Could someone explain to me how the author goes from 2nd to 3rd step
I think the intermediate step between 2 and 3 is basically to split up the commutator as
[y p_z, z p_x] - [y p_z,x p_z] - [z p_y,z p_x] + [z p_y, x p_z]
2nd term = 0
3rd term = 0
so leftover is
[L_x, L_y] = [y p_z, z p_x]...
for harmonic oscillator, V(x) = 1/2*m*w^2*x^2. here, the spring can be stretch or compress.
however, is if the spring can only stretch such that V(x) is infinity for x<0, then find energy level for this setup.
I don't understand the part about spring only being able to stretch. what does that...
you just need to read eigenvalue problem. that seems like where you are having trouble.
in Hψ = Eψ
ψ do NOT cancel out. if you like, you can write it in a different notation
H|ψ> = E|ψ>
where H should actually be written as "H hat" to make a distinction that it is an operator.
so...
Hψ = Eψ
is an eigenvalue problem
you can read about it here
http://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors
so,
H is the eigenvector and E is it's corresponding eigenvalue. eigenvalues are constants. for each eigenvalue you can find a corresponding eigenvector.
again, think of it...
oh i see. mathematically, i suppose e^(-i) part always goes away so they have to be equal.
in terms of physics, could you give a quick reasoning for why this is true? why is the probability of finding a particle in interval [a,b] the same as the probability of finding the particle at time t=0...
I'm asked to figure out how the so-called "coordinate space matrix elements" relate to "momentum space matrix elements <p|H|p'> but I don't understand what they are.
any idea on how <x|H|x'> is defined?
thanks in advance.
okay so I'm having a bit hard time understanding this:
i get that probability of finding a particle in between [a,b] is integral (over a,b) (Ψ(x,t)*)Ψ(x,t) dx.
however, can it also be integral (over a,b) of (Ψ(x,0)*)Ψ(x,0) dx?
if not, why?
i saw an example where Ψ(x,0) was given and problem...
I'm missing something obvious so please point out what I'm thinking wrong
SE equation is:
ih d/dt |> = H|>
the taking adjoint turns i -> -i and (d/dt) -> -(d/dt)
so adjoint of SE should be same as SE
however it isn't. adjoint of SE is
-ih d/dt |> = H|>
do we not take adjoint of d/dt, if...
i've tried several methods (that i can think of) for this problem and none of them seem to work.
also tried expanding as series and didn't get anywhere.
nvm what i said before. the only singularity inside our contour is z = 0. rest are outside and don't even matter.
and yes
integral = 2*pi*i * [sum of residues at each singularities]