Uncertainty Principle textbook equation

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Discussion Overview

The discussion revolves around the Heisenberg uncertainty principle in quantum mechanics, specifically the equation relating position and momentum uncertainties. Participants are examining different formulations of the principle as presented in textbooks and other sources, focusing on the interpretation of the terms involved.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant notes a discrepancy between their textbook's formulation of the uncertainty principle as ΔxΔp ≥ h-bar and the more commonly referenced ΔxΔp ≥ (h-bar)/2.
  • Another participant asserts that the latter formulation (h-bar/2) is correct when Δx and Δp are interpreted as standard deviations from their respective means.
  • A question is raised about whether there are circumstances under which ΔxΔp ≥ h-bar could be considered correct, particularly outside the context of standard deviations.
  • It is suggested that for order-of-magnitude estimates, the specific factor (2 or 1/2) may not significantly impact the results.
  • A participant states that the correct Heisenberg-Robertson uncertainty relation is ΔxΔp ≥ (h-bar)/2 and mentions that Gaussian wave packets are the only cases where equality holds.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the correct formulation of the uncertainty principle, with multiple competing views presented regarding the interpretation and context of the equations.

Contextual Notes

There is an assumption that Δx and Δp are understood in terms of standard deviations, but this is not universally accepted in the discussion. The implications of using different formulations in various contexts remain unresolved.

Daniel1992
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I have been going through my Physics textbook to brush up on my Quantum Mechanics before starting my next QM course next academic year and the Heisenberg uncertainty principle for position and momentum is written as ΔxΔp ≥ h-bar when I thought it was ΔxΔp ≥ (h-bar)/2. Other sources say it is the latter so am I missing something? Or is the textbook just wrong?
 
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The latter (hbar/2), assuming that Δx and Δp are interpreted as standard deviations from their respective means.
 
So are there circumstances when ΔxΔp ≥ h-bar is correct? Say when you are not dealing with standard deviations?
 
When you're interested mainly in order-of-magnitude estimates (powers of ten), a factor of 2 or 1/2 or something like that doesn't affect the result significantly.
 
The correct Heisenberg-Robertson uncertainty relation is
[tex]\Delta x \Delta p \geq \frac{\hbar}{2}.[/tex]
You can show that the Gaussian wave packets are the only ones, where the equality sign is valid.
 
OK, thanks for clearing that up.
 

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