Quantum Mechanics: Wheeler: Physics 6210

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1 Quantum Mehanis: Wheeler: Physis 60 Problems some modified from Sakurai, hapter. W. S..: The Pauli matries, σ i, are a triple of matries, σ, σ i = σ, σ, σ 3 given by σ = σ = σ 3 = i i Let stand for the identity matrix, and onsider the matrix of all omplex linear ombinations X = a 0 + a σ where a 0 and a, a, a 3 are omplex numbers.. Express these numbers in terms of the four traes, tr X and tr σ k X.. Show that X is Hermitian if and only if a 0 and a are real. 3. Show that X is traeless if and only if a 0 = 0. S..3: This problem is not too hard if we first review some fats about determinants and unitary matries. The determinant of a produt is the produt of the determinants, det AB = det A det B Now suppose B = A, so we have AA =. Then det AA = det det A det A = det A = det A Looking at the problem, we see that the result follows if we an show that the matries exp iϕ n σ and exp iϕ n σ are inverse to one another. This follows from a simple theorem relating unitary and Hermitian matries. Let a matrix U be written as the exponential of i times another matrix, H, so that U = e ih

2 Then U is unitary if and only if H is Hermitian. To prove this, note that unitarity requires U = U and it is not hard to show that U = e ih. Indeed, inserting a parameter λ, d e iλh e iλh = ihe iλh e iλh e iλh ihe iλh dλ and sine H ommutes with any funtion of itself, this vanishes. Therefore, e iλh e iλh is onstant. Setting λ = 0, shows that the onstant must be, and U = e ih. Therefore, unitarity implies U = U e ih = e ih The same trik, writing this as e iλh respet to λ shows that = e iλh and differentiating with ih e iλh = ihe iλh and using the original equality with λ = to remove the exponentials shows that H = H. Conversely, if H = H, the result is immediate. With these results at hand, the problem is straightforward. S..4: Pratie with Dira notation. S..6: Don t just show that it works give a derivation. Set up the eigenket ondition and dedue the onditions under whih it holds. S..8: These are important relationships, worth heking, and good pratie with the Dira notation. After you are done, find the matrix representations of S x, S y, and S z. They should look familiar. S..0: After you solved the problem, repeat it in the usual matrix notation. S..: For this problem, it s not hard to find the eigenvalues and eigenkets and the ondition that has to hold when H = 0, using standard matrix tehniques. It is helpful to reognize that the Hamiltonian is just the matrix Ĥ = H + H + H + Ĥ = H H H H Then it is permissible to work with the matrix form. It s not neessary, however you may just write the eigenket as a general linear ombination, E = α + β and substitute both Ĥ and E into the eigenvalue equation Ĥ E = E E

3 S..: Again, Dira notation or matries are ok. Remember that spin is a two-state system as well, so you an reast this as a spin problem. S..3: Another good physial problem. Work through step by step. S..4. Worked example problem: a three-state system. We are given the operator A = and are asked to find the eigenvalues and normalized eigenvetors. We an find the eigenvalues by solving 0 = det A λ = det λ 0 0 λ 0 λ = det 0 λ = λ λ λ = λ 3 + λ so the eigenvalues are λ = 0, ± Sine these are all different there is no degeneray. To find the eigenvetors we set v = a, b, and solve for eah value of λ. For λ =, so we have Av = λv a b b a + b = λ a b = λ b = λa = λ a + = λb a b 3

4 so either b = 0, = a when λ = 0, or a = = λ b. This gives the three vetors, b, a 0, b orresponding to, 0 and + respetively. Notie that eah of these vetors is orthogonal to the others. We use the remaining onstants, a or b, to normalize. Thus, = b b = b + + = b + + b = The last two ases are similar, so we have the three eigenvetors v = v 0 = v = 0 For part b, these ould orrespond to the spin eigenstates of a vetor partile, but there s no partiular reason to expet you to know that. S..5: The vanishing of an operator, suh as [Â, ˆB] = 0, means that it vanishes on every state. Completeness of a basis means that an arbitrary state an be expanded in the basis. S..6: The omment to S..5 is relevant here too. S..9: Calulate the expetations and substitute. When you re done with that, try to explain the values you get for part b. S..0: Just remember how to find the extrema of a funtion. Why does Sakurai ask for the maximum instead of the minimum? What is the minimum 4

5 value? You found it in the previous problem. Does maximizing the left side also maximize the right side? S..: This is a terrifi problem! Ask about it if you don t figure it out! S..3: A 3-state system. Work out all the details here. There s nothing that s not straightforward matrix manipulation, and the problem gives some lear insight into simultaneous eigenkets and degeneray. S..4: This problem foreshadows hapter three. Thinking about it now will save you effort later. a Do this by piking a state you know lies in the+z diretion: that is, the + > state. Now apply the operator and if it works it should give you a state that is definitely in the y, ± > diretion, depending on whether the rotation is lokwise or ounterlokwise. To test the operator ompletely, hek what happens to x, + > and y, + > states as well. b Just use the rotator of part a to rotate S y. Remember that to rotate an operator you have to do a similarity transformation, that is, if Û = + iσ x and you want to rotate an operator Ô, it is given by ÛÔÛ. S..6: Look at the states you re starting with and the ones you want to end up with. S..7: Good pratie with ontinuum basis hanges. S..8: This gives good insight into the anonial quantization proedure. Notie that Sakurai s lassial Poisson braket [A, B] lassial is [A, B] lassial,x,p = A B x p A p B x S..33: If this gives you trouble, go bak and review the last setion. 5