An Introduction to Thermal Physics · Problem A.9

Systems of Interacting Particles — Daniel Schroeder

Physics — Statistical Mechanics · gaussian-wavepacket

Problem

The formula for a ``properly constructed'' wavepacket is

\Psi(x) = Ae^{ik_0 x}e^{-ax^2},

where

A

a

, and

k_0

are constants. (The exponential of an imaginary number is defined in Problem A.7. In this problem, just assume that you can manipulate the

i

like any other constant.) ((a)) Compute and sketch

|\Psi(x)|^2

for this wavefunction. ((b)) Show that the constant

A

must equal

(2a/\pi)^{1/4}

. (Hint: The probability of finding the particle somewhere between

x = -\infty

and

x = \infty

must equal 1. See Section 1 of Appendix B for help with the integral.) ((c)) The standard deviation

\Delta x

can be computed as

\sqrt{\overline{x^2} - \bar{x}^2}

, and the average value of

x^2

is just the sum of all values of

x^2

, weighted by their probabilities:

\overline{x^2} = \int_{-\infty}^{\infty} x^2\,|\Psi(x)|^2\,dx.

Use these formulas to show that for this wavepacket,

\Delta x = 1/(2\sqrt{a})

. ((d)) The Fourier transform of a function

\Psi(x)

is defined as

\tilde{\Psi}(k) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} e^{-ikx}\,\Psi(x)\,dx.

Show that

\tilde{\Psi}(k) = (A/\sqrt{2a})\exp[-(k-k_0)^2/4a]

for a properly constructed wavepacket. Sketch this function. ((e)) Using formulas analogous to those in part (c), show that, for this wavefunction,

\Delta k = \sqrt{a}

. (Hint: The standard deviation does not depend on

k_0

, so you can simplify the calculation by setting

k_0 = 0

from the start.) ((f)) Compute

\Delta p_x

for this wavefunction, and check whether the uncertainty principle is satisfied.