Phy5645/Problem 1D sample

(Submitted by team 1. Based on problem 3.19 in Schaum's Theory and problems of Quantum Mechanics)

Consider a particle of mass m in a three dimensional potential: $V(x,y,z)=X(x)+Y(y)+Z(z)$

Using the Schroedinger's equation show that we can treat the problem like three independent one-dimensional problems. Relate the energy of the three-dimensional state to the effective energies of one-dimensional problem.

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The Schroedinger's equation takes the form: $-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}\psi (x,y,z)}{dx^{2}}}+(X(x)+Y(y)+Z(z))\psi (x,y,z)=E\psi (x,y,z)$

Assuming that $\Psi$ can be write like: $\psi (x,y,z)=\Phi (x)\Delta (y)\Omega (z)$

So,

$-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}\psi (x,y,z)}{dx^{2}}}+(X(x)+Y(y)+Z(z))\psi (x,y,z)=E\psi (x,y,z)$