How this calculator works
The uncertainty relation concerns standard deviations of repeated measurements on identically prepared quantum states. Their product cannot fall below hbar/2. This tool calculates that lower boundary, reached by suitable Gaussian states; a general state may have a larger product. Relating momentum spread to velocity spread with sigma_p = m sigma_v assumes a fixed nonrelativistic mass. The inverse mass is a mass compatible with equality, not a mass measurement or a universal limit on particle mass.
Formula
sigma_x sigma_p >= hbar/2 = h/(4pi). At equality: sigma_p = hbar/(2 sigma_x); sigma_x = hbar/(2 sigma_p). Nonrelativistic: sigma_v = sigma_p/m.
Worked example
A position standard deviation of 1 nm requires momentum standard deviation at least 5.272859088e-26 kg m/s. Allowing 2 nm halves that bound.
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Frequently asked questions
Are these measurement errors?
They are statistical spreads inherent to the quantum state, even for ideal measurement equipment.
Does every state attain equality?
No. Equality describes minimum-uncertainty states; many states have a larger uncertainty product.
Can I use velocity spread close to c?
The simple momentum-to-velocity spread conversion is nonrelativistic and is unsuitable near c.