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3uc1id
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[SOLVED] expectation value of P^2 for particle in 2d box
I am having difficulty in finding the right way to find this value. my book only give the 1d momentum operator as: ih(bar)*d/dx(partials). i see its much like finding the normalization constant. which i have done using a double integral. the problem has as a given soln psi(x,y)= Asin(k1x)sin(k2y). the dimensions are y ranges from 0 to l/2, and x ranges from 0 to l. i have found the normalization constant to be (2*sqrt2)/2 by doing a dble integral of psi*psi. I have also found the allowable energy lvls :E=((pi^2*h(bar)^2)/ml^2)*((n1^2/2)+2n2^2) where n1=n2 and are any whoile number for the quantum numbers. the main problem i am having is how to define the p operator for a 2-d box. can i use (-ih(bar)*(d/dx +d/dy))^2, and then evaluate it over a dble integral. but then i would have strange cross terms. i think i have to use the E value i have found but I am not sure. any help would be much appreciated.
I am having difficulty in finding the right way to find this value. my book only give the 1d momentum operator as: ih(bar)*d/dx(partials). i see its much like finding the normalization constant. which i have done using a double integral. the problem has as a given soln psi(x,y)= Asin(k1x)sin(k2y). the dimensions are y ranges from 0 to l/2, and x ranges from 0 to l. i have found the normalization constant to be (2*sqrt2)/2 by doing a dble integral of psi*psi. I have also found the allowable energy lvls :E=((pi^2*h(bar)^2)/ml^2)*((n1^2/2)+2n2^2) where n1=n2 and are any whoile number for the quantum numbers. the main problem i am having is how to define the p operator for a 2-d box. can i use (-ih(bar)*(d/dx +d/dy))^2, and then evaluate it over a dble integral. but then i would have strange cross terms. i think i have to use the E value i have found but I am not sure. any help would be much appreciated.