Heisenberg’s * uncertainty principle* states that the position and momentum of a particle cannot be determined simultaneously with unlimited precision.

The uncertainty not only applies to the position and momentum of a particle, but to any pair of complementary observables, e.g. energy and time. In general, the uncertainty principle is expressed as:

where and are Hermitian operators and and are their respective observables.

The derivation of eq12 involves the following:

- Deriving the
**Schwarz inequality** - Proving the inequality
- Showing that

__Step 1__

Let

where and are arbitrary square integrable wavefunctions and is an arbitrary scalar.

Since

Expanding eq13, we have

Since is an arbitrary scalar, substitute and in eq15:

Substituting eq14 in the above equation and rearranging, we have . Since

Eq16 is called the * Schwarz Inequality*.

__Step 2__

Let and , where is normalised, and and are Hermitian operators, which implies that and are also Hermitian operators (see this article for proof). The variance of the observable of is

Note that the 2^{nd} last equality uses the property of Hermitian operators (see eq36). Similarly,

Substitute eq17 and eq18 in eq16,

Let where . So, . Since , we have , which is

Substitute eq19 in eq20,

Next, we have

Similarly,

Substitute eq22 and eq23 in eq21

__Step 3__

We have used eq37 for the 2^{nd} equality and eq35 for the 3^{rd} equality. Substitute eq25 in one of the in eq24,

Therefore,

which is eq12, the general form of the uncertainty principle.

For, the observable pair of position and momentum , we have

Since