Question

[12 6. Consid er a particle of mass m moving in an infinitely deep square well potential of width a, whose wave function at time t 0 is where on Ce) is the normaized wave function of the n-th eigenstate of the Hamitonian of that particle The corresponding eigen-energy of the n-th state is 2ma?n 1,2,3,... (e) Find the average energy of the system (ie. the expectation value () (b) Write down the wave function p(z,t) at a later time t. (c) What is the average energy of the system at time t? [2 19]

Answer 1

E_1 = Ground state energy = pi^2 h^2 /2ma^2 E_2 = 1st excited state energy = 4 pi^2 h^2 /2ma^2 E_3 = 2nd excited state energy = 9 pi^2 h^2 /2ma^2 E_4 = 3rd excited state energy = 16 pi^2 h^2 /2ma^2 average energy of the system is (3/squareroot 30)^2 times E_1 + (4/squareroot 30)^2 times E_2 + (1/squareroot 6)^2 times E_4 = 9/30 pi^2 h^2 /2ma^2 + 16/30 middot 4 pi^2 h^2 /2ma^2 + 1/6 16 pi^2 h^2 /2ma^2 = pi^2 h^2 /2ma^2 [9/30 + 32/15 + 8/3] = 153 pi^2 h^2 /60 ma^2 Psi (a, t) = 3/squareroot 30 phi_1 (x) e^-i E_1 t/h + 4/squareroot 30 phi_2 (x) e^-ie_2 t/h + 1/squareroot 6 phi_2 9x) e^-iE_4 t/h = total wave function at time t. Since the average energy of the system does not depend on t. so it will not changes over time so at time t the average energy will by 153 pi^2 h^2 /60 ma^2