Question:
The wave function for the one dimensional harmonic oscillator with the potential energy, \frac{1}{2}kx^2, is given as
\Phi_0 = C\exp(-ax^2)
Find the constant C when the wave function is normalized. Note that you can use the normalized
Gaussian distribution: \frac{1}{\sqrt{2\pi}\sigma}\int^{\infty}_{-\infty}\exp(-\frac{x^2}{2\sigma^2})dx = 1.
Answer:
For the normalized wave function, it has to satisfy:
\int^{\infty}_{-\infty}\Phi_0\Phi^*_0 dx = C^2\int^{\infty}_{-\infty}\exp(-2ax^2) dx = 1
where \Phi^*_0 is the complex conjugate of \Phi_0. Compare this with the Gaussian normal distribution.
2a = \frac{1}{2\sigma^2} \\ \sigma = \frac{1}{\sqrt{4a}}
Plug it in the formula of Gaussian distribution.
\frac{1}{\sqrt{2\pi}(1/\sqrt{4a})}\int^{\infty}_{-\infty}\exp(-\frac{x^2}{2(1/4a)})dx = 1
We can see
C^2 = \frac{1}{\sqrt{2\pi}(1/\sqrt{4a})} = \frac{1}{\sqrt{2\pi}}\sqrt{4a} \\ = \sqrt{\frac{2a}{\pi}}
Therefore,
C = (2a/\pi)^{1/4}
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