Solving for C1 and C2: A Wave Function Boundary Condition

[zhillyz]

Homework Statement

A one-dimensional wave function associated with a localized particle can be written as

$\varphi (x) = \begin{cases} 1- \frac{x^2}{8}, & \text{if } 0<x<4, \\ C_1 - \frac{C_2}{x^2}, & \text{if} \,x \geq 4. \end{cases}$

Determine $C_1$ and $C_2$ for which this wave function will obey the boundary condition of continuity at x = 4.

Homework Equations

N\A

The Attempt at a Solution

So I am thinking the boundary condition is to make sure both equations hold at x = 4, and fed into the first equation it equals -1 so equate the second to -1 also and find values for $C_1 \text{and} C_2$ which would be 1 and 32 respectively? Is this correct because the question is worth 6marks which seems like a lot.

 

[gabbagabbahey]

Is $C_1=1$ and $C_2=32$ the only solution to $-1=C_1-\frac{C_2}{16}$?

You have two unknowns and one equation, so if you want a unique solution you will need one more independent equation for $C_1$ and $C_2$. What can you say about $\varphi'(x)$?

 

[zhillyz]

$16C_1+16 = C_2$ So for values of C_1 = 1,2,3,4 C_2 will = 32,48,64,80 respectively.

or

$C_2(n) = C_2(n-1) +16$

The first order differential of $\varphi$? Em that it would be part of the shrodinger equation?

 

[gabbagabbahey]

$16C_1+16 = C_2$ So for values of C_1 = 1,2,3,4 C_2 will = 32,48,64,80 respectively.

or

$C_2(n) = C_2(n-1) +16$

Who says that the constants have to be integers? There are an infinite number of solutions.

The first order differential of $\varphi$? Em that it would be part of the shrodinger equation?

You need to review your notes/textbook on the boundary conditions of the wavefunction. For a finite potential/barrier, the first derivative of the wavefunction must be continuous.

 

[paranormal]

I would like to provide some feedback on your attempt at solving this problem. Firstly, I appreciate your effort in trying to find a solution and your clear understanding of the boundary condition. However, I would suggest that you provide more details and steps in your solution to show a clear understanding of the problem and your approach to solving it. Additionally, it is important to justify your answer and explain why you chose certain values for C1 and C2. This will not only help with your understanding of the problem but also showcase your critical thinking skills. Lastly, it is always a good practice to double-check your answer and make sure it satisfies all the given conditions. Overall, your approach is on the right track, but I would suggest providing more detailed and well-explained steps to fully demonstrate your understanding of the problem.