- #1
Milsomonk
- 101
- 17
Homework Statement
The expectation value of <P^2>= -ħ∫ψ* ∂^2ψ/∂x^2 dx
For the Guassian wave-packet ψ(x)=(1/(π^1/4)(√d))e^-((x^2)/(2d^2))
Limits on all integrals are ∞ to -∞.
Homework Equations
<P^2>= -ħ∫ψ* ∂^2ψ/∂x^2 dx
ψ(x)=(1/(π^1/4)(√d))e^(iKx)-((x^2)/(2d^2))
The Attempt at a Solution
Ok, the previous question was to calculate <P> given <P>=-iħ∫ψ* ∂ψ/∂x dx. I did this by calculating the partial derivative of ψ and then taking the integral of ψ* ∂ψ/∂x, then multiplying by -iħ, I got the result <P>=ħK which seemed to make sense. So I tried to take the same method to this question;
Firstly working out the second derivative of ψ, since I already know the first derivative from the first question, I simply differentiated again and found it to be the following
∂^2ψ/∂x^2 = (∂ψ/∂x)(iK-(x/d^2))-(1/d^2)ψ Used Product rule, Plugged ∂ψ/∂x and ψ back in for simplicity.
Next I simplified the Integrand by factoring out ψ.
Giving the integral -iħ∫((x^2)/(d^4))-((2iKx)/(d^2))-(K^2)-(1/(d^2))ψ*ψ dx
I from the previous question that ψ*ψ is (1/((√π)(d)))e^-((x^2)/(d^2)).
So I split this into the sum of 4 separate integrals, two had x values multipied by e^-x values, I concluded that when plugging in infinited these would each go to zero.
so left with two integrals,
<P^2>=(-ħ/((√π)(d)))(-K^2∫e^-((x^2)/(d^2))-1/(d^2) ∫e^-((x^2)/(d^2))
It is given that ∫e^-((x^2)/(d^2)) between ∞ and -∞ is (√π)(d)
So I simplify this all and get <P^2> = ħ(K^2)+(ħ/d)
I have been through my calculations a number of times and can't find and error but I can't help but think that the ħ/d term doesn't really make sense. Any thought's would be much appreciated :)