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Seperation of variables, infinite cubic well

  1. Apr 28, 2010 #1
    Suppose one is to find the stationary states of a particle in an infinite cubic well. Inside the box the time independent SE is:

    [tex] - \frac{\hbar}{2m} \big( \frac{\partial ^2 \psi}{\partial x ^2 } + \frac{\partial ^2 \psi}{\partial z ^2 } + \frac{\partial ^2 \psi}{\partial z ^2 } \big)= E\psi [/tex]

    Using separation of variables: [tex] \psi = X(x)Y(x)Z(z) [/tex] we get:

    [tex] YZ\frac{\partial ^2 X}{\partial x^2} + XZ\frac{\partial ^2 Y}{\partial y^2} + XY\frac{\partial ^2 Z}{\partial z^2} = \frac{-2mE}{\hbar ^2} XYZ [/tex]

    After this one divides both sides by XYZ. My question is the following:
    When dividing by XYZ one must assume that XYZ is different from zero. However the solution we obtain has lots of zeroes. Is this not a problem?

    Thanks in advance
    Last edited: Apr 28, 2010
  2. jcsd
  3. Apr 28, 2010 #2

    Vanadium 50

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  4. Apr 28, 2010 #3


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    You just have to rule out XYZ being zero everhywhere.
    At an isolated zero, the limit of the ratio X"/X is well behaved for an eigenfunction.
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