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uncertainty principle
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(Example)
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We will find the Fourier transform
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(1) |
of the Gaussian bell-shaped function
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(2) |
where $C$ and $a$ are positive constants.
We get first $$F(\omega) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty Ce^{-at^2}e^{-i\omega t}\,dt \;=\; \frac{C}{\sqrt{2\pi}}\int_{-\infty}^\infty e^{-at^2-i\omega t}\,dt.$$ Completing the square in $$-at^2-i\omega t \,=\, -a\left(t^2+\frac{i\omega t}{a}\right) \,=\, -a\left(t+\frac{i\omega}{2a}\right)^2-\frac{\omega^2}{4a}$$ and substituting $\sqrt{a}\left(t+\frac{i\omega}{2a}\right) \,:=\,z$ , we may write
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(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 $$\frac{\partial I_y}{\partial y} \,=\, \int_{-\infty}^\infty\frac{\partial}{\partial y} e^{-(x+iy)^2}dx \,=\, -2i\int_{-\infty}^\infty e^{-(x+iy)^2}(x+iy)\,dx \,=\, i\!\sijoitus{x\,=-\infty}{\quad \infty}\!e^{-(x+iy)^2} \,=\, i\!\sijoitus{x\,=-\infty}{\quad \infty}\!e^{-x^2+y^2}e^{-2ixy} \,=\,0.$$ 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
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(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
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(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.
- 1
- . . & . . :. ``''. (1976).
- 2
- Ya. B. Zel'dovich and A. D. Myshkis: ``Elements of applied mathematics''. Nauka (Science) Publishers, Moscow (1976).
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Cross-references: conversely, spectrum, inverse, formula, relation, graphs, product, ordinate, axis, abscissa, interpretation, area under Gaussian curve, integral, point, passing through, real axis, parallel, complex plane, line, completing the square, positive, function, Gaussian, Fourier transform
There are 8 references to this entry.
This is version 7 of uncertainty principle, born on 2008-12-24, modified 2009-02-23.
Object id is 11381, canonical name is UncertaintyPrinciple.
Accessed 568 times total.
Classification:
| AMS MSC: | 42A38 (Fourier analysis :: Fourier analysis in one variable :: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type) |
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Pending Errata and Addenda
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