Quantum mechanic:Please compare the tasks with each other. I have asked the first task here and I have also received great solutions from you, but all without integral calculus. Now I have a very similar task here including solutions (and the solution path is quite different or with integral). I would like to see how to solve the task with this solution path from task 2 with A[...].

I.) A particle in a one-dimensional harmonic oscillator potential has as its initial state a superposition of energy eigenstates, namely the normalised ground state and the normalised second excited state a) Normalise up (x,0) b) Determine u (x.t) and |4 (x,t)|2 c) Calculate (x) and (p). d) Determine the expectation value of H. V(x, 0) = N [wo(I) + 1/2(I)] V (x, 0). (x, t) and |7 (z, t) |2 (I) ( P). H. V(x,0) = A[vo(z) + 1(z)] . (1) a) Normiere V(z, 0). Losung: Un (x) sind orthonormierte Energieeigenfunktionen (reell). Normierung: (2) da (3) Wahle nun A = 1/V2. Losung: . Energieeigenfunktionen Un(x) durch un(x)e Ent/h ersetzen. V(x,t) = vo(x)e-Both + vi(z)e-'Bulk] 1(z,t)? = V(x,t)'v(x,t) (5) (6) = (7) = [ (x) + vi(x) + 206(x)1(z) cos ((Eo - Ei)t/h)] (8) = 6(2) + vi(x) + 2vo(x)(z) cos(wt)] , mit Eo - E1 = -hw . (9) c) Berechne und

. Losung' Wir drucken z und p durch dic Auf- und Absteigcoperatoren a, und a- aus. Mit (mur # ip) (10) I = 1 4 (a + +a_ ) , P=iV 2 much (a, - a_ ) (11) Fur a, und a- gilt