Heisenberg inequality on the real line

A quantum mechanical principle discovered by Werner Heisenberg states that it is not possible to simultaneously determine the position and momentum of a particle. In fact, this principle is deeply mathematical and independent from the experimental precision of the instruments. We illustrate how it can be expressed for functions on the real line.

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 x \in \mathbb{R}.

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

\displaystyle \langle u , v \rangle = \int \overline{u ( x )} v ( x ) dx

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

\displaystyle \left \Vert u \right \Vert = \sqrt{\langle u , u \rangle}.

GABOR ATOM

We define the normalized function \phi ( x ) = \widetilde{\phi} / \left \Vert \widetilde{\phi} \right \Vert where

\displaystyle \widetilde{\phi} ( x ) = e^{i 2 \pi f_0 x} e^{- ( x - x_0 )^2}

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 x - x_0 with a modulation e^{i 2 \pi f_0 x}. 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 \phi ( x ) (upper) and the absolute value of its Fourier transform \left | \widehat{\phi}( f ) \right | (lower) for a choice of parameters x_0 = 6 and f_0 = 2.

As it can be observed, the atom is concentrated at x = x_0 and its Fourier transform at f = f_0. 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 \Delta x \Delta f \geq k where k 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 \left | \phi ( x ) \right |^2 = \overline{\phi ( x )} \phi ( x ) can be interpreted as a probability density function of the atom’s location, which satisfies the normalization condition \int \left | \phi ( x ) \right |^2 dx = \left \Vert \phi \right \Vert = 1. The expectation value of the position is given by

\displaystyle \int \overline{\phi ( x )} x \phi ( x ) dx

This suggests to introduce the position operator X

\displaystyle \boxed{( X \phi ) ( x ) = x \phi ( x )}

The expectation value can then be rewritten as

\displaystyle \int \overline{\phi ( x )} ( X \phi ) ( x ) dx

Algebraically

\displaystyle \langle X \rangle_{\phi} = \left \langle \phi , X \phi \right \rangle

In addition, X is Hermitian, that is to say \langle \phi , X \phi \rangle = \langle X \phi , \phi \rangle.

The quantity \left | \widehat{\phi} ( f ) \right |^2 = \overline{\widehat{\phi} ( f )} \widehat{\phi} ( f ) can be interpreted as a probability density function of the Fourier transform’s location, which satisfies the normalization condition \int \left | \widehat{\phi} ( f ) \right |^2 df = \left \Vert \widehat{\phi} \right \Vert = 1 as a consequence of the Parseval’s theorem. The expectation value of the frequency is given by

\displaystyle \int \overline{\widehat{\phi} ( f )} f \widehat{\phi} ( f ) df

However, it is well known that f \widehat{\phi} ( f ) = - \frac{i}{2 \pi} \widehat{\phi '} ( f ).

This suggests to introduce the derivation operator D

\displaystyle \boxed{( D \phi ) ( x ) = - \frac{i}{2 \pi} \phi '}

The expectation value can then be rewritten as

\displaystyle \int \overline{\widehat{\phi} ( f )} \widehat{D \phi} ( f ) df

Or equivalently

\displaystyle \int \overline{\phi ( x )} ( D \phi ) ( x ) dx

Algebraically

\displaystyle \langle D \rangle_{\phi} = \left \langle \phi , D \phi \right \rangle

In addition, D is Hermitian, that is to say \langle \phi , D \phi \rangle = \langle D \phi , \phi \rangle.

COMMUTATOR

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

\displaystyle X D f ( x ) = X (- \frac{i}{2 \pi} f' ) ( x ) = - \frac{i}{2 \pi} x f' ( x )

\displaystyle D X f ( x ) = D ( x f ) ( x ) = - \frac{i}{2 \pi} ( f + x f ') ( x )

\displaystyle (X D f - D X f)( x ) = \frac{i}{2 \pi} f ( x )

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

\displaystyle \boxed{[ X , D ] = X D - D X = \frac{i}{2 \pi}}

This relation is very similar to the usual quantum mechanical expression [ X , P ] = \frac{i h}{2 \pi}.

HEISENBERG INEQUALITY

We begin to derive the inequality in the case of two Hermitian operators U and V such that \langle U \rangle_{\phi} = \langle V \rangle_{\phi} = 0. The key point is the Cauchy-Schwarz inequality

\displaystyle \langle U \phi , U \phi \rangle \langle V \phi , V \phi \rangle \geq | \langle U \phi , V \phi \rangle |^2

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

\displaystyle | u \cdot v | = \Vert u \Vert \Vert v \Vert | \cos \theta | \leq \Vert u \Vert \Vert v \Vert

We use the Cauchy-Schwarz inequality in two steps.

First, the operator U being Hermitian it satisfies \langle U \phi , U \phi \rangle = \langle \phi , U^2 \phi \rangle = \langle U^2 \rangle_{\phi}. The same occurs for V which implies

\displaystyle \langle U^2 \rangle_{\phi} \langle V^2 \rangle_{\phi} \geq | \langle U \phi , V \phi \rangle |^2

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

\displaystyle | \langle U \phi , V \phi \rangle | \geq | \Im \langle U \phi , V \phi \rangle |

\displaystyle | \langle U \phi , V \phi \rangle | \geq | \Im \langle \phi , U V \phi \rangle |

\displaystyle | \langle U \phi , V \phi \rangle | \geq \left | \frac{1}{2 i} \left ( \langle \phi , U V \phi \rangle - \overline{\langle \phi , U V \phi \rangle} \right ) \right |

\displaystyle | \langle U \phi , V \phi \rangle | \geq \frac{1}{2} | \langle \phi , U V \phi \rangle - \langle \phi , V U \phi \rangle |

\displaystyle | \langle U \phi , V \phi \rangle | \geq \frac{1}{2} | \langle \phi , ( U V - V U ) \phi \rangle |

Therefore the Cauchy-Schwarz inequality implies

\displaystyle \langle U^2 \rangle_{\phi} \langle V^2 \rangle_{\phi} \geq \frac{1}{4} | \langle [ U , V ] \rangle_{\phi} |^2

Or more simply

\displaystyle \boxed{\langle U^2 \rangle \langle V^2 \rangle \geq \frac{1}{4} | \langle [ U , V ] \rangle |^2}

It remains to make the connection with the operators X and D. By substituting U = X - \langle X \rangle and V = D - \langle D \rangle

\displaystyle \langle U^2 \rangle = \langle ( X - \langle X \rangle )^2 \rangle = \sigma^{2}_{X}

\displaystyle \langle V^2 \rangle = \langle ( D - \langle D \rangle )^2 \rangle = \sigma^{2}_{D}

where \sigma_X and \sigma_D are dispersions of X and D respectively.

The commutator satisfies

\displaystyle [ U , V ] = [ X - \langle X \rangle , D - \langle D \rangle ] = [ X , D ]

This leads to

\displaystyle \sigma_{X}^{2} \sigma_{D}^{2} \geq \frac{1}{4} | [ X , D ] |^2

And finally to the Heisenberg inequality

\displaystyle \boxed{\sigma_{X} \sigma_{D} \geq \frac{1}{4 \pi}}

This relation is very similar to the usual quantum mechanical expression \sigma_x \sigma_p \geq \frac{h}{4 \pi}. 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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One thought on “Heisenberg inequality on the real line

  1. Sudheer Gopinathan says:

    I happened to come across your blog by chance. The articles have the required mathematical rigor and are informative.

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