5.272859e-25
kg·m/s
578,838.189244
m/s
1.526066e-19
J
2.000000e-10
m
5.272859e-35
J·s
5.272859e-25
kg·m/s
578,838.189244
m/s
1.526066e-19
J
2.000000e-10
m
5.272859e-35
J·s
The Heisenberg Uncertainty Calculator quantifies the fundamental quantum mechanical limit on simultaneously knowing a particle's position and momentum. Werner Heisenberg formulated this principle in 1927, and it stands as one of the most profound statements in all of physics: the universe itself places irreducible limits on how precisely certain pairs of physical quantities can be known at the same time.
The uncertainty principle states that the product of the uncertainties in position and momentum must always be greater than or equal to hbar/2, where hbar (h-bar) is the reduced Planck constant equal to h/(2*pi) = 1.0546 x 10^-34 J·s. This is not a statement about experimental imprecision or technological limitations; it reflects a fundamental property of nature described by quantum mechanics.
A common misconception is that the uncertainty principle results from the act of measurement disturbing the particle. While measurement disturbance is real, the uncertainty principle is deeper: it arises from the wave nature of matter. A particle confined to a small region necessarily has a spread of momenta (just as a wave packet confined in space must contain a range of frequencies), and vice versa. Attempting to pin down position more precisely always increases the spread of momenta.
The uncertainty principle has enormous practical and conceptual consequences. It explains why electrons do not spiral into the atomic nucleus (confining an electron to nuclear dimensions would require enormous momentum uncertainty, hence enormous kinetic energy, pushing it out). It underlies the zero-point energy of quantum oscillators, the stability of matter, the operation of tunnel diodes, and the line widths of atomic spectra through the energy-time uncertainty relation.
There is also a complementary energy-time uncertainty relation: delta_E * delta_t >= hbar/2. This explains the natural line width of spectral lines (a state that decays quickly has an uncertain energy) and has implications for virtual particles in quantum field theory.
This calculator takes the position uncertainty and particle mass as inputs and computes the minimum allowed momentum uncertainty and the corresponding minimum velocity uncertainty.
The Heisenberg uncertainty principle gives delta_x * delta_p >= hbar/2. The minimum momentum uncertainty is delta_p_min = (hbar/2) / delta_x. The corresponding velocity uncertainty is delta_v_min = delta_p_min / mass. The reduced Planck constant hbar = h/(2*pi) = 1.0546 x 10^-34 J·s.
If delta_v_min is much smaller than the particle's actual speed, classical mechanics is a good approximation. For an electron confined to 0.1 nm (atomic scale), delta_v_min ~ 10^6 m/s, comparable to orbital speeds, confirming quantum effects dominate. For a 1 g ball confined to 1 mm, delta_v_min ~ 10^-28 m/s, utterly negligible classically.
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Results
For an electron confined to the Bohr radius (0.053 nm), the minimum velocity uncertainty is about 1.1 x 10^6 m/s, consistent with known electron orbital speeds.
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Results
A proton confined to a nucleus (1 fm) has minimum velocity uncertainty ~3 x 10^7 m/s (about 10% of c), indicating relativistic quantum effects are needed.
hbar (h-bar) is the reduced Planck constant, equal to h/(2*pi) = 1.0546 x 10^-34 J·s. It appears in many quantum mechanical formulas.
Not entirely. While measuring one quantity can disturb another, the uncertainty principle is deeper: it reflects the wave nature of matter. A particle with a definite position must have a spread of momenta by the mathematics of Fourier analysis.
Yes, but the uncertainties involved are negligibly small for everyday objects. For a 1 kg ball the quantum uncertainties are far below any measurable scale.
delta_E * delta_t >= hbar/2. A quantum state that exists for a short time (small delta_t) has a correspondingly large uncertainty in its energy, explaining natural spectral line widths.
Confining a particle to a small region increases its momentum uncertainty. Since kinetic energy = p^2/(2m), a larger spread of momenta means larger average kinetic energy, which is why electrons do not collapse into the nucleus.
No. This would violate the principle. A particle can only have a definite position or a definite momentum, never both at the same time in quantum mechanics.
Even at absolute zero temperature, quantum systems retain minimum energy due to the uncertainty principle. A particle in a potential well cannot be perfectly at rest (zero momentum) and perfectly located simultaneously.
Quantum tunneling is related to the wave nature of particles (same origin as uncertainty). Particles can penetrate potential barriers that classical physics would forbid due to their wave-like spread in space.
The inequality delta_x * delta_p >= hbar/2 is exact for Gaussian wave packets. For other distributions the product can be larger. The lower bound hbar/2 is the theoretical minimum.
Werner Heisenberg derived it in 1927 using matrix mechanics. A more rigorous proof using wave mechanics was given by Kennard the same year. Robertson later generalized it to arbitrary observables.
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