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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 u (x,0) b) Determine ( (x.t) and |4 (x.t)|2 c) Calculate (x) and (p). d) Determine the expectation value of H. V(x, 0) = N [vo(I) + 1/2(I)] V (z, 0) (x, t) and |V(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 v.(x) durch un(x)e-Ent/h ersetzen. "(x,t) = o(x)e-'Both + vi(z)e-'But/h] 1(z,t)? = V(x,t)' V(x, t) (5) (6) = (7) [:(x) + vi(x) + 20o(x)1(z) cos ((Eo - Ex)t/h)] (8) 1 [vo(x) + vi(x) + 2vo(x)(x) cos(wt)] , mit Eo - E1 = -hw . (9) c) Berechne und Losung: Wir drucken a und p durch die Auf- und Absteigeoperatoren a, und a- aus. Mit (mur Fip) (10) much (as - a_) (11) Fur a, und a- gilt