Recent content by cdot

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    Element of surface area in spherical coordinates

    My question is actually about how the expression for dS given in the problem statement (##dS=2πrsinθ\sqrt{dr^2+r^2dθ^2}##) was arrived at in the first place, given the equations stated above (which was all that we were assumed to know as of attempting this problem). The only example problems up...
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    Element of surface area in spherical coordinates

    I don't see how that helps me though
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    Element of surface area in spherical coordinates

    Doesn't it depend on the surface? If you're integrating over x-y plane its just dxdy
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    Element of surface area in spherical coordinates

    Just realized I forgot the ##d\theta## in the area element on a sphere ##(dS=r^2sin\theta d\phi d\theta) ##
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    Element of surface area in spherical coordinates

    r,θ,ϕ For integration over the ##x y plane## the area element in polar coordinates is obviously ##r d \phi dr ## I can also easily see ,geometrically, how an area element on a sphere is ##r^2 sin\theta d\phi ## And I can verify these two cases with the Jacobian matrix. So that's where I'm at...
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    Expanding the original commutator on the LHS

    Ahhh that makes sense now. For practice with proof by induction as you explained it, does this work? If ##[X,P] = i\hbar## show that ##[X,P^n] = nP^{n-1}i\hbar## Let ##C(n) = [X,P^n]## ##~~~~~~C(1) = [X,P] = i\hbar## ## \text{If } C(k) = kP^{k-1}i\hbar \text{ is true,} ## ##...
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    Expanding the original commutator on the LHS

    Thanks for making me feel stupid by asking if I was serious.
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    Expanding the original commutator on the LHS

    Does this constitute a legitimate proof by induction?: $$Xe^{iaP} = e^{iaP}(X-\hbar a)$$ $$X^ne^{iaP} = X^{n-1}Xe^{iap} = X^{n-1}e^{iap}(X-\hbar a) = X^{n-2}Xe^{iap}(X-\hbar a) = X^{n-2}e^{iap}(X-\hbar a)^2$$ $$= \dots$$ $$=X^{0}e^{iap}(X-\hbar a)^{n} = e^{iap}(X-\hbar a)^n$$
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    Expanding the original commutator on the LHS

    $$\left(X - \hbar a \right)^n = e^{-niaP} X^n e^{niap}.$$
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    Expanding the original commutator on the LHS

    Homework Statement Using [x,eiap]=-ħaeiap show that xneiap = eiap(x-ħa)n Homework Equations [x,eiap]=-ħaeiap From which it follows that, xeiap = eiap(x-ħa) The Attempt at a Solution [xn,eiap] = [xxn-1,eiap] = [x,eiap]xn-1 + x[xn-1,eiap]...
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    Undergrad Can [A,B^n] always equal 0 if [A,B] equals 0?

    Nevermind got it! Thanks for your help
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    Undergrad Can [A,B^n] always equal 0 if [A,B] equals 0?

    PeroK that makes a lot of sense. What Cryo did above also makes perfect sense. That's what I originally did, I just wasn't sure that constituted an actual proof. Also, how do you get your superscripts/subscripts to look nice on here. It doesn't seem to be working for me? Ex. x^2
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    Undergrad Can [A,B^n] always equal 0 if [A,B] equals 0?

    So I realize that [A,B] = 0 means AB = BA and I actually just walked myself through a case of proof by induction to show that if two linear operators A and B both commute with their commutator ( I understand what this means ) then [A,B^n]=nB^n-1[A,B]. I don't have that much experience with...
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    Undergrad Can [A,B^n] always equal 0 if [A,B] equals 0?

    This is not a homework problem. It was stated in a textbook as trivial but I cannot prove it myself in general. If [A,B]=0 then [A,B^n] = 0 where n is a positive integer. This seems rather intuitive and I can easily see it to be true when I plug in n=2, n=3, n=4, etc. However, I cannot prove it...
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    General commutation relations for quantum operators

    (This is not a homework problem). I'm an undergrad physics student taking my second course in quantum. My question is about operator methods. Most of the proofs for different commutation relations for qm operators involve referring to specific forms of the operators given some basis. For...