Use definitions (Ax)² = (n|x²\n) - (n|x|n))² (Ap)2 = (n\p²|n) - (n\p|n))? ((n|p|n))? To calculate ApAx from state |n)
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- Consider a system of N non - interacting particles. Each particle is fixed in position and can sit in two possible states which, for convenience, we will cal "spin up" |t) and "spin down" |↓). We take the energy of these states to be E=0, E_1 = \epsi the system has N_↑ particles with spin up and N_↓ = N - N_t particles with spin down. Find the final state of the system if E_< < E_↑The Lennard-Jones potential, (E = 48[-(0/r)6+ (o/r)¹2]), is a good approximation that describes realistic potential energy of 2 atoms, where o is collision distance and ris the distance between two atoms. Explain the physical meaning when (1) r = o and (ii) ro=1.1220.Show that at limit ħo kgT, the following expression, h €(w, T) = c exp(Bho)-1 reduces to the classical form given by: e(w, T)= Ac ³(kgT)w²: ² = 0 [₁²³ (²+²)]. Ac
- 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?iii) Consider a 2D square potential energy well with sides L (length) containing six electrons. The potential energy is infinite at the sides and zero inside. The h? single-particle energies are given by 8mL +n), where n and ny are integers. If a seventh electron is added to the system when it is in its ground state find the least energy the additional electron can have?