Q-Source said:
Well, in a way yes. There is a vanishingly small probability that the planet could break it's orbit even though it does not have the necessary energy to do so. Hopefully that will makes more sense after the rest of this post.
Yeah. The thing is, that applying quantum mechanics in such a way gets mathematically complex in a very big hurry. So it's just easier to approximate things using classical mechanics, since it so closely approximates what is right in those cases anyway. It's the same thing with classical mechanics and relativity. If you are dealing with speeds that are only a very small fraction of c, it isn't worth doing the more complicated work of dealing with relativity because you are going to get practically the same results anyway (out to a large number of decimal places).
Ok...here I go...quantum tunneling. This is a good way to show that quantum mechanics is probabilistic, and at the same time give an example of something that classical mechanics can not explain. Say I'm shooting some electrons at a thin physical barrier. A barrier that the electrons do not actually have the energy to pass through. If I do this over and over, I will eventually notice that, to my surprise, a certain percentage of the electrons that I'm shooting actually DO pass through the barrier. This should never happen according to classical mechanics. The particles don't have enough energy to pass through the barrier, so they don't pass through the barrier. According to quantum mechanics however, we do actually expect this to be the case. I think the best way to cover this is through a visual representation, and since I can't draw a picture I will point you to here http://phys.educ.ksu.edu/vqm/html/qtunneling.html
All you have to do is hit the redraw graphs button. The top box is a representation of the particle energy and the barrier conditions (note that the particle energy is lower than the energy necessary to go over the barrier). The bottom box is a graph of the particle's wave function. You will notice that it continues on past the position of the barrier, but at a much lower amplitude. The probability of finding the particle at any particular position is a function of the amplitude of the wave function (bigger amplitude, bigger probability). So it is much more likely that the particle will be on the side of the barrier that it originates from, but there is some small probability that it will pass through the barrier and be located on the other side, despite it not having enough energy to actually pass through the barrier. Now, if you mess with the width of the barrier in the barrier properties, and redraw the graph, you will notice that by increasing the barrier width by even a small amount, the amplitude on the right hand side of the graph quickly approaches zero (it does remain nonzero, it's just not easy to see on the graph). That just goes to show how small the probability usually is.
Now, apply this to a macroscopic event...let's say you are throwing a tennis ball at a brick wall. The ball would need some amount of energy to pass through the wall and end up on the other side. An energy which we will assume that you are not going to generate with your throw. Now, we could deal with this event classically quite easily. However, we could also do it quantum mechanically by doing an analysis of every single particle in the tennis ball in the same manner that we would do for the single electron in the example above. You should be able to see why I would rather do this classically...a tennis ball has a lot of particles, and that means a whole lot of calculations. If I were to do this, and then add up all the wave functions, I would have a wave function that would adequately describe the macroscopic actions of the tennis ball. Now, when I throw it at the brick wall, the height and width of my energy barrier are MUCH bigger than those in the example in the applet (we are talking several orders of magnitude at least). So, obviously the amplitude of the wave function on the other side of the barrier is going to be VERY small. So in reality, if you throw a tennis ball at a brick wall...it actually might have a 0.000000000000000000000000000000.....0000001% chance of going through and ending up on the other side, even though you didn't throw it hard enough to go through (according to classical mechanics). That probability is so small though, that it never actually happens.
