Solving for Constants and Graphing Wave Function: Modern Physics Exam Guide

[OhNoYaDidn't]

This was in my Introduction to modern physics exam, but i don't quite know what i should do here... My teacher said there was an easy trick. Can you guys help me?

An electron is described by the following wave function
ψ(x)=(ax+b for 0<x<L,
cx+d for L<x<3L,
0 for x<0 V x>3L)

a) Determine the constants a, b and c, and sketch the graph of Ψ.

The Attempt at a Solution

I tried using the boundary conditions, but everything turns out being zero. Guys, please help me, I'm lost here.

 

[Fightfish]

The gist of the problem as you pointed out, is to match the boundary conditions between the different regions. However, I don't see why this should turn out to be zero everywhere. Could you show your work that led you to this conclusion?

 

[OhNoYaDidn't]

Ok, so applying the boundary conditions of the wave functions.
ψ(0-)= ψ(0+)
ψ(L-)= ψ(L+)
ψ(3L-)= ψ(3L+)
i get
b=0
a=-2c
d=-3L
---
Then i use the continuity of the derivative:

and i get
a = 0 = c = b, only d=-3L.

Is this right? I hope I'm not doing any ridiculous mistake.

 

[Fightfish]

Ok, so applying the boundary conditions of the wave functions.
ψ(0-)= ψ(0+)
ψ(L-)= ψ(L+)
ψ(3L-)= ψ(3L+)
i get
b=0
a=-2c
d=-3L

d = -3Lc. I presume that is a typo since u managed to arrive at a = -2c.

From this you should be able to figure out the shape of the wavefunction. The values of a, c and d (which are related to each other), can be determined through imposing normalisation.

It is okay for the derivative not to be continuous. This can happen when the potential function itself is discontinuous to begin with. In this case since the potential is not given, we assume that the wavefunction given is not the trivial (zero) solution.

 

[HalfLostAndSearching]

Hello! As a fellow scientist, I understand your struggle with solving for constants and graphing wave functions. The first step in solving for the constants a, b, and c is to use the boundary conditions, as you have already attempted. However, it seems that you may have made a mistake in your calculations. I would suggest going back and carefully checking your work to see if you missed any steps or made any errors.

If you are still having trouble, here is a tip that may help: remember that the wave function must be continuous and differentiable at all points. This means that the value of ψ(x) and its derivative must be the same on both sides of the boundary points (x=0, x=L, and x=3L). Using this information, you should be able to set up a system of equations and solve for the constants.

Once you have determined the values of a, b, and c, you can then sketch the graph of ψ(x). Remember to consider the shape of the wave function in each region and how it connects at the boundary points. It may also be helpful to use a graphing calculator or software to visualize the function.

I hope this helps. Good luck on your exam! Remember to always double check your work and don't be afraid to ask for help if you need it. Science is all about collaboration and problem solving.