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## Homework Statement

Use the variation method to find a approximately value on the ground state energy at the one dimensional harmonic oscillator, H = -ħ^2/(2m) * d^2/dx^2 + 1/2mω^2*x^2

## Homework Equations

H = -ħ^2/(2m) * d^2/dx^2 + 1/2mω^2*x^2

u(x) = Nexp(-ax^2)

<H> = <u|Hu>

## The Attempt at a Solution

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Normilized N is N = (2a/π)^(1/4)

If i start to calculate Hu = N(-ħ^2/(2m)(4a^2*x^2 - 2a) + 1/2*m*ω^2*x^2)exp(-ax^2)

Then

<H> = ∫u

^{*}Hu dx = (2a/π)^(1/2)∫[0 to inf](-ħ^2/(2m)(4a^2*x^2 - 2a) + 1/2*m*ω^2*x^2)exp(-ax^2) dx

if i divide the integral into 3 different integrals i got the integral to be equal to -3ħ^2*a/(4m) + mω/(16*√a)) which is incorrect. It should be equal to ħ^2/(2m)*a + 1/8*m*ω^2*1/a why ?