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- Homework Statement
- please see below

- Relevant Equations
- please see below

(a) I guess I should find ##C_n## by normalizing ##\psi_n##.

$$∫_{∞}^∞|C_nψn(x)|^2 dx=C_n^2 \frac{2}{a}∫_0^a sin^2(\frac{πnx}{a})dx=1$$

$$C_n^2 \frac{2}{a}[\frac{a}{2}−\frac{a}{4πn}sin(\frac{2πna}{a})]=1⇒C_n=1$$

(b) $$Hψ_n(x)=\frac{-ħ^2}{2m}\frac{\partial^2}{\partial x^2}\sqrt{\frac{2}{a}}sin(\frac{πnx}{a})$$

$$Hψ_n(x)=\frac{-ħ^2}{2m}\frac{(\pi n)^2}{a^2}\sqrt{\frac{2}{a}}sin(\frac{πnx}{a})$$

##\frac{-ħ^2}{2m}\frac{(\pi n)^2}{a^2}## gives the ##n##th energy value with probability ##\frac{1}{n^2}##

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