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The solution we demonstrated is called a Hermite polynomial,
![$\displaystyle H(\eta)=(-1)^ne^{\eta^2}\frac{d^n}{d\eta^n}e^{-\eta^2}$](img60.png) |
(2.7) |
Properties of this function can be found with repeated activation of the derivatives,
A continuation of this activation would thus find,
![$\displaystyle H_n(\eta)=-2(n-1)H_{n-2} + 2\eta H_{n-1} \ \rightarrow \ H_{n+1}=-2nH_{n-1}+2\eta H_n$](img65.png) |
(2.9) |
This implies that,
The normalization equation is, via integration by parts,
![$\displaystyle \int \psi^*_n(\eta) \cdot \psi_n(\eta) dx=A^2\sqrt{\frac{\hbar}{m...
...}\int\left(\frac{d^{n-1}}{d\eta^{n-1}}e^{-\eta^2}\right)\cdot H'_n(\eta)d\eta=1$](img69.png) |
(2.11) |
Use of Equation 2.10 and repeated integration by parts will
diminish the stand-alone derivative and the Hermite polynomial until
we have,
![$\displaystyle A^2\sqrt{\frac{\hbar}{m\omega}}2^nn!\int e^{-\eta^2}=A^2 \sqrt{\frac{\hbar}{m\omega}}2^n n!\sqrt{\pi} = 1$](img70.png) |
(2.12) |
Thus we have
![$\displaystyle \psi_n(x)=\left(\frac{m\omega}{\pi \hbar}\right)^{1/4}\frac{e^{-\...
...2/2}}{\sqrt{2^nn!}}\left((-1)^n e^{\eta^2}\frac{d^n}{d\eta^n}e^{-\eta^2}\right)$](img71.png) |
(2.13) |
Where a most general solution is
Here
are coefficients which can only be determined when the specific physical situation is given.
Next: The Spherical Harmonic Oscillator
Up: Elementary quantization of the
Previous: Asymptotic behavior and the
Timothy D. Jones
2007-01-29