Psi(x)= Asin(2πx), for what value of A is the wave function normalised
1=∫ΨΨ∗dx=A2∫01sin2(2πx)dx=A22∫01(1−cos(4πx)dx=1=\int \Psi \Psi^* dx=A^2\int _0^1sin^2(2\pi x)dx=\frac{A^2}{2}\int_0^1(1-cos(4\pi x)dx=1=∫ΨΨ∗dx=A2∫01sin2(2πx)dx=2A2∫01(1−cos(4πx)dx=
=A22(x−14πsin(4πx))∣x=0x=1=A22, ⟹ =\frac{A^2}{2}(x-\frac{1}{4\pi}sin(4\pi x))\vert_{x=0}^{x=1}=\frac{A^2}{2},\implies=2A2(x−4π1sin(4πx))∣x=0x=1=2A2,⟹
A=2,A=\sqrt2,A=2,
Ψ=2sin(2πx)\Psi=\sqrt2 sin(2\pi x)Ψ=2sin(2πx)
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