Book edition 2nd Edition

Author(s) Randy Harris

Pages 633 pages

ISBN 9780805303087

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Short Answer

Question: Starting with the assumption that a general wave function may be treated as an algebraic sum of sinusoidal functions of various wave numbers, explain concisely why there is an uncertainty principle.

Answer:

The more sinusoidal wave functions are required to express a function, the less spreading it has in space, and vice versa. This follows immediately from the Fourier theorem and has a connection to the Heisenberg uncertainty principle.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Uncertainty relation

Mathematically, the uncertainty relation can be stated as $Δ x Δ p ⩾ \frac{h}{2}$ .

Step 2: Explanation

Fourier established this theory, which connects the function spreading in space $Δ x$ to the range of wavenumber $Δ k$ , long before the Heisenberg uncertainty principle. The relation states that $Δ x Δ k ⩾ \frac{1}{2}$ .

In other words, if we require a function with a very small spatial bandwidth, we must cover a wide variety of wavelengths or wavenumbers; $(λ = \frac{2 π}{k})$ as a result, we must combine many sinusoidal functions.

If we only take note that the momentum is equal to , then this may be directly connected to the Heisenberg connection. Consequently, if we substitute that back in, we will reestablish our original relationship $Δ x Δ k ⩾ \frac{h}{2}$ . So, the same conclusion is drawn: the more wavenumbers added together (greater uncertainty in momentum), the less uncertainty there is in location, and the opposite is also true.

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Modern Physics

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Short Answer

Question: Starting with the assumption that a general wave function may be treated as an algebraic sum of sinusoidal functions of various wave numbers, explain concisely why there is an uncertainty principle.

Step by Step Solution

Step 1: Uncertainty relation

Step 2: Explanation

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Modern Physics

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Short Answer

Question: Starting with the assumption that a general wave function may be treated as an algebraic sum of sinusoidal functions of various wave numbers, explain concisely why there is an uncertainty principle.

Step by Step Solution

Step 1: Uncertainty relation

Step 2: Explanation

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