Factor 2 in Heisenberg Uncertainty Principle: Which formula is correct?Some websites and textbooks refer to Δ x Δ p ≥ ℏ 2 as the correct formula for the uncertainty principle whereas other sources use the formula Δ x Δ p ≥ ℏ .Question: Which one is correct and why?The latter is used in the textbook "Physics II for Dummies" (German edition) for several examples and the author also derives that formula so I assume that this is not a typing error.This is the mentioned derivation: sin ⁡ θ = λ Δ y assuming θ is small: tan ⁡ θ = λ Δ y de Broglie equation: λ = h p x ⇒ tan ⁡ θ ≈ h p x ⋅ Δ y but also: tan ⁡ θ = Δ p y p x equalize tan ⁡ θ: h p x ⋅ Δ y ≈ Δ p y p x ⇒ h Δ y ≈ Δ p y ⇒ Δ p y Δ y ≈ h ⇒ Δ p y Δ y ≥ h 2 π ⇒ Δ p Δ x ≥ h 2 π

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Factor 2 in Heisenberg Uncertainty Principle: Which formula is correct?Some websites and textbooks refer to  Δ  x  Δ  p  ≥      ℏ    2  as the correct formula for the uncertainty principle whereas other sources use the formula  Δ  x  Δ  p  ≥  ℏ  .Question: Which one is correct and why?The latter is used in the textbook &quot;Physics II for Dummies&quot; (German edition) for several examples and the author also derives that formula so I assume that this is not a typing error.This is the mentioned derivation:  sin  ⁡  θ  =      λ          Δ      y      assuming   θ is small:  tan  ⁡  θ  =      λ          Δ      y      de Broglie equation:  λ  =      h          p      x        ⇒  tan  ⁡  θ  ≈      h                  p        x            ⋅      Δ      y      but also:  tan  ⁡  θ  =            Δ              p        y                    p      x      equalize   tan  ⁡  θ:      h                  p        x            ⋅      Δ      y        ≈            Δ              p        y                    p      x        ⇒      h          Δ      y        ≈  Δ      p    y    ⇒  Δ      p    y    Δ  y  ≈  h  ⇒  Δ      p    y    Δ  y  ≥      h          2      π        ⇒  Δ  p  Δ  x  ≥      h          2      π

lascosasdeali3v · Accepted Answer

The strongest limit without loss of generality is  Δ  p  Δ  x  ≥      1    2    ℏ  ,this is always true. Whilst   Δ  p  Δ  x  ≥  ℏ might often be true, it is not always true.The       1    2   is often omitted, because, as mentioned in the comments, often only the magnitude of the right-hand-side is important, and not its precise value. Also, it might be omitted for brevity/simplicity.A further reason is historical: Heisenberg&#039;s original statement of his uncertainty principle was a rough estimate that omitted       1    2  . Only later was his estimate refined with a formal calculation and the       1    2   added.

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