum state, with all its parts, emerges, but our conscious experience branches into separate macro-coherent streams? Do all the streams except one somehow collapse away? Was there some sort of (non-local) hidden fact-of-the matter beforehand, so that only one of the streams is fully real?

You can find more on this by searching this site and others for "quantum measurement", Bell Inequalities", "Many Worlds", "Copenhagen", etc.

Mike W.


(published on 07/31/2014)