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. Here we illustrate how it can be expressed for functions on the real line.

THE REAL LINE

The real line represents the set of all real numbers, which is a one dimensional Euclidean space. In particular, it 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. A useful case concerns square-integrable functions that 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 mathematical tools from 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 in order 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 natural question that arises is whether it would be 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 shrink to reduce its dispersion over the real line. As expected by the Heisenberg inequality, the dispersion is larger for the Fourier transform.

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 is larger for the atom.

OPERATORS X & 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 of the position is naturally 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 can then be rewritten as

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

In a more algebraic way

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

In addition, it can be pointed out that $X$ 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 of the frequency is naturally 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 can then be rewritten as

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

Or equivalently

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

In a more algebraic way

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

In addition, it can be pointed out that $D$ 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 $X$ .

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

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

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

Thus, the default of commutativity 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 close 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 given by 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 very simple geometrical formula for two vectors $u$ and $v$ in the plane

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

Now we transform the two sides of the Cauchy-Schwarz inequality.

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, we transform the right hand side of the Cauchy-Schwarz inequality. The magnitude of a complex number being greater than the magnitude of its imaginary part we have

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

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

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

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

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

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

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

Then, we make the connection with the operators $X$ and $D$ by substituting $U = X - \langle X \rangle$ and $V = D - \langle D \rangle$ . It comes

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

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

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 close 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 Response to Heisenberg inequality on the real line

Sudheer Gopinathan says:

February 15, 2011 at 18:58

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

Heisenberg inequality on the real line

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