1. The problem statement, all variables and given/known data 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. 2. Relevant equations <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) 3. 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. I have: <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!