Solving Quantum Problems: The Double Slit Experiment Explained

[philosophy42]

hello all,
i'm a college freshman and have a physics problem that i am haing a tough time figuring. Here is the problem



"Neutrons traveling at 0.400m/s are directed through a double slit apparatus having 1.00-mm separation. An array of detectors is placed 10.0 m from the slits. A) How far off axis is the first zero intensity point on the detector array? B) WHen a neutron traches a given dectector, can we say which slit the neutron passed through? C) Write a wave function that describes a neutron in this expereiment."

I understand parts of the question, such that we cannot tell which slit the neutron passed through because of the Heisenberg uncertainty principal. I am having trouble because I believe that i need to know the wavelength of the neutrons however am only given the speed and donnot know how to get the wavelength from the speed without to frequency. To find the first dark spot on the detection panel i was going to use the equation, y(dark) = (m = 1/2)* ((wavelength * L(dist from slit to dectector))/(distance between slits d)) am i way off here and how am i suppose to figure out wavelength, or because I am dealing with the duality of the particle being botha wave and a particle am i not suppose to have a wavelenth and figure the problem another way. Any help would be wonderful, thank you.

 

[M98Ranger]

Hi there,

I can understand your confusion with this problem. The double slit experiment is a classic example of quantum mechanics and can be quite challenging to wrap your head around. Let me try to break it down for you and hopefully it will make more sense.

First, let's address part A of the question. The first zero intensity point on the detector array refers to the point where there is no detection of neutrons. This occurs when the waves from the two slits interfere destructively, canceling each other out. In order to calculate the distance of this point from the center, we need to use the equation you mentioned, y(dark) = (m = 1/2)* ((wavelength * L)/(d)). In this equation, m represents the number of bright fringes away from the center. In this case, we are looking for the first dark fringe, so m = 1. L represents the distance from the slits to the detector array, which is given as 10.0 m in the problem. And d is the distance between the slits, which is given as 1.00 mm. The missing piece here is the wavelength, which we can calculate using the speed of the neutrons. The formula for wavelength is wavelength = h/mv, where h is Planck's constant (6.626 x 10^-34) and mv is the momentum of the neutrons. So, we can use the given speed of 0.400 m/s to calculate the momentum (p = mv) and then use that to find the wavelength. Once you have the wavelength, you can plug it into the equation to find y(dark).

Moving on to part B of the question, you are correct in saying that we cannot determine which slit the neutron passed through. This is because of the Heisenberg uncertainty principle, which states that we cannot know both the position and momentum of a particle with absolute certainty. In this experiment, we can know the position of the neutron (where it lands on the detector) but we cannot know its momentum (which slit it passed through). This is due to the wave-like nature of particles at the quantum level.

Lastly, for part C, we need to write a wave function that describes the neutron in this experiment. The wave function for a particle in a double slit experiment is given by Ψ(x) = Acos(kx) + Bsin(kx), where A and B are constants