Question

6. (40 points) A particle is trapped inside a circular ring. (a) solve for the allowed energy levels of the particle and (b) its normalized wavefunction.

Answer 1

Let o particle of scesses is confined to a arde of cirumference L. The co-ordinate on the circle is called x. x belongs to o to L xn xth Which means the points and ath are same points so we can write & 4 (&+L) - 4ra) –– @ the partice confined on the circle has Vix)=0 Time independent Schrodinger ware age is 22y + 20 (EHO - xe already know that value of should be greater than or equal to min. value of potential i.e. E o. In our case E7,0 but En0 requires & constant. How we can write egr cas. d²y = - 2 mt 4 2ME which is greater than or. a2 dx² tet K² equal to zeto .. E= 54² Schrodinger can becomes d2y = – K²4 ate becomes dx² The solution is given by elke 4(x) = that But equation ik (2th) requires ikke

nez elkha, kh = 2000 un ke – 21h solutions are given by to the Schrodinger equations ! 7 (2) = Nulkox constant and can be where & is normalisation determined as. - 5 4 + (x) 8. (x) da u ²de - ² > NEJI ika ztina .: %)= S teer* The energy values are 5 kr - t²4T²² 2012 ²2. En 20 2mL2 m2