Orders of magnutide in Heisenberg's uncertainty principle

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Discussion Overview

The discussion revolves around the relationship between the orders of magnitude of momentum and its uncertainty as described by the Heisenberg uncertainty principle. Participants explore theoretical implications and calculations related to quantum mechanics, particularly in the context of an electron confined in a box.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions the assumption that momentum and its uncertainty are of the same order of magnitude, asking for clarification on the circumstances under which this assumption holds.
  • Another participant provides an example involving an electron in a box, discussing how the uncertainty in position relates to the uncertainty in momentum as the box size decreases, leading to an increase in momentum uncertainty.
  • This participant also raises a concern about the assumption that momentum increases with increasing uncertainty in momentum, suggesting that momentum could theoretically be zero while uncertainty increases.
  • A later reply attempts to connect the uncertainty in momentum to the expectation values of momentum and kinetic energy, indicating a proportional relationship but does not resolve the initial concerns raised.

Areas of Agreement / Disagreement

Participants express differing views on the assumptions regarding momentum and its uncertainty, with no consensus reached on the validity of these assumptions or the implications of the examples provided.

Contextual Notes

Participants reference specific conditions (e.g., the size of the box) and relationships (e.g., between momentum and kinetic energy) that may not be universally applicable, leaving some assumptions and dependencies unresolved.

Heirot
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Is there any a priori connection beetween the orders of magnitude of e.g. momentum, and its uncertainty? Why do we always assume that the momentum is the same order of magnitude as its uncertainy? I'm referring to all those "back of the envelope" calculations.

Thanks
 
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Heirot said:
Is there any a priori connection beetween the orders of magnitude of e.g. momentum, and its uncertainty? Why do we always assume that the momentum is the same order of magnitude as its uncertainy? I'm referring to all those "back of the envelope" calculations.

Thanks
I'm not quite sure what you mean, could you perhaps expand? Under which circumstances are you referring to?
 
For example, let's put an electron in a box of length L. Since the electron is in the box, we know that delta x = L (or L/2, I'm not quite sure). Using uncertainty relations, we have delta p >= hbar / 2L. As L gets smaller, delta p gets bigger. So, the textbooks conclude, at one point delta p gets so big that there's enough energy to create a positon - electron pair! But energy increases with p, not delta p! For all we now, p could be zero all the time. Why do we assume that p increases as delta p increases?
 
Heirot said:
For example, let's put an electron in a box of length L. Since the electron is in the box, we know that delta x = L (or L/2, I'm not quite sure). Using uncertainty relations, we have delta p >= hbar / 2L. As L gets smaller, delta p gets bigger. So, the textbooks conclude, at one point delta p gets so big that there's enough energy to create a positon - electron pair! But energy increases with p, not delta p! For all we now, p could be zero all the time. Why do we assume that p increases as delta p increases?
The uncertainty in momentum can be related to the expectation value (average) of the momentum and momentum squared, which in turn can be related to the expectation value of the kinetic energy. It turns out that for a particle in a box, the expectation value of the kinetic energy of a particle is directly proportional to the expectation value of the square of the momentum and hence also directly proportional to the uncertainty in momentum.

See http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/uncer2.html#c1" for more information.
 
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