Question

An hydrogen atom starts out in the following linear combination of the stationary states ψί00 and 211- r(r, 0) (V100 + ψ211). (a) Construct ψ(r,t). Simplify it as much as possible. No explicit form of ψ'ın İs required. (b) What is the expectation value of energy expressed in units of eV?

Answer 1

(e) psi(r, 0) = 1/squareroot 2 (psi_100 + psi_211) (a) psi(r, t) = 1/squareroot 2 (psi_100 e^-E_1 ti/h + psi_211 e^-E_2 ti/h) (b) < psi(r, t) Vertical-bar E^^ Vertical-bar psi(r, t) > < E > = sigma^N_n = 1 P_n E_n where P_n rightarrow probability of energy E_n < E > = 1/2 E_1 + 1/2 E_2 E_n = 13.6/n^2 = 1^2 E_1 = 13.6/1 = 13.6 ev E_2 = 13.6/4 = 3.4 ev < E > = 1/2 (13.6) + 1/2 (3.4) < E > = 9.5 ev \r\n (b) < psi Vertical-bar r Vertical-bar psi > is time independent because the time factor comes as e^-iEt/h, which will be cancel out while canculating expentation value L^^= L_x i + L_y j + L_z k^^L^^times L^^= Vertical-bar i L_x L_x j L_y L_y k L_z L_z Vertical-bar= i(L_y L_z - L_z L_y) + j(L_x L_z - L_z L_x) + k^^(L_x L_y - L_y L_x) we know that L_x L_y - L_y L_x = ihL_z, similar for others L^bar times L^bar = ihL proved