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Question-139002




Question Number 139002 by BHOOPENDRA last updated on 21/Apr/21
Commented by Dwaipayan Shikari last updated on 22/Apr/21
Probability  ∫_(−(π/2)) ^(π/2) ∣Ψ(x)∣^2 dx=1  =∫_(−(π/2)) ^(π/2) A^2 cos^4 xdx=A^2 Γ((5/2))Γ((1/2))=A^2 ((3π)/4)  ⇒1=A^2 ((3π)/4)⇒A=±(2/( (√(3π))))
$${Probability}\:\:\int_{−\frac{\pi}{\mathrm{2}}} ^{\frac{\pi}{\mathrm{2}}} \mid\Psi\left({x}\right)\mid^{\mathrm{2}} {dx}=\mathrm{1} \\ $$$$=\int_{−\frac{\pi}{\mathrm{2}}} ^{\frac{\pi}{\mathrm{2}}} {A}^{\mathrm{2}} {cos}^{\mathrm{4}} {xdx}={A}^{\mathrm{2}} \Gamma\left(\frac{\mathrm{5}}{\mathrm{2}}\right)\Gamma\left(\frac{\mathrm{1}}{\mathrm{2}}\right)={A}^{\mathrm{2}} \frac{\mathrm{3}\pi}{\mathrm{4}} \\ $$$$\Rightarrow\mathrm{1}={A}^{\mathrm{2}} \frac{\mathrm{3}\pi}{\mathrm{4}}\Rightarrow{A}=\pm\frac{\mathrm{2}}{\:\sqrt{\mathrm{3}\pi}} \\ $$
Commented by Dwaipayan Shikari last updated on 22/Apr/21
∫_(−(π/2)) ^(π/2) A^2 cos^4 x dx=1  =∫_0 ^(π/2) A^2 cos^4 x dx=(1/2) ⇒∫_0 ^(π/4) A^2 cos^4 xdx=(1/4)  or 25%  Probability=(1/4)
$$\int_{−\frac{\pi}{\mathrm{2}}} ^{\frac{\pi}{\mathrm{2}}} {A}^{\mathrm{2}} {cos}^{\mathrm{4}} {x}\:{dx}=\mathrm{1} \\ $$$$=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {A}^{\mathrm{2}} {cos}^{\mathrm{4}} {x}\:{dx}=\frac{\mathrm{1}}{\mathrm{2}}\:\Rightarrow\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} {A}^{\mathrm{2}} {cos}^{\mathrm{4}} {xdx}=\frac{\mathrm{1}}{\mathrm{4}}\:\:{or}\:\mathrm{25\%} \\ $$$${Probability}=\frac{\mathrm{1}}{\mathrm{4}} \\ $$

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