The Heisenberg uncertainty principle
In Schroedinger's equation the momentum was replaced by a derivative. This means that if a wave function is confined to a small volume, it must rise from zero at the boundary to a finite value, then fall back to zero at the opposite boundary. Small volumes necessarily imply larger derivatives and higher momenta. This is analogous to the DeBroglie wavelength being inversely related to the momentum. One consequence of the Schroedinger equation (derived in a different way by Werner Heisenberg) is the Heisenberg uncertainty principle:
Dp Dx > h / (4 p) .
where Dx and Dp are the uncertainty in the momentum and position, which can be defined precisely using the basic statistical concept of standard deviation. (The numerical factors in the uncertainty principle are often stated incorrectly in elementary text books.)
The more confined a particle is to a given position, the more uncertain is its momentum. Thus a particle in its lowest energy state in a box has a range of momentum and therefore a non-zero kinetic energy. Unlike in classical physics, it cannot simply "lie still" with zero velocity and zero kinetic energy. Similarly, an electron in a attractive Coulomb potential (e.g. a Hydrogen atom) must have more kinetic energy for smaller "orbits". If it were not for this aspect of Schroedinger's equation, atoms would collapse to a point.
Another profound change from the classical viewpoint follows directly from the uncertainty principle: Position and momentum (hence velocity) cannot be precisely known simultaneously; so particles do not move according to any kind of well-defined paths, such as the circular or elliptical "orbits" that are often drawn in cartoon style to represent the motion of electrons in atoms.
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