Quantum mechanics
Uncertainty principle

One consequence of the basic quantum formalism is the uncertainty principle. In its most familiar form, this states that no preparation of a quantum particle can imply simultaneously precise predictions both for a measurement of its position and for a measurement of its momentum. Both position and momentum are observables, meaning that they are represented by Hermitian operators. The position operator X ^ {\displaystyle {\hat {X}}} and momentum operator P ^ {\displaystyle {\hat {P}}} do not commute, but rather satisfy the canonical commutation relation:

Given a quantum state, the Born rule lets us compute expectation values for both X {\displaystyle X} and P {\displaystyle P}, and moreover for powers of them. Definingthe uncertainty for an observable by a standard deviation, we have

and likewise for the momentum:

The uncertainty principle states that

Either standard deviation can in principle be made arbitrarily small, but not both simultaneously. This inequality generalizes to arbitrary pairs of self-adjoint operators A {\displaystyle A} and B {\displaystyle B}. The commutator of these two operators is

and this provides the lower bound on the product of standard deviations:

Another consequence of the canonical commutation relation is that the position and momentum operators are Fourier transforms of each other, so that a description of an object according to its momentum is the Fourier transform of its description according to its position. The fact that dependence in momentum is the Fourier transform of the dependence in position means that the momentum operator is equivalent (up to an i / ℏ {\displaystyle i/\hbar } factor) to taking the derivative according to the position, since in Fourier analysis differentiation corresponds to multiplication in the dual space. This is why in quantum equations in position space, the momentum p i {\displaystyle p_{i}} is replaced by − i ℏ ∂ ∂ x {\displaystyle -i\hbar {\frac {\partial }{\partial x}}}, and in particular in the non-relativistic Schrödinger equation in position space the momentum-squared term is replaced with a Laplacian times − ℏ 2 {\displaystyle -\hbar ^{2}}.
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