Question

In studying quantum mechanics, we find that all waves have object-like qualities and all objects have wave-like qualities. That technically means that all objects must obey the Heisenberg Uncertainty Principle. Let's assume the object is a 62 kg physics student and she moves through a 0.77 m wide door. We know that as she moves through the door, she must be somewhere inside its frame.

1) What is the uncertainty in her velocity?

Δv_student =

2) So it looks like we really won't care about the quantum mechanical effects involved unless we're worried about very, very small numbers. Another way of looking at the fact that all objects have wave-like properties is diffraction. Waves diffract as they move through a small opening and technically quantum mechanics means that all objects should diffract like a wave when going through a \"slit\". Again, let's assume the object is our 62 kg physics student and she moves through the \"slit\" of a 0.77 m wide door. We know that the diffraction of a wave first becomes significant when the width of the aperture is comparable to the wavelength of the waves being diffracted. At what speed should the student be moving in order to have significant diffraction as she enters the door?

v_student =

3) Lets assume she \"enters\" the room by moving 1 meter from the outside of the door to inside. At this speed, how long would it take for the student to enter to room?

t_enter =

Please help with this one as I have no clue on this one. Which formula should we use as well. Thanks!

Answer 1

1] Uncertainty in position of the student is atmost 0.77 m.

use Uncertainty principle to determine the uncertainty in momentum

=>

therefore, uncertainty in velocity is:

2]

Here, the De-Broglie wavelength can be taken to be approximately 0.77 m

then, using De-Broglie relation, the student's momentum will be:

so, the speed must be:

3] Use distance = speed x time

d = 1 meter and v = 1.38 x 10^-35 m/s

=> t = d/v = 1/(1.38 x 10^-35) = 7.2 x 10³⁴ s.

this is the time it will take to cover the 1 meter distance.