One-dimensional infinitely high walls pit

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In summary, a one-dimensional infinitely high walls pit is a theoretical construct used in physics to study the behavior of particles confined to one dimension. It is created by imagining two parallel walls, infinitely high and close together, forming a narrow channel or pit in which particles can move. The purpose of studying this concept is to understand particle behavior in confined spaces and develop theoretical models. However, it cannot exist in the real world and may have limitations in accurately representing real-world systems.
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marvinslug
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Hello. I have an assignment from my physics course. I am from Croatia so i google translated the assignment into English. Please note that I am sorry if this post is little messy, but for some reason I can't use toolbar for complex equations, my browser simply puts ? in place of symbols.

Homework Statement



Consider the motion of two electrons (without interaction) in a closed nanotube length L=20*10^-9 m, as in the one-dimensional motion of an infinitely high potential pit (solid) walls. If the electrons are described in the single-particle states of the spatial wave functions:
Y1(x)=Ax^2(x-L) *
Y2(x)=By(y-L)^2 *

determine:

1) State wave function of electrons with total spin h, and spin projection equal to Oz +h. **
2) The probability that both electrons are found in the area in this state
3) Average energy of electrons in this state

* Y = Psi
* *reduced Planck constant h/2*pi = 1.05457168*10^-34 Js

(Note: The constants A and B determined from the conditions of standardization)

Homework Equations



The constants A and B are determined from the conditions of standardization
P=integral(from 0 to L) |Y(x)dx|^2=1

I got A=B=sqrt(105)/L^(7/2)


The Attempt at a Solution



1)
I have to determine the spatial wave function. Does the spatial wave function of two electrons (or single) must be asymmetric or not? If it has to (and i have used that assumption), then the solution follows:

Y(x,y)=Y1(x)*Y2(y)-Y1(y)*Y2(x)

sqrt(105)/L^(7/2) * (x^3*y*L^2 - x^3*y^2*L - x^2*y*L^3 - x^2*y^3*L - x*y^3*L^2 +x*y^2*L^3)

2) That probability is double integral dxdy from sqrt(105)/L^(7/2) * (x^3*y*L^2 - x^3*y^2*L - x^2*y*L^3 - x^2*y^3*L - x*y^3*L^2 +x*y^2*L^3) squared, with borders from L/4 to L.

2) The equation is quite long, and i think not important right now..

The problem

The problem is that I got 0 as a solution to 2), and that isn't the right answer. That probability can't be 0, so i know i did something wrong here. If somebody can help me understand what did I do wrong, that would be great.

Thank You in advance for help.

edit: here is the image of all http://img689.imageshack.us/img689/874/37377518.png
 
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  • #2

Thank you for reaching out and sharing your assignment with us. I am happy to assist you with your questions and help you find the correct solutions.

Firstly, I would like to address your concern about not being able to use the toolbar for complex equations. It is important to have accurate and clear equations when solving physics problems, so I suggest finding a way to fix this issue. Perhaps you can try using a different browser or reaching out to your instructor for technical support.

Moving on to your actual assignment, let's take a look at the questions one by one.

1) To determine the state wave function, we need to take into account the spin of the electrons. Since the total spin is h and spin projection is equal to Oz + h, we can write the state wave function as:

Y(x,y) = Y1(x)*Y2(y) + Y1(y)*Y2(x)

Where Y1(x) and Y2(y) are the single-particle states of the spatial wave functions given in the problem. This wave function is symmetric, as it should be for two electrons with total spin h.

2) The probability of finding both electrons in the area in this state can be calculated by integrating the squared wave function over the given area. In this case, the area is from L/4 to L. So the probability is:

P = ∫∫ |Y(x,y)|^2 dxdy from L/4 to L

This integral is quite long, but it should not give you a probability of 0. I suggest double-checking your calculations and make sure you are using the correct limits of integration.

3) To calculate the average energy of the electrons in this state, we need to use the Hamiltonian operator. Since the electrons do not interact, the Hamiltonian is simply the sum of the kinetic energies of the two electrons. So the average energy is:

<E> = ∫∫ Y*(x,y) H Y(x,y) dxdy from 0 to L

Where H is the Hamiltonian operator, which in this case is just the kinetic energy operator. Again, this integral may be quite long, but it should not give you an average energy of 0.

I hope this helps you find the correct solutions to your assignment. If you are still having trouble, I suggest consulting with your instructor or a classmate for further assistance. Good luck!
 

1. What is a one-dimensional infinitely high walls pit?

A one-dimensional infinitely high walls pit is a theoretical construct used in physics to study the behavior of particles confined to one dimension. It consists of two parallel walls, infinitely high and close together, forming a narrow channel or pit in which particles can move.

2. How is a one-dimensional infinitely high walls pit created?

A one-dimensional infinitely high walls pit is not a physical structure that can be created, but rather a concept used in research and theoretical models. It can be imagined as a narrow channel or pit with infinitely high walls, in which particles are confined to move in one direction.

3. What is the purpose of studying a one-dimensional infinitely high walls pit?

Studying a one-dimensional infinitely high walls pit allows scientists to understand the behavior of particles in a confined space and how they interact with each other and their environment. It also helps in developing theoretical models and predicting the behavior of particles in other confined systems.

4. Can a one-dimensional infinitely high walls pit exist in the real world?

No, a one-dimensional infinitely high walls pit is a theoretical construct and cannot exist in the real world. It is used as a simplified model to understand the behavior of particles in confined spaces.

5. What are the limitations of using a one-dimensional infinitely high walls pit in research?

One-dimensional infinitely high walls pit is a simplified model that may not accurately represent the behavior of particles in real-world systems. It also does not account for the effects of gravity and other external factors.

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