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i'm trying to see how does an HO, traveling with constant speed v looks like. suppose a unitless system

[tex]H = P^2+(X-vt)^2[/tex]

define

[tex]Y = X-vt[/tex]

then

[tex]H = P^2+Y^2[/tex]

i can see that [P,Y] = -i (unitless - no h-bar) so i guess it means that P and Y are conjugate space/momentum operators. therefore the solution for this is, using ehrenfest theorem

[tex]<Y> = Y(0)cos(t)+P(0)sin(t)[/tex]

[tex] <P> = P(0)cos(t)-Y(0)sin(t)[/tex]

where Y(0) is the expectation value of Y at t = 0 and the same for P(0).

now, going back to X, assuming the state is a square-integrable one

[tex]Y(0) = X(0) - v*0 = X(0)[/tex]

[tex]<Y> = <X-vt> = <X> - vt<state|state> = <X>-vt[/tex]

[tex]<X> = X(0)cos(t)+P(0)sin(t)+vt[/tex]

[tex] <P> = P(0)cos(t)-X(0)sin(t)[/tex]

now this makes some sense, <X> really oscillates around a value increasing with rate v, but P seems unchanged. I'd expect P to have a constant part as well, with size v since that is the constant velocity.

i've written the ehrenfest theorem equations for the original P,X and i've noticed that if I set P_new = P-v is solves those equations. so where did I get it wrong?

thanks a lot

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# Traveling harminc oscillator

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