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Natural Science Forum / Physics / Particle Physics / September 2005Expectation value of time dependent solutions
My quantum mechanics book states that for a free particle modelled by a top hat wave packet: <x> = Integral of (from -inf to +inf) | Psi(x,0) | ^2 x dx where Psi (x,0) represents the state of the particle at t=0 Earlier on in the text it states the more general: <x> = Integral of (from -inf to +inf) | Psi(x,t) | ^2 x dx Solutions to the Schroedinger equation are all of the form Psi(x,t) = f(x) g(t) I'm unclear as to how the time component can be ignored inside the integral. I can understand that maybe it equals zero as the integral is evaluated. Maybe they just missed out all the steps. Any thoughts welcome.  Signaturezincnews123 at tiscali.c123o.u123k To reply to address don't click. Cut and paste, change at to symbol then delete all 123's ------------------------------------ Dear Zinc, > Solutions to the Schroedinger equation are all of the form > Psi(x,t) = f(x) g(t) Maybe all the solutions you encountered were of this form. However, in general this does not need to be the case. > My quantum mechanics book states that for a free particle modelled by > a top hat wave packet: [quoted text clipped - 6 lines] > > <x> = Integral of (from -inf to +inf) | Psi(x,t) | ^2 x dx > I'm unclear as to how the time component can be ignored inside the integral. It can't be ignored, but maybe it is stated somewhere in the text that in the first case the expectation value of the position at time t=0 was the thing they wanted to obtain. Best wishes, Chris > Solutions to the Schroedinger equation are all of the form > Psi(x,t) = f(x) g(t) Almost never. If you plug a solution of this form into the Schrodinger equation, you will find that the time dependence is necessarily of the form e^{iwt}. That means you have a stationary state, one where probabilities (which are determined by |Psi|^2) are independent of t. This is an exceptional situation; it's certainly not true in general. (If you *do* have a solution of this form, then the time dependence in <x> trivially drops out.) Steve Carlip |
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