Question

Diffraction effects become significant when the width of an aperture is comparable to the wavelength of the waves being diffracted. (a) At what speed will the de Broglie wavelength of a 65-kg student be equal to the 0.76-m width of a doorway? (b) At this speed, how much time will it take the student to travel a distance of 1.0 mm? (For comparison, the age of the universe is approximately 4 * 1017 s.)

Answer 1

To develop this problem it is necessary to apply the concepts related to Broglie hypothesis.

The hypothesis defines that

$\lambda = \frac{h}{p}$

Where,

P = momentum

h = Planck's constant

The momentum is also defined as,

P = mv

Where,

m = mass

v = Velocity

PART A) Replacing at the first equation

$\lambda = \frac{h}{mv}$

Our values are given as,

$h = 6.626*10^{-34}Js$

$m = 65Kg$

$\lambda = 0.76m$

Re-arrange to find v, we have:

$v = \frac{h}{m\lambda}$

$v = \frac{6.626*10^{-34}}{65*0.76}$

$v = 1.341*10^{-35}m/s$

PART B) From the kinematic equations of movement description we know that velocity is defined as displacement over a period of time, that is

$v = \frac{x}{t}$

Re-arrange to find t,

$t = \frac{d}{v}$

$t = \frac{0.001}{ 1.341*10^{-35}}$

$t = 7.455*10^{31}s$

$7.455*10^{31} > 4*10^{17} \rightarrow$the age of the universe.