Introduction to Quantum Mechanics · Problem 3.35

Formalism — David J. Griffiths

Physics — Quantum Mechanics · uncertainty_principle

Problem

Consider the wave function

\Psi(x, 0) = \begin{cases} \frac{1}{\sqrt{2n\lambda}} e^{i2\pi x/\lambda}, & -n\lambda < x < n\lambda, \\ 0, & \text{otherwise}, \end{cases}

where

n

is some positive integer. This function is purely sinusoidal (with wavelength

\lambda

) on the interval

-n\lambda < x < n\lambda

, but it still carries a range of momenta, because the oscillations do not continue out to infinity. Find the momentum space wave function

\Phi(p, 0)

. Sketch the graphs of

|\Psi(x, 0)|^2

and

|\Phi(p, 0)|^2

, and determine their widths,

w_x

and

w_p

(the distance between zeros on either side of the main peak). Note what happens to each width as

n \to \infty

. Using

w_x

and

w_p

as estimates of

\Delta x

and

\Delta p

, check that the uncertainty principle is satisfied. *Warning:* If you try calculating

\sigma_p

, you're in for a rude surprise. Can you diagnose the problem?