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The one-dimensional Schrindinger wave equation for a particle in a potential field

The one-dimensional Schrindinger wave equation for a particle in a potential field \(V=\) \(\frac{1}{2} k x^{2}\) is

$$ -\frac{h^{2}}{2 m} \frac{d^{2} \psi}{d x^{2}}+\frac{1}{2} k x^{2} \psi=E \psi(x) $$

(a) Lsing \(\xi=\alpha x\) and a constant \(\lambda\), we have

$$ a=\left(\frac{m k}{A^{2}}\right)^{1 / 4}, \quad A=\frac{2 L}{A}\left(\frac{m}{k}\right)^{1 / 2} $$

show that

$$ \frac{d^{2} y(\xi)}{d \xi^{2}}+\left(\lambda-\xi^{2}\right) \psi(\xi)=0 $$

(b) Substituting

$$ \psi(\xi)=y(\xi) e^{2} / 2 $$

show that \(y(t)\) satisfies the Hermite di fferential equation.

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