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}, &amp; \text{if } 0&lt;x&lt;4, \\<br /> C_1 - \frac{C_2}{x^2}, &amp; \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&#039;(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.