Lecture Notes Prepared by Murila Musungu (ay (x, 2) w(x, t)dx =- [ 4(x,t) ay ( x, tax.(xvi) ox dx Replacing equation (xvi) into equation (xv) we have d(x) th 4." (x , t) ay ( x, t ) 2 dx + 4 (x, t) 4(x,t) dx dt 2m dx ax d(x) 2th I* (x, t) dy (x, t) dx = 4* ( x , t ) - ay(x,tax dt 2m ax mi ax d(x) 1 ha y (x,t) Tax (x, t) dx... (xvil) dt m By definition (P) = [ (x,t) Tax h d (x, t) dx d(x) 1 - 1 ( 4 (x,t) ha w(x,t ) dx = dt m. d(x) (P) dt m (b) Consider the time derivative of (P) d(P) " w(x,t) dx =- h d yr (x, t) a ay ( x , tax dt d(P) d' (x, t) ay ( x, t) dx + day(x, t) dx dt dt ax But d "(at 46. therefore d(P) dt dt dy (x, t) ay(x, dx + [ y(x. tax - at ax Consider Schrodinger equation and its conjugate as shown in equation (iii) and (iv) above we have them as at (x, t) in a2 at 2 maxi " (x, t ) - V(x, t ) (x, t )....(2) ay* (x, t) in a2 dt 2mayil (x,t ) + , V(x, t )y" (x, t ) .... (3 ) Replace (2) and (3) into (1) we have 38 BPY 4103 -QUANTUM MECHANICS 1