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Found in: Page 133

Modern Physics

Book edition 2nd Edition
Author(s) Randy Harris
Pages 633 pages
ISBN 9780805303087

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.

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.

See the step by step solution

Step 1: Uncertainty relation

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

Step 2: Explanation

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

In other words, if we require a function with a very small spatial bandwidth, we must cover a wide variety of wavelengths or wavenumbers; $\left(\lambda =\frac{2\pi }{k}\right)$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 $\Delta x\Delta k⩾\frac{h}{2}$. 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.