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MCS,PHD
Argosy University/ Phoniex University/
Nov-2005 - Oct-2011
Professor
Phoniex University
Oct-2001 - Nov-2016
*Problem 2.1 Prove the following three theorems:
(a)   For normalizable solutions, the separation constant E must be real . Hint: Write E (in Equation 2.7) as Eo + t r (with Eo and r real), and show that if Equation 1.20 is to hold for all t , r must be zero.
(b)  The time-independent wave function v, (x) can always be taken to be real (unlike \ll (x . t ), which is necessarily complex). This doesn't mean that every solution to the time-independent Schrodinger equation is real; what it  says is that if you 've got one that is not, it can always be expressed as a linear combination of solutions (with the same energy) that are. So you might as i,vell stick to v,'s that are real. Hint: If 1/f (x ) satisfies Equation 2.5, for  a given E, so too does its complex conjugate, and hence also the real linear
combinations (1/1 + v,*) and i (VJ - v,*).
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( c) If V (x ) is an even function (that is, V ( -x ) = V (x )) then v,(x) can always be taken to be either even or odd. Hint: If 1/f (x) satisfies Equation 2.5, for
a  given  E,  so  too  does  v,(-x),  and  hence  also  the even  and  odd  linear combinations  1/f (x) + 1/1(-x ).
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