# Is it intuitive that the Energy levels...

by pivoxa15
Tags: energy, intuitive, levels
 P: 2,268 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?
 HW Helper Sci Advisor P: 1,204 Well you know that the physicists always say that to probe smaller distances requires higher energy particles. Carl
 PF Patron Sci Advisor Emeritus P: 11,137 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...
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## Is it intuitive that the Energy levels...

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)
P: 228
 Quote by yeahright 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)
What?!? Can you explain this a little more...
HW Helper
The kinetic energy is a measure of the curvature of the wavefunction, right? (since $p^2/2m = - \hbar^2 {\partial^2 \over \partial x^2}$). 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.