Problem 45

  From a cliff 9.5 m above a lake, a young woman (mass = 41 kg) jumps from rest, straight down into the water. At the instant she strikes the water, what is her de Broglie wavelength?

SOLUTION: The definition of de Broglie wavelength is

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

and momentum is

p = mv

so

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

We're given the young woman's mass, and we know Planck's constant, so if we find the velocity of the woman as she strikes the water, we'll know her de Broglie wavelength. The velocity is given by the kinematic equations which we all know and love, in particular

v² = v_o² + 2 a y

$$v = \sqrt{2 g y}$$

since the woman jumps from rest. Now we plug in to get

$$\lambda = \frac{6.63 \times 10^{-34} J \dot s} {(41 kg) \sqrt{2 (9.8 m/s^2)(9.5 m)}}$$

$$\lambda = 1.2 \times 10^{-36} m$$

which is why you don't observe the waviness of people.