Help in Normalization of a Wave Function

[Vaibhav DixiT]

Hello Guys,

I am trying to Normalize the following wave function but I am getting stuck in between (Maybe maths is the problem here for me). Can anyone please provide some hints.

The Wave Function is


ψ = e ^{- |x|} sin (α x)Please help.

 

[blue_leaf77]

You need to show which part you get stuck in.

 

[Vaibhav DixiT]

Since ψ² here is an even function. I am trying the following relation:_-∞^∞∫|ψ|²dx = 2 ₀^∞∫|ψ|²dx

for x ∈ [0,∞) , ψ = e^-xsin( ∝x)

|ψ|² = e^-2xsin²(∝x) = e^-2x {1- cos(2∝x)}/2

So essentially I am trying to find ∝, so that:

₀^∞∫e^-2x {1- cos(2∝x)}dx = 1

I am trying to do integration by parts but the e^-2x cos(2∝x) is being a problem for me.

 

[blue_leaf77]

That kind of integral will require the integration by parts step twice. I think it is better to direct you to the related link rather than rewriting everything here: http://www.analyzemath.com/calculus/Integrals/exp_sine_cosine.html

 

[Vaibhav DixiT]

thanks a lot!

Exactly what I needed.