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Next: The Spherical Harmonic Oscillator Up: Elementary quantization of the Previous: Asymptotic behavior and the

Normalization of wave function

The solution we demonstrated is called a Hermite polynomial,

$\displaystyle H(\eta)=(-1)^ne^{\eta^2}\frac{d^n}{d\eta^n}e^{-\eta^2}$ (2.7)

Properties of this function can be found with repeated activation of the derivatives,

$\displaystyle H_n(\eta)=$   $\displaystyle (-1)^n e^{\eta^2} \frac{d^{n-1}}{d\eta^{n-1}}\left(-2\eta e^{-\eta^2}\right)$  
    $\displaystyle =(-1)^ne^{\eta^2}\frac{d^{n-2}}{d\eta^{n-2}}\left(-2e^{-\eta^2}-2\eta \frac{d}{d\eta}e^{-\eta^2}\right)$  
    $\displaystyle =(-1)^ne^{\eta^2}\frac{d^{n-3}}{d\eta^{n-3}}\left(-2\frac{d}{d\et...
...-\eta^2}-2\frac{d}{d\eta}e^{-\eta^2}-2\eta\frac{d^2}{d\eta^2}e^{-\eta^2}\right)$ (2.8)

A continuation of this activation would thus find,

$\displaystyle H_n(\eta)=-2(n-1)H_{n-2} + 2\eta H_{n-1} \ \rightarrow \ H_{n+1}=-2nH_{n-1}+2\eta H_n$ (2.9)

This implies that,

$\displaystyle \frac{d}{d\eta}H_n(\eta)$ $\displaystyle =2\eta(-1)^n e^{\eta^2}\frac{d^n}{d\eta^n}e^{-\eta^2} + (-1)^ne^{\eta^2}\frac{d^{n+1}}{d\eta^{n+1}}e^{-\eta^2}$  
  $\displaystyle =2\eta H_n(\eta) - H_{n+1}(\eta) = 2nH_{n-1}(\eta)$ (2.10)

The normalization equation is, via integration by parts,

$\displaystyle \int \psi^*_n(\eta) \cdot \psi_n(\eta) dx=A^2\sqrt{\frac{\hbar}{m...
...}\int\left(\frac{d^{n-1}}{d\eta^{n-1}}e^{-\eta^2}\right)\cdot H'_n(\eta)d\eta=1$ (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,

$\displaystyle A^2\sqrt{\frac{\hbar}{m\omega}}2^nn!\int e^{-\eta^2}=A^2 \sqrt{\frac{\hbar}{m\omega}}2^n n!\sqrt{\pi} = 1$ (2.12)

Thus we have

$\displaystyle \psi_n(x)=\left(\frac{m\omega}{\pi \hbar}\right)^{1/4}\frac{e^{-\...
...2/2}}{\sqrt{2^nn!}}\left((-1)^n e^{\eta^2}\frac{d^n}{d\eta^n}e^{-\eta^2}\right)$ (2.13)

Where a most general solution is

$\displaystyle \Psi = \sum_{n=0}^{\infty} C_n \psi_n$

Here $ C_n$ are coefficients which can only be determined when the specific physical situation is given.


next up previous
Next: The Spherical Harmonic Oscillator Up: Elementary quantization of the Previous: Asymptotic behavior and the
Timothy D. Jones 2007-01-29