Thursday, May 6, 2010

More Wave Function Stuff

A few comments about the last post:

If you're not familiar with the density matrix formulation, you might be suspicious about my claim. Does the statistical interpretation really do away with wave function collapse, or did I somehow hide a collapse inside the formalism? But it really does the job. There is only one postulate needed to prove that the final mixed state is the correct description of the beam state, that is the postulate that expectation values (average values) of measurable quantities are given by the usual quantum mechanical expression.

In fact, the statistical interpretation is essentially what Max Born originally proposed when he introduced the probability interpretation of the wave function way back in the early days of quantum mechanics. So why did the collapse idea become so prominent?

For one simple reason: in the statistical interpretation, the wave function is only something that keeps track of what we know about the system. It is not an object/entity that exists physically in space and time.

This is a radical departure from normal physics. Physicists are used to thinking of their abstractions - electric and magnetic fields, for instance - as things that really exist "out there." And it makes sense, intuitively, that if there are rules about the how the world behaves, they ought to be rules about the things that exist objectively in the universe. And these rules are expressed mathematically.

But if the wave function doesn't exist "out there", if it is only an expression of what we know, then why should it obey a mathematical equation? Why should our knowledge about the system follow a strict mathematical law?

So many physicists preferred to think of the wave fucntion as something that exists "out there." But when you do that, you suddenly discover the collapses. Every time you learn something new about the system (perform a measurement), you need to discard your old wave function and replace it with a new one. Hence, all the to-do about collapsing wave functions.

Notice, though, that the collapses happen precisely when we learn something about the system. That is, they happen when our information changes. So, to me, it makes eminent sense to think of the wave function as embodying our knowledge of the system, rather than thinking of it as an independently existing object. It is then perfectly natural that it should change whenever our information about the system changes.

But then we are left with two rather puzzling questions:
  1. Why should our knowledge about a physical system obey a mathematical equation?
  2. What is the physical reality that the wave function describes - and why can't we just write an equation for it?
I don't know the answers. As far as (1), I would just comment that it's not so clear why the physical objects themselves should obey mathematical equations, either. But we are so used to the idea that it doesn't really seem strange - until we stop to think about it. And as for (2) - maybe this hints at the true message of quantum mechanics: that we can only have imperfect knowledge of any physical system. The ultimate reality might be inaccessible to us.

No comments:

Post a Comment