Uncertainty Principle textbook equation

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The discussion centers on the Heisenberg uncertainty principle in quantum mechanics, specifically the relationship between position (Δx) and momentum (Δp). The confusion arises from two different formulations: ΔxΔp ≥ h-bar and ΔxΔp ≥ (h-bar)/2. The latter is the correct expression, particularly when Δx and Δp are interpreted as standard deviations from their means. The equality holds true for Gaussian wave packets, which are the only cases where this condition is met. The conversation also notes that for rough estimates, the difference between using h-bar and h-bar/2 may not significantly impact results, particularly in order-of-magnitude calculations.
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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