"Exercise 1. Verify spin operators and expectation values." zp = (1,0) zm = (0,1) xp = (zp + zm) / sqrt(2) xm = (zp - zm) / sqrt(2) yp = (zp + i zm) / sqrt(2) ym = (zp - i zm) / sqrt(2) Sx = hbar / 2 ((0,1),(1,0)) Sy = hbar / 2 ((0,-i),(i,0)) Sz = hbar / 2 ((1,0),(0,-1)) check(Sx == hbar / 2 (outer(xp,conj(xp)) - outer(xm,conj(xm)))) check(Sy == hbar / 2 (outer(yp,conj(yp)) - outer(ym,conj(ym)))) check(Sz == hbar / 2 (outer(zp,conj(zp)) - outer(zm,conj(zm)))) S2 = dot(Sx,Sx) + dot(Sy,Sy) + dot(Sz,Sz) check(S2 == 3/4 hbar^2 ((1,0),(0,1))) cp = Xp + i Yp cm = Xm + i Ym s = (cp,cm) check(dot(conj(s),Sx,s) == 1/2 hbar (cp conj(cm) + conj(cp) cm)) check(dot(conj(s),Sy,s) == 1/2 i hbar (cp conj(cm) - conj(cp) cm)) check(dot(conj(s),Sz,s) == 1/2 hbar (cp conj(cp) - cm conj(cm))) check(dot(conj(s),S2,s) == 3/4 hbar^2 (cp conj(cp) + cm conj(cm))) "ok" "Exercise 2. Verify expected spin vector." clear s = (1/3 - 2/3 i, 2/3) check(dot(conj(s),s) == 1) Sx = hbar / 2 ((0,1),(1,0)) Sy = hbar / 2 ((0,-i),(i,0)) Sz = hbar / 2 ((1,0),(0,-1)) S = (Sx,Sy,Sz) check(dot(conj(s),transpose(S),s) == hbar / 2 (4/9, 8/9, 1/9)) "ok" "Exercise 3. Verify spin measurement probabilities." clear s = (1/3 - 2/3 i, 2/3) zp = (1,0) zm = (0,1) xp = (zp + zm) / sqrt(2) xm = (zp - zm) / sqrt(2) yp = (zp + i zm) / sqrt(2) ym = (zp - i zm) / sqrt(2) Pr(a,b) = dot(conj(a),b) dot(conj(b),a) check(Pr(xp,s) == 13/18) check(Pr(xm,s) == 5/18) check(Pr(yp,s) == 17/18) check(Pr(ym,s) == 1/18) check(Pr(zp,s) == 5/9) check(Pr(zm,s) == 4/9) "ok" "Exercise 4. Verify indistinguishable spin states." clear s = (1/3 - 2/3 i, 2/3) Sx = hbar / 2 ((0,1),(1,0)) Sy = hbar / 2 ((0,-i),(i,0)) Sz = hbar / 2 ((1,0),(0,-1)) S = (Sx,Sy,Sz) x = 2 / hbar dot(conj(s),Sx,s) y = 2 / hbar dot(conj(s),Sy,s) z = 2 / hbar dot(conj(s),Sz,s) cp = sqrt((z + 1) / 2) cm = sqrt((1 - z) / 2) (x + i y) / sqrt(x^2 + y^2) check(cp == sqrt(5) / 3) check(cm == (2 + 4 i) / (3 sqrt(5))) chi = (cp,cm) check(dot(conj(s),transpose(S),s) == dot(conj(chi),transpose(S),chi)) "ok" "Exercise 5. Verify spin commutators." clear Sx(psi) = -i hbar (y d(psi,z) - z d(psi,y)) Sy(psi) = -i hbar (z d(psi,x) - x d(psi,z)) Sz(psi) = -i hbar (x d(psi,y) - y d(psi,x)) psi = Psi() check(Sy(Sz(psi)) - Sz(Sy(psi)) == i hbar Sx(psi)) check(Sz(Sx(psi)) - Sx(Sz(psi)) == i hbar Sy(psi)) check(Sx(Sy(psi)) - Sy(Sx(psi)) == i hbar Sz(psi)) S2(psi) = Sx(Sx(psi)) + Sy(Sy(psi)) + Sz(Sz(psi)) check(S2(Sx(psi)) - Sx(S2(psi)) == 0) check(S2(Sy(psi)) - Sy(S2(psi)) == 0) check(S2(Sz(psi)) - Sz(S2(psi)) == 0) Sp(psi) = Sx(psi) + i Sy(psi) Sm(psi) = Sx(psi) - i Sy(psi) check(Sp(Sm(psi)) - Sm(Sp(psi)) == 2 hbar Sz(psi)) "ok" Run