# To find the expectation value of momentum

1. Apr 13, 2012

### johnny1

1. The problem statement, all variables and given/known data

At time t=0 a particle is described by a one dimensional wavefunction
(capital)ψ(x,0)= (2a/)^(1/4) e^(-ikx)e^(-ax^2)
(three lines)=(2a/)^(1/4) e^(-ikx-ax^2)--------equation 1
k and a are real positive constants

2. Relevant equations

I think this is the one
<p subscript(x)> = ∫(from-inf to inf)[(capitalψ)*(x,t)(-i(hbar)(∂/∂x))capitalψ(x,t)]dx
for normalization ∫from-inf to inf e^-x^2 dx =√π
and ∫from-inf to inf e^-ibx dx=√π *e^(-b^2/4) where b is a real constant

3. The attempt at a solution

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I verified that equation 1 is normalized
by completing the square and letting t=u/a I then integrate upon which I arrive at √(π/a) *e^(k^2/4a)

I know that *e(k^2/4a) is a phase factor which is irrelevant to the equation
so
integration of e^(-ikx-ax^2)=√(π/a)

since ∫-inf to inf IΨI^2dx=(Ψ*)(Ψ)=Ae(-ax^2)*Ae(-ax^2)=(A^2)e(-2ax^2)
and if e(-2ax^2)=√(π/2a)
then A^2=√(π/2a)
hence A=(2a/π)^(1/4)

Was it necessary to normalize
before I begin in attempting to answer the question

attempting to find<p subscript(x)>
<psubscriptx> =∫(Ψ*)(-ihbar(∂/∂x))(Ψ)=(e(-ax^2))(-ihbar((1/2)e^(-4ax^2))(e(-ax^2))
and here is where I am really worried
I dont know if for <psubscriptx> I am plugging in correctly

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1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

Last edited: Apr 13, 2012
2. Apr 13, 2012

### fzero

That is a real quantity, so it's not just a phase factor.
It looks like it.

You didn't compute the derivative in the last expression.