Physical significance of nodes?

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Discussion Overview

The discussion revolves around the physical significance of nodes in quantum mechanics, particularly in the context of a particle in a potential well. Participants explore the implications of nodes, their relationship to energy, and the nature of these points in space where a particle cannot be located.

Discussion Character

  • Exploratory
  • Debate/contested
  • Technical explanation

Main Points Raised

  • One participant questions the physical significance of nodes, asking why a particle cannot be located at these points.
  • Another participant argues that nothing is special about the nodes, suggesting that their existence is dependent on the energy of the particle rather than the properties of the points in space.
  • A classical analogy is presented, comparing the behavior of a planet in orbit to the behavior of a particle in quantum mechanics, indicating that the points of nodes do not hold special significance.
  • One participant challenges the assertion about energy, proposing that the Hamiltonian can be modified without changing the eigenbasis, implying that energy is not unique and that the basis states are more fundamental.
  • There is a mention of the concept of perfect cancellation of right-moving and left-moving waves in relation to nodes.

Areas of Agreement / Disagreement

Participants express differing views on the significance of nodes, with some agreeing that nodes are not special while others contest the relationship between nodes and energy. The discussion remains unresolved regarding the implications of energy changes on the nature of nodes.

Contextual Notes

The discussion includes assumptions about the relationship between energy and nodes that are not fully explored, and the implications of modifying the Hamiltonian are not resolved. There is also a lack of consensus on the interpretation of nodes in quantum mechanics.

Sheldon Cooper
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Hi,

II have an issue with nodes in QM, like suppose in a well, the number of nodes depend on (n-1), the thing is that, what is so special(physically) about the point that the particle cannot be located there?

Thanks is advance
 
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Sheldon Cooper said:
the thing is that, what is so special(physically) about the point that the particle cannot be located there?
Nothing is special about those points. You can make any point be a node or not be a node just by changing the energy of the particle, which tells us that it's the energy that matters and not some properties of the points in space where the nodes happen to be.

An analogy from classical physics: A planet will pass through some points in space and not others as it moves around the sun. That doesn't make the points along the orbit special; the planet is just going where Newton's laws say it will go and not going anywhere else.
 
Nugatory said:
Nothing is special about those points. You can make any point be a node or not be a node just by changing the energy of the particle, which tells us that it's the energy that matters and not some properties of the points in space where the nodes happen to be.

An analogy from classical physics: A planet will pass through some points in space and not others as it moves around the sun. That doesn't make the points along the orbit special; the planet is just going where Newton's laws say it will go and not going anywhere else.

While I do agree that nodes aren't special, I think you're mistaken about the energy bit, you can always refine a Hamilton by adding cI where I is unity and c is a real constant, this will change all observable energies by c but will not alter the eigenbasis, that is energy is not unique, basis are
 
nashed said:
While I do agree that nodes aren't special, I think you're mistaken about the energy bit, you can always refine a Hamilton by adding cI where I is unity and c is a real constant, this will change all observable energies by c but will not alter the eigenbasis, that is energy is not unique, basis are
Yes, that's right... I could have said ##p^2## to be more accurate.
 
Perfect cancellation of right moving and left moving waves.
 

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