Question

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.

Short answer

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.

Step-by-step solution

Uncertainty relation

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

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.