Module 6 : PHYSICS OF SEMICONDUCTOR DEVICES
Lecture 30 : Quantum Mechanical Concepts
The equation may be solved by separation of variables by substituting
 
$\displaystyle \psi(x,y,z) = \psi_x(x)\psi_y(y)\psi_z(z)$
  in eqn. (A) and dividing the resulting equation by $ \psi_x\psi_y\psi_z$. We get
 
$\displaystyle \frac{1}{\psi_x}\frac{\partial^2\psi_x}{\partial x^2} +\frac{1}{... ...{\partial y^2} + \frac{1}{\psi_z}\frac{\partial^2\psi_z}{\partial z^2} + k^2= 0$
  Since $ k^2$ is constant and the first three terms depend upon $ x,y,z$ separately, the above equation can be satisfied for all values of $ x,y,z$ only if each of the three terms is constant, i.e.
 
$\displaystyle \frac{1}{\psi_x}\frac{\partial^2\psi_x}{\partial x^2}+ k_x^2$ $\displaystyle =$ 0
$\displaystyle \frac{1}{\psi_y}\frac{\partial^2\psi_y}{\partial y^2} + k_y^2$ $\displaystyle =$ 0
$\displaystyle \frac{1}{\psi_z}\frac{\partial^2\psi_z}{\partial z^2} + k_z^2$ $\displaystyle =$ 0
 
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