Finding a Wave Function

[bannadonna]

On My Last Straw Trying to Find a Wave Function

I am horribly confused as how to I can actually find a wave function for any given problem. The specific wave function I am trying to find right now is that of a neutron passing through a double slit apparatus. Here is how I have the problem set up thus far:

d= spacing between slits= 1.00 x 10^-3 m (or 1.oo mm)
L= distance between slit apparatus and the detector array = 10.0 m
y= distance on detector array from major maximum (the one in the middle)
m= # of high intensity bands away from middle with the middle one at m=0
v= velocity of the neutron = .400 m/s
w= mass of neutron= 1.67 x 10^-27 kg
h=Planck's constant = 6.63 x 10^-34 Js

I know that the wave function should demonstrate where the destructive interference is going on by being zero when y=(m+.5)(hL/vwd). It should also show the bright spots as maximums and minimums at y=(mhL)/(vwd). I figured that since wave functions are a lot like amplitude functions, this will be a cosine function. My best guess is ... what's that pitchfork sign? ... anyway, that the wave function equals cos(2(pi)ywv/h). Sorry, I don't know how to put a pi in the equation. :uhh:
This cosine function maxes at y=0, which is good; that is what the interference pattern does. But a lot of trig functions could do that, and I cannot justify why mine is a good one. Is it a wave function that describes this neutron? If it is, why? If not, how do I find a function that does?

 

[anathema]

I understand your frustration in trying to find a wave function for your specific problem. The process of finding a wave function can be complex and daunting, but with the right approach and understanding, it can be achieved.

Firstly, it is important to understand the concept of a wave function. A wave function is a mathematical representation of a particle's behavior in a given system. It describes the probability of finding the particle at a certain position and time. In your case, the neutron passing through the double slit apparatus can be described by a wave function that takes into account its position and time.

To find the wave function for your specific problem, you can use the Schrödinger equation, which is a fundamental equation in quantum mechanics that describes the behavior of particles. It takes into account the particle's energy, potential energy, and time. By solving this equation, you can obtain the wave function for your neutron passing through the double slit apparatus.

In your case, the Schrödinger equation can be written as:

iħ∂ψ/∂t = (-ħ^2/2m)∂^2ψ/∂x^2 + V(x)ψ

Where ħ is the reduced Planck's constant, m is the mass of the neutron, ψ is the wave function, t is time, x is the position, and V(x) is the potential energy.

To solve this equation, you can use various mathematical techniques, such as separation of variables or the Fourier transform method. It may also be helpful to consult with a colleague or a mentor who has experience in solving similar problems.

Once you have obtained the wave function, you can then use it to calculate the probability of finding the neutron at a certain position on the detector array. This will help you understand the interference pattern and the behavior of the neutron passing through the double slit apparatus.

In summary, finding a wave function for a specific problem requires a thorough understanding of the concept and the use of mathematical techniques. I hope this helps in your search for the wave function for the neutron passing through the double slit apparatus. If you need further assistance, do not hesitate to reach out to other scientists or experts in the field for guidance. Good luck!

 

[wais]

I can understand your frustration and confusion in trying to find a wave function for a specific problem. It can be a complex and daunting task, especially when dealing with a problem like a neutron passing through a double slit apparatus. However, it is important to remember that finding a wave function is not an exact science and there may be multiple possible solutions.

Based on the information provided, it seems like you have a good understanding of the variables involved and the general form of the wave function. However, as you mentioned, there could be many different trigonometric functions that could fit the criteria you have set. In order to determine the correct wave function, you may need to consider additional factors such as the boundary conditions and the specific properties of the neutron.

One approach you could take is to look at similar problems and their corresponding wave functions. This could give you some insights and help you to narrow down the possibilities. You could also consult with a physics professor or seek out additional resources for guidance.

Remember that finding a wave function is not a simple task, but with patience and persistence, you will eventually find a solution that fits the problem. Keep exploring and don't give up on your search.