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Question-138828




Question Number 138828 by BHOOPENDRA last updated on 18/Apr/21
Answered by Dwaipayan Shikari last updated on 22/Apr/21
Ψ=Asin((πx)/l)  ∫_0 ^l ∣Ψ∣^2 dx=1⇒∫_0 ^l A^2 sin^2 ((πx)/l)dx=1       ((πx)/l)=u⇒1=(l/π).(du/dx)  ⇒A^2 (l/π)∫_0 ^π sin^2 u du=1  ⇒A^2 ((π/2))=(π/l)⇒A=(√(2/l))
$$\Psi={Asin}\frac{\pi{x}}{{l}} \\ $$$$\int_{\mathrm{0}} ^{{l}} \mid\Psi\mid^{\mathrm{2}} {dx}=\mathrm{1}\Rightarrow\int_{\mathrm{0}} ^{{l}} {A}^{\mathrm{2}} {sin}^{\mathrm{2}} \frac{\pi{x}}{{l}}{dx}=\mathrm{1}\:\:\:\:\:\:\:\frac{\pi{x}}{{l}}={u}\Rightarrow\mathrm{1}=\frac{{l}}{\pi}.\frac{{du}}{{dx}} \\ $$$$\Rightarrow{A}^{\mathrm{2}} \frac{{l}}{\pi}\int_{\mathrm{0}} ^{\pi} {sin}^{\mathrm{2}} {u}\:{du}=\mathrm{1} \\ $$$$\Rightarrow{A}^{\mathrm{2}} \left(\frac{\pi}{\mathrm{2}}\right)=\frac{\pi}{{l}}\Rightarrow{A}=\sqrt{\frac{\mathrm{2}}{{l}}} \\ $$

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