Uncertainty Principle textbook equation

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SUMMARY

The Heisenberg uncertainty principle in Quantum Mechanics is correctly expressed as ΔxΔp ≥ (h-bar)/2, where Δx represents the uncertainty in position and Δp represents the uncertainty in momentum. This formulation applies when Δx and Δp are interpreted as standard deviations from their respective means. The equation ΔxΔp ≥ h-bar is applicable in contexts that do not involve standard deviations, particularly for order-of-magnitude estimates. The Gaussian wave packets are the only cases where equality holds in this relation.

PREREQUISITES
  • Understanding of Quantum Mechanics principles
  • Familiarity with the Heisenberg uncertainty principle
  • Knowledge of standard deviations in statistical analysis
  • Basic concepts of wave packets in quantum physics
NEXT STEPS
  • Study the derivation of the Heisenberg-Robertson uncertainty relation
  • Explore Gaussian wave packets and their properties
  • Research applications of the uncertainty principle in quantum mechanics
  • Learn about the significance of standard deviations in quantum measurements
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Students of Quantum Mechanics, physicists, and anyone interested in the mathematical foundations of the Heisenberg uncertainty principle.

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
\Delta x \Delta p \geq \frac{\hbar}{2}.
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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