Heisenberg’s Uncertainty Principle

This is a fundamental principle of Quantum mechanics. Loosely speaking, it states that:

It is impossible to specify precisely both the linear momentum and the position of a particle at a single moment in time.

We can illustrate this principle by a consideration of some wavefunctions.

We have established that the wavefunction for a particle in free space is of the form:

We will consider a situation in which the constant B is equal to zero, so that the wavefunction becomes:

The probability density for this wavefunction is given by:

The probability density is independent of x, so the particle is equally likely to be found at any point – its position is totally unpredictable. This is what we would expect. With equal potential everywhere, there is no reason that the particle should be found at any one position more frequently than at any other.

The linear momentum of the particle is given by the linear momentum operatorpx:

which implies that the linear momentum of the particle is numerically equal to k. So when the linear momentum is precisely specified, the position of the particle is totally unspecified.

Similarly, if a particle is precisely located at one point in space, it can be shown that its wavefunction will not be an eigenfunction of the linear momentum operator, so no information about its linear momentum can be obtained. i.e. if the position is precisely specified, its linear momentum cannot be.

The uncertainty principle may be formulated more rigorously as:

Δp is the uncertainty in the linear momentum parallel to the axis q , and Δq is the uncertainty in the position along this axis. These uncertainties are defined as the root mean square deviations of the properties from their mean values, and are given by:

Note that when the uncertainty in either quantity is zero, the uncertainty relation may only be satisfied if the uncertainty in the other quantity is infinite.

Note also that p and q in the uncertainty relation refer to the same direction in space. The uncertainty relation does not forbid, for example, simultaneous specification of the position of a particle along the x axis and its linear momentum in the y direction.