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# uncertainty principle

We will find the Fourier transform

$\displaystyle F(\omega):=\frac{1}{\sqrt{2\pi}}\int_{{-\infty}}^{\infty}f(t)e^{% {-i\omega t}}\,dt$ | (1) |

of the Gaussian bell-shaped function

$\displaystyle f(t)\;=\;Ce^{{-at^{2}}}$ | (2) |

We get first

$F(\omega)=\frac{1}{\sqrt{2\pi}}\int_{{-\infty}}^{\infty}Ce^{{-at^{2}}}e^{{-i% \omega t}}\,dt\;=\;\frac{C}{\sqrt{2\pi}}\int_{{-\infty}}^{\infty}e^{{-at^{2}-i% \omega t}}\,dt.$ |

$-at^{2}-i\omega t\,=\,-a\left(t^{2}+\frac{i\omega t}{a}\right)\,=\,-a\left(t+% \frac{i\omega}{2a}\right)^{2}-\frac{\omega^{2}}{4a}$ |

and substituting $\sqrt{a}\left(t+\frac{i\omega}{2a}\right)\,:=\,z$, we may write

$\displaystyle F(\omega)\,=\,\frac{C}{\sqrt{2\pi}}\int_{{-\infty}}^{\infty}e^{{% -a\left(t+\frac{i\omega}{2a}\right)^{2}}}e^{{-\frac{\omega^{2}}{4a}}}\,dt\,=\,% \frac{C}{\sqrt{2\pi a}}e^{{-\frac{\omega^{2}}{4a}}}\int_{l}e^{{-z^{2}}}\,dz,$ | (3) |

where $l$ is a line of the complex plane parallel to the real axis and passing through the point $z=\frac{i\omega}{2\sqrt{a}}$. Now we can show that the integral

$I_{y}\,:=\,\int_{l}e^{{-z^{2}}}\,dz=\int_{{-\infty}}^{\infty}e^{{-(x+iy)^{2}}}% \,dx$ |

does not depend on $y$ at all. In fact, we have

$\frac{\partial I_{y}}{\partial y}\,=\,\int_{{-\infty}}^{\infty}\frac{\partial}% {\partial y}e^{{-(x+iy)^{2}}}dx\,=\,-2i\int_{{-\infty}}^{\infty}e^{{-(x+iy)^{2% }}}(x+iy)\,dx\,=\,i\!\operatornamewithlimits{\Big/}_{{\!\!\!x\,=-\infty}}^{{\,% \quad\infty}}\!e^{{-(x+iy)^{2}}}\,=\,i\!\operatornamewithlimits{\Big/}_{{\!\!% \!x\,=-\infty}}^{{\,\quad\infty}}\!e^{{-x^{2}+y^{2}}}e^{{-2ixy}}\,=\,0.$ |

Hence we may evaluate $I_{y}$ as

$I_{y}\,=\,I_{0}=\int_{{-\infty}}^{\infty}e^{{-x^{2}}}\,dx\;=\;\sqrt{\pi}$ |

(see the area under Gaussian curve). Putting this value to (3) yields the result

$\displaystyle F(\omega)\;=\;\frac{C}{\sqrt{2a}}e^{{-\frac{\omega^{2}}{4a}}}.$ | (4) |

Thus, we have gotten another Gaussian bell-shaped function (4) corresponding to the given Gaussian bell-shaped function (2).

Interpretation. One can take for the breadth of the bell the portion of the abscissa axis, outside which the ordinate drops under the maximum value divided by $e$, for example. Then, for the bell (2) one writes

$Ce^{{-at^{2}}}=Ce^{{-1}},$ |

whence $t=\frac{1}{\sqrt{a}}$ giving, by evenness of the function, the breadth $\Delta t=\frac{2}{\sqrt{a}}$. Similarly, the breadth of the bell (4) is $\Delta\omega=4\sqrt{a}$. We see that the product

$\displaystyle\Delta t\cdot\Delta\omega=8$ | (5) |

has a constant value. One can show that any other shape of the graphs of $f$ and $F$ produces a relation similar to (5). The breadths are thus inversely proportional.

If $t$ is the time and $f$ is the action of a force on a system of oscillators with their natural frequencies, then in the formula

$f(t)\;=\;\frac{1}{\sqrt{2\pi}}\int_{{-\infty}}^{\infty}F(\omega)e^{{i\omega t}% }\,d\omega$ |

of the inverse Fourier transform, $F(\omega)$ means the amplitude of the oscillator with angular frequency $\omega$. We can infer from (5) that the more localised ($\Delta t$ small) the external force is in time, the more spread ($\Delta\omega$ great) is its spectrum of frequencies, i.e. the greater is the amount of the oscillators the force has excited with roughly the same amplitude. If one, conversely, wants to better the selectivity, i.e. to compress the spectrum narrower, then one has to spread out the external action in time. The impossibility to simultaneously localise the action in time and enhance the selectivity of the action is one of the manifestations of the quantum-mechanical uncertainty principle, which has a fundamental role in modern physics.

# References

- 1 Я. Б. Зельдович & А. Д. Мышкис: Элементы прикладной математики. Издательство ‘‘Наука’’. Москва (1976).
- 2 Ya. B. Zel’dovich and A. D. Myshkis: ‘‘Elements of applied mathematics’’. Nauka (Science) Publishers, Moscow (1976).

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## Comments

## Uncertainty principle

There exists a very general formulation of the mathematical uncertainty theorem

in the frame of wavelet theory. It applies to any function f(t), not only to the

gaussian one.

Let F(a) be the Fourier transform of f(t) (a stands for "omega"). The uncertainty

theorem states that the "time-width" D_t(f) and the "spectrum-width" D_a(F) satisfy

the following inequality:

D_t(f)D_a(F) >= 1/2

Equality is reached for a gaussian function.

f(t) is any complex function of the real variable t, satisfying some conditions

detailed at the bottom.

By analogy with "classical" quantum theory, the width of a function f is defined

as follows:

Let d_t(f) be the distribution function |f(t)|^2/||f||, where ||f|| is the norm of

f, that is the integral between +- infinity of |f|^2. The integral of the

distribution between +- infinity is of course 1. Then t_0 is the "average" value

of t treated as a random variable with distribution d_t(f). The variance of t will

be the average value of |t-t_0|^2, and the width of f is defined as the square root

of this variance. Similarly for F(a). But, if f(t) is real, the average a_0 is 0.

Like in quantum theory, the proof relies on the Cauchy-Schwarz inequality.

f(t) and its Fourier transform F(a) must satisfy the following conditions:

1 - f(t) is in L_2

2 - tf(t) is in L_2

3 - f'(t) (the derivative) is in L_2

4 - F(a) is in L_2

5 - aF(a) is in L_2

These conditions are probably not independent. It seems to me that the second one

implies all the others; can you prove it, or give a counter-example?