ecoanuncios7x

2022-10-02

Factor 2 in Heisenberg Uncertainty Principle: Which formula is correct?

Some websites and textbooks refer to

$\mathrm{\Delta}x\mathrm{\Delta}p\ge \frac{\hslash}{2}$

as the correct formula for the uncertainty principle whereas other sources use the formula

$\mathrm{\Delta}x\mathrm{\Delta}p\ge \hslash .$

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:

$\mathrm{sin}\theta =\frac{\lambda}{\mathrm{\Delta}y}$

assuming $\theta $ is small:

$\mathrm{tan}\theta =\frac{\lambda}{\mathrm{\Delta}y}$

de Broglie equation:

$\lambda =\frac{h}{{p}_{x}}$

$\Rightarrow \mathrm{tan}\theta \approx \frac{h}{{p}_{x}\cdot \mathrm{\Delta}y}$

but also:

$\mathrm{tan}\theta =\frac{\mathrm{\Delta}{p}_{y}}{{p}_{x}}$

equalize $\mathrm{tan}\theta $:

$\frac{h}{{p}_{x}\cdot \mathrm{\Delta}y}\approx \frac{\mathrm{\Delta}{p}_{y}}{{p}_{x}}$

$\Rightarrow \frac{h}{\mathrm{\Delta}y}\approx \mathrm{\Delta}{p}_{y}\Rightarrow \mathrm{\Delta}{p}_{y}\mathrm{\Delta}y\approx h$

$\Rightarrow \mathrm{\Delta}{p}_{y}\mathrm{\Delta}y\ge \frac{h}{2\pi}$

$\Rightarrow \mathrm{\Delta}p\mathrm{\Delta}x\ge \frac{h}{2\pi}$

Some websites and textbooks refer to

$\mathrm{\Delta}x\mathrm{\Delta}p\ge \frac{\hslash}{2}$

as the correct formula for the uncertainty principle whereas other sources use the formula

$\mathrm{\Delta}x\mathrm{\Delta}p\ge \hslash .$

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:

$\mathrm{sin}\theta =\frac{\lambda}{\mathrm{\Delta}y}$

assuming $\theta $ is small:

$\mathrm{tan}\theta =\frac{\lambda}{\mathrm{\Delta}y}$

de Broglie equation:

$\lambda =\frac{h}{{p}_{x}}$

$\Rightarrow \mathrm{tan}\theta \approx \frac{h}{{p}_{x}\cdot \mathrm{\Delta}y}$

but also:

$\mathrm{tan}\theta =\frac{\mathrm{\Delta}{p}_{y}}{{p}_{x}}$

equalize $\mathrm{tan}\theta $:

$\frac{h}{{p}_{x}\cdot \mathrm{\Delta}y}\approx \frac{\mathrm{\Delta}{p}_{y}}{{p}_{x}}$

$\Rightarrow \frac{h}{\mathrm{\Delta}y}\approx \mathrm{\Delta}{p}_{y}\Rightarrow \mathrm{\Delta}{p}_{y}\mathrm{\Delta}y\approx h$

$\Rightarrow \mathrm{\Delta}{p}_{y}\mathrm{\Delta}y\ge \frac{h}{2\pi}$

$\Rightarrow \mathrm{\Delta}p\mathrm{\Delta}x\ge \frac{h}{2\pi}$

lascosasdeali3v

Beginner2022-10-03Added 10 answers

The strongest limit without loss of generality is

$\mathrm{\Delta}p\mathrm{\Delta}x\ge \frac{1}{2}\hslash ,$

this is always true. Whilst $\mathrm{\Delta}p\mathrm{\Delta}x\ge \hslash $ might often be true, it is not always true.

The $\frac{1}{2}$ 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's original statement of his uncertainty principle was a rough estimate that omitted $\frac{1}{2}$. Only later was his estimate refined with a formal calculation and the $\frac{1}{2}$ added.

$\mathrm{\Delta}p\mathrm{\Delta}x\ge \frac{1}{2}\hslash ,$

this is always true. Whilst $\mathrm{\Delta}p\mathrm{\Delta}x\ge \hslash $ might often be true, it is not always true.

The $\frac{1}{2}$ 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's original statement of his uncertainty principle was a rough estimate that omitted $\frac{1}{2}$. Only later was his estimate refined with a formal calculation and the $\frac{1}{2}$ added.

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