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:

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

Assuming that $\psi \!$ can be write like:

$\psi (x,y,z)=\Phi (x)\Delta (y)\Omega (z)\!$

So,

Failed to parse (unknown function "\begin{array}"): {\displaystyle \begin{array} -\frac{\hbar^2}{2m} \left[ \frac{d^2\Phi(x)}{dx^2} \Delta(y) \Omega (z) + \Phi(x)\frac{d^2\Delta(y)}{dy^2} \Omega (z) + \Phi(x) \Delta (y)\frac{d^2\Omega(z)}{dz^2} \right] \\ & + \left[X(x)+Y(y)+Z(z)\right]\Phi(x) \Delta(y) \Omega (z) = E\Phi(x) \Delta(y) \Omega (z) \end{array} }

Dividing by $\psi (x,y,z)\!$

Failed to parse (unknown function "\begin{array}"): {\displaystyle \begin{array} -\frac{\hbar^2}{2m} \frac{1}{\Phi(x)} \frac{d^2\Phi(x)}{dx^2} + X(x) -\frac{\hbar^2}{2m} \frac{1}{\Delta(y)} \frac{d^2\Delta(y)}{dy^2} + Y(y) \\ & -\frac{\hbar^2}{2m} \frac{1}{\Omega(z)} \frac{d^2\Omega(z)}{dz^2} + Z(z) = E \end{array} }

Perfectly we can separate the right hand side in three parts, where only one depends of $x\!$, only one of $y\!$ and only one of $z\!$. Then each of these parts must be equal to a constant. So:

$-{\frac {\hbar ^{2}}{2m}}{\frac {1}{\Phi (x)}}{\frac {d^{2}\Phi (x)}{dx^{2}}}+X(x)=E_{x}$

$-{\frac {\hbar ^{2}}{2m}}{\frac {1}{\Delta (y)}}{\frac {d^{2}\Delta (y)}{dy^{2}}}+Y(y)=E_{y}$

$-{\frac {\hbar ^{2}}{2m}}{\frac {1}{\Omega (z)}}{\frac {d^{2}\Omega (z)}{dz^{2}}}+Z(z)=E_{z}$

where $E_{x}\!$, $E_{y}\!$ and $E_{z}\!$ are constant and $E=E_{x}+E_{y}+E_{z}\!$

Hence the three-dimensional problem has been divided in three one-dimensional problems where the total energy $E\!$ is the sum of the energies $E_{x}\!$, $E_{y}\!$ and $E_{z}\!$ in each dimension.