The raising and lowering operators generate new solutions to the Schrodinger equ
ID: 2077486 • Letter: T
Question
The raising and lowering operators generate new solutions to the Schrodinger equation, but these new solutions ate not correctly normalized. Thus o + Psi_n is proportional to Psi_n+1 and a_Psi_n is proportional to Psi_n-1, but we'd like to know the precise proportionality constants. Use integration by parts and the Schrodinger equation to show that. Integral^infinity_-infinity |a + Psi_n|^2 dx = (n + 1) hw and integral^infinity_-infinity |a - Psi_n|^2 dx = nhw and hence a + Psi_n = Squareroot (n + 1) hw Psi_n+1. a-Psi_n = Squareroot nhw Psi_n-1. (If desired, we can add factors of i to the right hand sides to keep the wave functions real).Explanation / Answer
Solution :-
The raising and lowering operators generate new solutions to the Schr odinger equation, but these new solutions are not correctly normalized. Thus a+n is proportional to n+1 and an is proportional to n1 but we would like to know the precise proportionality constants. Use integration by parts and the Schrodinger equation to show that
() |a+n|2dx=(n+1) h
() |a-n|^2dx = nh
and hence (with i's to keep the wave function real)
a+n = i Sqrt ((n+1) h)) *n+1
an = -i Sqrt (nh) *n-1