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

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

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.