Question

1) Consider a particle with mass m confined to a one-dimensional infinite square well of length L. a) Using the time-independent Schrödinger equation, write down the wavefunction for the particle inside the well. b) Using the values of the wavefunction at the boundaries of the well, find the allowed values of the wavevector k. c) What are the allowed energy states En for the particle in this well? d) Normalize the wavefunction

Answer 1

Vyoo ^ TOTA Consider motion of the partice of a Y notion of the partice of mas, m along rectangular box with perfectly rugid & infinitely hard walls. let lbe the length between the walls so that the motion of the particle is 2 confined between n=0 to n l . An infinite potentia - exist at the walls of the box such that an infinite 3 amount of work has to be done by the particle in order to leave the box which is not possible lic, particle cannot exist at uso and uzl .4) - POTTED Time independent Schrödinger wave equation is - h² ²4 + vy = EU am 2012 - As v= 0. I inside from n= 0 to a - h² 2² 4 = & 4 - 11) am ana a²4 + 2ME 4 = 0 2 2 2 2²4 +K ² Y=0 - (2) an? where 2m t V ha a Equation Solution (2) is a differential equation and its solution can be given as. 4 = A sin ka + B coska - (3) Applying boundary conditions i 4 (u=o) = 0 Eqr (3) becomes. 4 (n=o) = A.O+B.1=0 В= 0 3

Substituting W(2) - in (3) A sin ka - (1) Applying boundary condition wla= = 0 at a= l. A sin ke= 0 A to i sin ke=0 or Kl=na l Wauefunction becomes y (2) = A sin nia - 6) b Allowed values - K= n of wavelector k üchene na 1, 2, 3, is given. (c) For allowed energy eigenst ates - K=no z v This n = 2ME - l v tz Squarung both sides 2mE - I E = n²r2 h ² zmez , n = 1, 2, 3, 4

d og Normalization of wanafunction . Useng fl 14 (2) 1 ² dn = 1 je Az sin? non du=1 A? To [-cos (20021] det! 5 [- sin (2009) l / ant Rnokl zna e Thus normalized W (W) = manfunction sin intal is quen by 12