Calculating the de Broglie wavelength?

Question

Calculate the de Broglie wavelength for a Br2 molecule moving at 206 m/s and the textbook weighing 7.27 kg dropping at 8.94 m/s. How do the wavelengths compare to the size of the object?

Answer 1

For matter, we have the de Broglie wavelength as:

wavelength = h / momentum

with h being Planck's constant, 6.626 * 10 ^ - 34 J * s. We will start with the Br2 molecule - first, when solving these I prefer to convert the mass into kilograms, where we can use that definition of h.

So, for the mass in kilograms, we have:

m (Br2 ) =(79.904 * 2 g / mol ) * (1 kg / 1000 g) * (1 / 6.022 * 10^ 23 / mol) = 2.654 * 10^-25 kg

Then, the momentum is p = m * v = (2.654 * 10^-25 kg) * ( 206 m/s) = 5.467 * 10^-23 N * s

Plug this into de Broglie's law:

wavelength = (6.626 * 10^-34 J * s) / ( 5.467 * 10^-23 N * s )

wavelength = 1.212 * 10^-11 m

Awesome. Now for the textbook:

momentum = 7.27 kg * 8.94 m/s

momentum = 65 N*s

Again, plug into de Broglie's equation:

wavelength = 6.626 * 10^-34 J * s / ( 65 N*s )

wavelength = 1.019 * 10^-35 m

We can see that for the object with the larger mass, the de broglie wavelength is much smaller - meaning that any matter interference effects will be very hard to measure. The consequences of this are relevant for experiments, in the fact that any interference measurements either have to be very precise, or use very low-mass objects.