Why Would a Finite Well Fluctuate?

Click For Summary

Discussion Overview

The discussion revolves around the behavior of a finite potential well in quantum mechanics, particularly focusing on how changes in the well's dimensions affect energy levels and the implications for ground state energy. Participants explore concepts such as quantum tunneling, the uncertainty principle, and the nature of bound states.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants suggest that shortening a finite well increases energy levels due to the uncertainty principle, questioning whether the ground state can have energy exceeding the potential of the well.
  • Others clarify that if energy exceeds the height of the well, it indicates a non-bound state, challenging the initial premise of the ground state being squeezed into a higher energy.
  • There is a debate regarding the applicability of the uncertainty principle to finite square wells, with some arguing it may not be rigorously applicable due to the nature of confinement.
  • Participants mention quantum tunneling, with some asserting that if energy is above the barrier, it does not qualify as tunneling.
  • Concerns are raised about the implications of negative energy states and their relation to the probability of fluctuations around the ground state.
  • One participant emphasizes that a finite square well should always possess at least one bound state, regardless of its depth or width.

Areas of Agreement / Disagreement

Participants express multiple competing views regarding the implications of energy levels in finite wells, the validity of the uncertainty principle in this context, and the nature of quantum tunneling. The discussion remains unresolved with no consensus on several key points.

Contextual Notes

Some limitations include the dependence on definitions of energy states, the uncertainty principle's applicability, and the conditions under which quantum tunneling is defined. These aspects are not fully resolved within the discussion.

RedX
Messages
963
Reaction score
3
If you make the length of a finite well shorter, then the energy levels should increase, because for example the uncertainty principle. But can the ground state, when squeezed, have such high energy that the energy is greater than the potential of the well? Does that even make sense?
 
Physics news on Phys.org
what do you mean by "energy is greater than the potential of the well"? Do you mean E=K+V>V? But that's always true. Or do you mean total energy is higher than the height of the well, e.g. E=K+V>|V|, in this case since E>0 it's not a bound state let alone ground state.
And I think the uncertainty principle is not a very good way to analyze finite square well, because the particle is not strictly confined inside the well, you can't know for sure the narrower the well is, the smaller delta x is (I don't know if this is true in finite square well case, but anyway you shouldn't simply conclude that without calculation). But I think probably it's ok (but still not rigourously correct) to use it analyze infinite square well,because the particle is not allowed to go outside.
 
I should have been more careful and instead of saying "energy is greater than the potential of the well", I should have said, like you mentioned, "greater than the height of the well."
 
Yes, it's called quantum tunneling.
 
statespace101 said:
Yes, it's called quantum tunneling.

No, if the energy is already higher than a barrier,it's not called a tunneling.
 
Anyway, I don't think OP's concern will come true, ground state in finite square well indicates a negative E (conventionally take the potential to be 0 outside the well), and a finite square well always at least possesses one bound state, e.g. the ground state, no matter how shallow or narrow the well is.
 
If the energy is negative wouldn't that mean there's still a non-vanishing probability that the well will fluctuate around the ground state? Then again it would be impossible to even get a probability right?
 
statespace101 said:
If the energy is negative wouldn't that mean there's still a non-vanishing probability that the well will fluctuate around the ground state? Then again it would be impossible to even get a probability right?

Why would the well fluctuate? Finite square well is a potential well with definite height
 

Similar threads

Graduate Ground state energy of a particle-in-a-box in coordinate scaling
  • · Replies 4 ·
Replies
4
Views
2K
Graduate Ground state energy of cube-like potential wells
  • · Replies 3 ·
Replies
3
Views
2K
Graduate 2-electron system under compression
  • · Replies 12 ·
Replies
12
Views
1K
Undergrad Volume of momentum space of an ideal gas
  • · Replies 8 ·
Replies
8
Views
2K
Graduate Overlap of Ground States in Quantum Field Theory
  • · Replies 5 ·
Replies
5
Views
3K
Graduate Exact symmetry, quantum states, and symmetric dynamics
  • · Replies 8 ·
Replies
8
Views
675
Undergrad The wave function in the finite square well
  • · Replies 10 ·
Replies
10
Views
2K
Graduate Ground state calculation with a time averaged wave function
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad How can there be an Uncertain Momentum at a Specific Energy?
  • · Replies 13 ·
Replies
13
Views
2K
High School Energy-time uncertainty relation for a volume and macroscopic objects?
  • · Replies 2 ·
Replies
2
Views
1K