Consider ﬁrst a free particle (Potential energy zero everywhere). When the particle at
a given time is prepared in a state ψ (x) it has
<x> = 0 and <p> = 0.
The particle is now prepared in Ψ (x, t = 0) = ψ (x) exp (ikx)
Give <p> at time t = 0.
It can be shown that in quantum mechanics <p> is independent of time for a free
particle, in analogy to Newton’s ﬁrst law in classical mechanics. Give <x> as a
function of time.
<p> = [itex]\int\Psi*(-i[/itex][itex]\hbar[/itex]d/dx)[itex]\Psi[/itex]dx (Should be a partial derivative
I just don't know how to write that, also integral is from -infinity to +infinity, same for all
I write further down too)
The Attempt at a Solution
The attempt at a solution:
Basically substituting straight into the equation I gave above:
<p> = ∫ψ(x)exp(-ikx).-i[itex]\hbar[/itex].d/dx[ψ(x)exp(ikx)]dx
I guessed that I have to use the product rule for the second half of that so I get:
<p> = ∫ψ(x)exp(-ikx).-i[itex]\hbar[/itex].dψ(x)/dx.exp(ikx)]dx+∫ψ(x)exp(-ikx).-i[itex]\hbar[/itex].ψ(x)ik.exp(ikx)]dx
This is where my complete lack of experience shows if not already. Simplifying the first integral, the exponents add to zero for the exp() terms so they cancel. Also, dψ/dx.dx can be simplified to dψ. This leaves the term in the fist integral as -i[itex]\hbar[/itex]ψ.dψ.
Simplifying the term in the second integral I use the same exponent rule to cancel those and also --i.i = 1. That leaves, in the second integral, the term [itex]\hbar[/itex]kψ2dx.
I've been doing educated guessing up to here but now I really start guessing.
<p> = -i[itex]\hbar[/itex]∫ψ(x)dx+[itex]\hbar[/itex]k∫ψ2(x)dx
And by some math voodoo and because I saw somewhere that the integral of psi2dx = 1, I get zero for the first integral and <p> = [itex]\hbar[/itex]k. I suspect I am wrong at many different points for so many different reasons. I hope this doesn't hurt too many people reading it.
As for the second part, finding <x> as a function of t, I have no idea where to start so even a pointer or two would be appreciated for that. If anyone could point out the numerous mistakes that are no doubt in my attempt at the first part I would also be very grateful. Thanks!