Notes4 - Lesson Module 4 I Heisenberg Uncertainty Principle...

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Lesson Module 4 I. Heisenberg Uncertainty Principle A. Fourier Transform A common technique for solving differential equations is the Fourier Transform that converts the solution of a difficult differential equation into an easier to solve algebraic equation. The Fourier transform F(k) of a function f(x) is found by - = dx e f(x) 1 F(k) x k i and the original function can be obtained using the inverse Fourier transform - = dk e F(k) 1 f(x) x k i - In addition to the advantage of being able to convert calculus based equations (differential equations) into easier to solve algebra equations, Fourier transforms also sometimes are more aligned with the way experimental data can be obtained. For instance, the frequency response of an electrical circuit is more easily obtained than its time response. B. Uncertainty with the Fourier Transform Any two variables related by the Fourier transform (example: x and k) obey the uncertainty relationship 2 1 Δk Δx This relationship is due to the fact that the Fourier transform is really equivalent to an expansion of a function f(x) in terms of a series of sinusoidal waves with different wave numbers. If f(x) is a single sine wave that starts at negative infinity and continues to positive infinity then only a single wave is needed. In the case, the wave number is known with absolute certainty, but the position is totally uncertain. If we try and restrict the domain of the sinusoid then additional waves of different wave numbers must be added to destructively interfere in the regions we wish to exclude. In this case, the uncertainty in the domain is decreased but the uncertainty in the wave number is increased. This uncertainty relationship was has important applications in many fields of engineering including electrical
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engineering. Since time and frequency are variables related by a Fourier transform, using a band pass or other type of frequency filter to reduce noise in your signals will always lead to reduced ability to distinguish between signals arriving at different times. If you want to design circuits with excellent time resolution then you must have a very wide pass band for your electronics. Since all electronic components have built in capacitance and inductance, you have inherent filters in your designs. Thus, proper selection of component materials is essential in such circuits. This is uncertainty relationship is called time-frequency reciprocity by electrical engineers doing filter design. C. Quantum Mechanics and Uncertainty The uncertainty relationship between variables related by Fourier transforms was well known by Physicists long before quantum mechanics. However, an uncertainty in the wave number of a baseball had no physical meaning in classical mechanics. deBroglie's duality relationship however gives the uncertainty relationship important consequences in quantum mechanics.
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