Question

3. For a particle moving in an infinite, one-dimensional, symmetric square well of width 2a, show that the (normalized) wave functions are of the form ?-kx).va. cos?x): "-1. 3.5 ,.. COS ? -?? r")(x)=?sin n-r | ; n-2, 4, 6 Express the state ?(x)=N sin,(rx/a) as a linear superposition eigenstates, and find its normalization constant N. of the above HINT sin39-3sin ?-4sin'?

Answer 1

COMMENT IF YOU HAVE ANY DOUBT

Psi(x) = N sin^3 (pi x /a) sin 3 theta = 3 sin theta - 4 sin^3 theta sin^3 theta = 3 sin theta - sin 3 theta /4 Psi(x) = N/4 (3 sin m pi /a - sin 3 n pi /a) = N(3/4 squareroot a Psi_2 - 1/4 squareroot a Psi_6) for normalization integral^a_- a Psi Psi dx = 1 integral^a_-a Vertical-bar Psi Vertical-bar ^2 dx = N^2 /16 integral^a_-a (3 sin m pi /a - sin n pi /a)^2 dx = N^2/16 [integral^a_-a 9 sin m pi /a sin n pi /a dx + integral^a_-a sin 3 m pi /a sin 3 n pi /a dx + 2 integral^a_-a 3 sin m pi /a (- sin 3 n pi /a) dx] = 1 (i) integral^a_-a 9 sin m pi /a sin n pi /a dx = 9 . a (ii) integral^a_-a sin 3 m pi /a sin 3 n pi /a dx = a (iii) integral^a_-a sin 3 m pi /a sin 3 n pi /a dx = 0