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I am trying to normalise the wave function $\psi$ for a free particle, with initial boundary conditions.

$$\Psi(x,0)=Ae^{-2|x|}.$$

When trying to normalise it, I keep getting $\infty$ which clearly isn't right. I am using this equation to normalise it but I think I am missing some form of understanding of how free particle normalisation works which is why I keep getting an incorrect value.

$\int_{-\infty }^{\infty }\left|\Psi (x)^2 \right|dx=1$

Maybe I should change the boundaries, in lectures we are using between $0>x>a$, or between $-a/2>x>a/2$?

I am not sure how to normalise the wave function but I am sure the solution is pretty simple.

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1 Answer 1

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Since there is $|x|$ in the exponent, we split the integral into two parts via $$\int _{-\infty}^{\infty}dx |\psi(x,0)|^2=\int_{-\infty}^0 dx A^2e^{2x}+\int_0^\infty dx A^2 e^{-2x}$$ These finite-valued integrals then give a finite value for $A$ to normalise the wavefunction.

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