Mike2 said:
OK then... what physical thing is being described by the imaginary portion of Schrodinger's eq?
This is actually a very subtle and interesting question!
Well, sort of.
When you separate time and space variables, the space equation is purely real (though it may have complex solutions of course). On the other hand, the time equation is of the form:
i\hbar\frac{\partial T}{\partial t} = E T(t)
where E is the separation of variables constant and as we all know, turns out to be the energy of the solution. Before I go ranting on about the solutions, compare the above to the time equation derived from the electromagnetic wave equation:
\frac{\partial^2 T}{\partial t^2} = -\omega^2 T(t)
It looks like the latter is just the second order form of the first. However, note that sin(wt) and cos(wt) can be solutions to the latter independently. That is, both:
T_{1}(t) = A_{1}\cos{\omega t}, T_{2}(t) = A_{2}\sin{\omega t}
are solutions to the EM time equation. On the other hand, T1 and T2 are *not* solutions of the QM time equation. A particular linear combination is, though - if A2 = A1, then:
T_{3}(t) = T_{1}(t) - iT_{2}(t) = A_{1}(\cos{\omega t}-i\sin{\omega t})
is a solution provided \omega = \frac{E}{\hbar}. But the solution above can be recognised as our old friend:
T_{3}(t) = A_{1} e^{-i\omega t}
the complex exponential. Notice that physically, the electromagnetism solutions can be made to change phase forwards or backwards in time (for a given "forwards" direction) - they are symmetric in that respect, whereas the first-order QM equation only allows one direction of time evolution - to obtain time-reversed solutions one must change the sign of the i in the TDSE.
Now, why does this matter?
Well, let's examine the *observable* probability distribution. If a solution to the TISE is described by a space-solution multiplied by a time-varying sine (say), our probability distribution would look like: (* denotes complex conjugation)
\Psi^{*}\Psi = \psi(\vec{r})^{*}\psi(\vec{r})\sin^{2}{\omega t}
That is, the probability distribution would *change* periodically in time. This would be fine, say, for linear combinations of stationary states, but certainly not for single stationary states. On the other hand, when the complex exponential is used, the time variance with the probability distribution is not present. This is one way of explaining what the 'stationary' in 'stationary state' means. Indeed, if we push degeneracy to one side, every stationary state is forced by the Schrodinger equation to have a time-independent probability distribution.
So both the EM and QM time equations allow us to assign a number called 'frequency' to the periodic phase changes that the wave functions go through. However, the fact that the TISE is first order in time forces us to use a particular kind of function to represent this periodic aspect of a quantum mechanical object.
Of course, the EM equation admits the complex exponentials (in both time directions) as solutions too, but I'm not sure whether that is interesting here.
As a side note, I should point out that the complex exponential here is one of a more general class of
unitary operators that *implement* symmetries on the underlying state space. Every unitary operator has the form:
U = e^{iS}
where S is the infinitessimal generator of the operator. What this exponential actually *means* is basically the subject of the functional calculus. Unitary operators have the property that they don't change the probability distribution. The simplest subclass of such operators are the constant phase factors e^{i\lambda}, where lambda is a real number.
This is sort of what I meant when I said the framework of Quantum Mechanics is strongly effected by the fact that the operators are complex. As has been mentioned, we could switch to a two-component formalism such as is used in electromagnetism, and we would then get real operators, but there isn't really much point.
Regards,
Kane O'Donnell
)
I think what confuses the issue is what we percieve if light appears as both a wave and a particle whichever way we look at it we have to take into account that we might not in fact be being objective, we are all just conditioned by DNA to see what isn't there? Or are we? I'm fed up of people asking this very question and being bombarded with so called proof? we must use philosophy to discuss physics; that why it's called a Doctorate of Philosophy, you are all just being sophists, sophistry is fine but don't say its prooven till the results are irrefutable, then question your own frame of reference and proove it wrong 
