**REAL LINE**

The real line represents the set of all real numbers, which is a one dimensional Euclidean space that can be described by a single coordinate variable .

Such a variable can be used to define complex valued functions on the real line. In particular, square-integrable functions define a Hilbert space with inner product

Intuitively, this is an extension of the dot product in Euclidean geometry. The natural norm is given by

.

**GABOR ATOM**

We define the normalized function where

This kind of function is called a Gabor atom. It was invented by Dennis Gabor when trying to apply mathematics of quantum mechanics to sound theory. This function combines a translation with a modulation . Of course, the term *atom* does not refer to a physical atom but to an elementary waveform used to decompose sounds or other signals.

The following figure shows the real part of (upper) and the absolute value of its Fourier transform (lower) for a choice of parameters and .

As it can be observed, the atom is concentrated at and its Fourier transform at . However, there is significant dispersion for each curve, which means that both the atom and its Fourier transform are spread over the real line.

**HEISENBERG DISPERSION**

The question arises whether it is possible to reduce the dispersion simultaneously for both the atom and its Fourier transform. The answer given by Heisenberg is no, because the product of the two dispersions is constrained by an inequality of the form where is some constant.

In the following figure, the atom has been shrunk to reduce its dispersion over the real line. As expected by the Heisenberg inequality, the dispersion of the Fourier transform becomes larger.

Similarly, in the following figure, the atom has been stretched to reduce the dispersion of its Fourier transform. As expected by the Heisenberg inequality, the dispersion of the atom becomes larger.

**OPERATORS X and D**

The quantity can be interpreted as a probability density function of the atom’s location, which satisfies the normalization condition . The expectation value of the position is given by

This suggests to introduce the position operator

The expectation value can then be rewritten as

Algebraically

In addition, is Hermitian, that is to say .

The quantity can be interpreted as a probability density function of the Fourier transform’s location, which satisfies the normalization condition as a consequence of the Parseval’s theorem. The expectation value of the frequency is given by

However, it is well known that .

This suggests to introduce the derivation operator

The expectation value can then be rewritten as

Or equivalently

Algebraically

In addition, is Hermitian, that is to say .

**COMMUTATOR**

An important property of and is that they do not commute, which means that applying then is not equivalent to applying then .

The noncommutativity can be measured through the concept of commutator defined as follows

This relation is very similar to the usual quantum mechanical expression .

**HEISENBERG INEQUALITY**

We begin to derive the inequality in the case of two Hermitian operators and such that . The key point is the Cauchy-Schwarz inequality

In fact, this can be seen as a generalization of a simple geometrical formula for two vectors and in the plane

We use the Cauchy-Schwarz inequality in two steps.

First, the operator being Hermitian it satisfies . The same occurs for which implies

Second, the magnitude of a complex number being greater than the magnitude of its imaginary part we have

Therefore the Cauchy-Schwarz inequality implies

Or more simply

It remains to make the connection with the operators and . By substituting and

where and are dispersions of and respectively.

The commutator satisfies

This leads to

And finally to the Heisenberg inequality

This relation is very similar to the usual quantum mechanical expression . Thus, the origin of the Heisenberg inequality is strongly mathematical and very independent from the physical world unlike many common beliefs.

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