Is it intuitive that the Energy levels

In summary, the energy levels of an electron trapped inside a 1D infinite well are depedent on the length of the well. The longer the well, the less its energy will be for each state.
  • #1
pivoxa15
2,255
1
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?
 
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  • #2
Well you know that the physicists always say that to probe smaller distances requires higher energy particles.

Carl
 
  • #3
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...
 
  • #4
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)
 
  • #5
yeahright said:
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...
 
  • #6
pivoxa15 said:
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?

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.
 
  • #7
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?
 
Last edited:

1. What is the concept of energy levels in science?

Energy levels refer to the specific states of energy that an atom or molecule can possess. These states are characterized by different amounts of energy and are represented by quantum numbers.

2. How do energy levels affect the behavior of particles?

The energy levels of particles determine their behavior and interactions with other particles. For example, electrons in an atom can only exist in certain energy levels and can move between levels by gaining or losing energy.

3. Is it intuitive that particles have discrete energy levels?

No, it is not necessarily intuitive that particles have discrete energy levels. This concept comes from the principles of quantum mechanics, which describe the behavior of particles at a microscopic level.

4. How are energy levels related to the stability of atoms and molecules?

The energy levels of particles contribute to their overall stability. In general, particles in lower energy levels are more stable than those in higher energy levels. This is why electrons tend to occupy the lowest available energy levels in an atom.

5. Can energy levels be changed or manipulated?

Yes, energy levels can be changed or manipulated through various processes such as absorption or emission of energy, collisions with other particles, or exposure to external forces. These changes can alter the behavior and properties of particles.

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