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A one-dimensional wave function, determine C1 and C2 so that boundary conditions are

  1. Aug 11, 2012 #1
    1. The problem statement, all variables and given/known data

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

    [itex]\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}[/itex]

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

    2. Relevant equations

    N\A

    3. 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 [itex] C_1 \text{and} C_2[/itex] which would be 1 and 32 respectively? Is this correct because the question is worth 6marks which seems like a lot.
     
  2. jcsd
  3. Aug 11, 2012 #2

    gabbagabbahey

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    Re: A one-dimensional wave function, determine C1 and C2 so that boundary conditions

    Is [itex]C_1=1[/itex] and [itex]C_2=32[/itex] the only solution to [itex]-1=C_1-\frac{C_2}{16} [/itex]?

    You have two unknowns and one equation, so if you want a unique solution you will need one more independent equation for [itex]C_1[/itex] and [itex]C_2[/itex]. What can you say about [itex]\varphi'(x)[/itex]?
     
  4. Aug 12, 2012 #3
    Re: A one-dimensional wave function, determine C1 and C2 so that boundary conditions

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

    or

    [itex] C_2(n) = C_2(n-1) +16[/itex]

    The first order differential of [itex]\varphi[/itex]? Em that it would be part of the shrodinger equation?
     
  5. Aug 12, 2012 #4

    gabbagabbahey

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    Re: A one-dimensional wave function, determine C1 and C2 so that boundary conditions

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

    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.
     
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