A particle of mass m moves along the x-direction such that V(x)=½Kx^2. Is the state u(¥)=B¥exp(+¥2/2), where ¥ is Hx (H = constant), an energy eigenstate of the system?. What is probability per unit length for measuring the particle at position x=0 at t=t0>0?
find the following:
if a particle (s=1/2) is prepared such that it is in the spin up state |f>=|z+>
what do the following mean? [<f|(Sz - <z+|Sz|f>1)^2|f>]^½ and
[<f|(Sx - <f|Sx|f>1)^2|f>]^½
The middle term sandwiched between the states is squared and the whole term being square...
i don't see how a minus sign can be missing as don't you have to take the complex conjugate of the eigenvalue for the time evolution operator in order to find the matrix element? so shouldn't it be fine after the edits
also i kind of got lost in what this all means now, so what exactly does...
Well starting with the integrals
i subsituted for <x|p> = (1/sqrt(2pih))e^(ipx/h) <p'|x'> = (1/sqrt(2pih))e^(-ip'x'/h)
<p|U(t,to)|p'> = e^(itp^2/2m)<p|p'>
therefore when you plug everything in p = p' because of delta dirac function so one of the ingrals cancels and you are left with 3...
Ah i see so for the matrix element using momentum basis and momentum eigenstate i get:<p|u|p'> = U*<p|p'>U is just the eigenvalue found earlier which is just the exponential in terms of momentum eigenvalue
therefore when you find the matrix element using position, you have to use compltness...
Ah i see, so the point was to show that you can express the hamiltonian in terms of the momentum eigenvalue (by showing that the hamiltonian operator can be expressed in terms of momentum operator) so that you can find the time evolution operator (essentially due to the hamiltonian) using a...
Oh I see so its similar to my orginal plan which was to use the power series expansion
e^x = sum(1 + x...) (i'll only use first 2 terms)
therefore you should get:
1 - iHt/h for your eigenvalue
which is what i had previously and doesn't seem to make much sense
hat{H}| p>
so would that just be the same as the case with the momentum operator but now its (h^2)/2m multiplied by the momentum eigenstate squared? so the question is asking to find the eigenvalue associated with the momentum eigenstate when you use the time evolution operator on the...
Maybe iam just struggling with dirac notation, but wouldn't the first just give you an eigenstate proportional to the momentum operator by a scaler, eigenvalue. Is it the eigenvalue you are interested in or the eigenstate |p>.
\hat{H} = (ih/2m)(d/dx) hat{p}
Are you saying that since the hamiltonian generates the time evolution of eigenstates, we can express the time evolution operator in terms of eigenstates |p> using the hamiltonian?
\hat{H} | p\rangle = {E}| p\rangle
so you would just have to find the constant eigenvalue and since the potential is zero wouldn't you just end up with a second order ODE and solve for the eigenfunction (momentum). But iam not sure how that would apply to the time operator in terms...