For a 1D infinite well, The energy levels of an electron trapped inside is dependent on the length of the well. The longer the length, the less its energy will be for each state. I am aware how the formula is derived. The main form of the formula is a solution of Schrodinger's equation which books say is not derived from anything more fundalmental. But is the fact that the energy levels are depedent on L intuitive? If so why? Could you say that a longer well would mean that the energy of the electron is distributed more evenly for each position x in the well? Hence the energy of the electron is lower at each x in the well for a particular state in a longer well?
Well you know that the physicists always say that to probe smaller distances requires higher energy particles. Carl
Depends on your intuition. If you are stuck with a classical intuition, it will not help you any. Here's one kind of intuition : The smaller the box, the greater the momentum uncertainty...
yeah right. only if your " box " happens to be an atom - in which case - what are you putting in it again? if not, any basic QM text will tell you that for the same potential you could choose position or momentum eigenstates (or eigenstates of any other operator) which would have, respectively, 0 uncertainty in position and momentum. (moral: math works even if intuition runs awry)
The kinetic energy is a measure of the curvature of the wavefunction, right? (since [itex] p^2/2m = - \hbar^2 {\partial^2 \over \partial x^2} [/itex]). If you narrow the well, the wavefunction has to "bend" more (recall that it must be zero at the two endpoints) which explains why the energy is larger.
So the reason why E is depedent on L is because of UP. When I said intuitive, I meant classically intuitive. Obviously, since the UC is needed, the answer is that it is not intuitive. Classically, wouldn't it be the case that the energy of an electron is fixed from the start, no matter what the size of the well it is in? Hence intuitively E should not depedent on L. E=E until the electron is given potential or kinetic energy via a force. Looks like my explanation... 'Could you say that a longer well would mean that the energy of the electron is distributed more evenly for each position x in the well? Hence the energy of the electron is lower at each x in the well for a particular state in a longer well?' is wrong in the classic sense. Is it wrong in a QM sense as well?