Uncertainty Principle Momentum

In quantum mechanics, the Heisenberg uncertainty principle states by precise inequalities that certain pairs of physical properties, like position and momentum, cannot simultaneously be known to arbitrary precision. That is, the more precisely one property is measured, the less precisely the other can be measured. In other words, the more you know the position of a particle, the less you can know about its velocity, and the more you know about the velocity of a particle, the less you can know about its instantaneous position.

Published by Werner Heisenberg in 1927, the principle means that it is impossible to determine simultaneously both the position and velocity of an electron or any other particle with any great degree of accuracy or certainty. Moreover, his principle is not a statement about the limitations of a researcher’s ability to measure particular quantities of a system, but it is a statement about the nature of the system itself as described by the equations of quantum mechanics.

Delta x Delta p congruent to h

Wave Particle Duality

One way to understand the complementarity between position and momentum is by wave particle duality. If a particle described by a plane wave passes through a narrow slit in a wall like a water-wave passing through a narrow channel, the particle diffracts and its wave comes out in a range of angles. The narrower the slit, the wider the diffracted wave and the greater the uncertainty in momentum afterwards. The laws of diffraction require that the spread in angle Δθ is about λ / d, where d is the slit width and λ is the wavelength. From the de Broglie relation, the size of the slit and the range in momentum of the diffracted wave are related by Heisenberg’s rule:

Heisenberg’s Microscope

One way in which Heisenberg originally argued for the uncertainty principle is by using an imaginary microscope as a measuring device. He imagines an experimenter trying to measure the position and momentum of an electron by shooting a photon at it.

If the photon has a short wavelength, and therefore a large momentum, the position can be measured accurately. But the photon scatters in a random direction, transferring a large and uncertain amount of momentum to the electron. If the photon has a long wavelength and low momentum, the collision doesn’t disturb the electron’s momentum very much, but the scattering will reveal its position only vaguely.

If a large aperture is used for the microscope, the electron’s location can be well resolved ; but by the principle of conservation of momentum, the transverse momentum of the incoming photon and hence the new momentum of the electron resolves poorly. If a small aperture is used, the accuracy of the two resolutions is the other way around.

The trade-offs imply that no matter what photon wavelength and aperture size are used, the product of the uncertainty in measured position and measured momentum is greater than or equal to a lower bound, which is up to a small numerical factor equal to Planck’s constant. Heisenberg did not care to formulate the uncertainty principle as an exact bound, and preferred to use it as a heuristic quantitative statement, correct up to small numerical factors.

Wave Packet

In quantum physics, a particle is described by a wave packet, which gives rise to this phenomenon. Consider the measurement of the position of a particle. It could be anywhere the particle’s wave packet has non-zero amplitude, meaning the position is uncertain – it could be almost anywhere along the wave packet. To obtain an accurate reading of position, this wave packet must be ‘compressed’ as much as possible, meaning it must be made up of increasing numbers of sine waves added together. The momentum of the particle is proportional to the wavelength of one of these waves, but it could be any of them. So a more precise position measurement-by adding together more waves-means the momentum measurement becomes less precise

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