|Sep25-04, 12:25 PM||#1|
what is a probabilty wave?
In my last physics course, we spent the last couple of weeks introducing quantum mechanics, covering the schroedinger wave equation and simple examples like a particle in a box. So I don't know much about QM, and this may be a stupid question.
Anyway, we were taught that the particle can be thought of as a wave, described by the wave equation. Then they said the interpretation of the wave equation is that the integral of its magnitude squared over a region is the probablity of finding the particle in that region. So which is it? Is the particle some speck that can be anywhere, but is likely to be in the regions of high probability? Or is it the wave itself?
|Sep25-04, 01:03 PM||#2|
As you said, a particle can be viewed at as a wave. This vision is the main consequence of the double slit experiment where the "logic" adding of probabilities is no longer respected due to the interference term. I am sure you heard of this before.
deBroglie set up some relations (in his PhD-thesis) which connect the wavelike properties to the particle-like properties : E = hv and p = h/l where v is the frequency and l is the wavelength... E and p are energy and momentum of the particle...
Now the wavefunction is the solution of the Schrödinger-equation and this function contains all physical info of a certain QM-system. The observables like energy are now operators that work on this wavefunction. As a result of this you will get a number that is a possible energy-value. When you square this number you will get the probability that the system exhibits this energy-)value. This is a big difference with classical physics, now in QM the energy-value is not "really" important. It is the square that is important because it expresses the chance of the state having this particular energy-value. These energy-values are the socalled eigenvalues of the energy-operator. This operator is the Hamiltonian.
Just like this you need to see that the wave-function does contain all physical info, yet in order to acquire "numbers" describing the physical state you need to square the wavefunction. This gives you the probability of finding some particle at some given place...just like in the energy-eigenvalue-example, here above...
The reason for this fact (the square gives the probability of something) comes from the wave nature of this approach. In wave-physics, the square of the wave equation gives the intensity corresponding to the wave...This approach was also used in the double slit experiment and it turned out that the square of the wavefunction of the electrons gave the right probability-distribution that was observed at the detector-screen.
Well, keep in mind that this experiment was a thought experiment and cannot be executed in reality because the used apparatus would have to be extremely accurate...
ps : i suggest you check this double-slit-experiment in your QM-textbooks or on the net. Just google away, there is enough available info out there...
|Sep25-04, 01:07 PM||#3|
Welcome to Quantum Mechanics !
Hold your breath. A quantum particle is supposed to be described by a quantum state (your wavefunction, solution of the Schroedinger equation) and to be "nowhere" until a measurement of its position is performed.
Then the *probability* of finding the particle in a certain volume is given by what your professor told you (the square of the wavefunction, integrated over that volume). You're not supposed to ask where it is if you do not measure the position ; the punishment being incoherent answers.
So, yes, we've given up on saying where the particle is, or even on saying that the particle must be somewhere. One remark however: the particle IS NOT the wave function. The wave function is the state description of the quantum state of the particle. In the same way that the position and speed of a dust particle is not the particle itself in classical mechanics, but a description of its dynamical state.
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