We have derived four coupled first-order differential equations describing stellar structure. Let us rewrite them here: dP(r) GM(r)p(r) (3.56) dr r2 dM(r) = 4rr²p(r), dr (3.57) dT(r) 3L(r)« (r)p(r) (3.58) dr 4Jt p24acT(r)³’ dL(r) (3.59) 4Jt r? p(r)e(r). dr We can define four boundary conditions for these equations, for example, (3.60) M(r = 0) = 0, %3D %3D (3.61) L(r = 0) = 0, %3D (3.62) P(r = r.) = 0, (3.63) M(r = r.) = M,, %3D

ext cenb. Consider a system whose states are given in term of complete and orthonormal set of kets |1>, |2 >, [3 >,14 > as follows: 1 1p >= |1 > + 기2 > +2|3 > + 기4 > i 214> Where the kets |n > are eigenstates of an observable A defined on the system as follows: 2 A]n > = na|n > with n = 1,2,3,4 and with a a constant number. have 4) eiyen vealue 1. If A is measured, which values will be found and with which probabilities? 2. Find the expectation value of A for the state |Ø >. 3. Assume that the state 14> is found after the measurement of A. If A is measured again immediately, which states will be found and with which probabilities? 4. Find the expectation value of A if the system is in the state |4 >. 5. Assume B another observable defined on the system, which is compatible with A. Write the uncertainty inequality between A and B. 6. If B is measured, which states will be found and with which probabilities?

What does population vector, Π=(P1,P2,P3r,P3w)T mean ? How do this formula describe the overall state probability? (there are state 1, state 2 and state 3w, 3r)

(a) Prove the "vertical angle hypothesis" (I. 15): opposing angles are congruent if two lines cut each other. (Hint: You'll need to use postulate 4 about right angles in this case.) ) (b) Complete the proof of the Exterior Angle Theorem using section (a): illustrate why beta < alpha