Prove that the following are true:\ a. For a normalizable solution, the separation constant

E

must be real. Do this by writing

E

in Eqn. 2.7 as

E_(0)+i\\\\Gamma

where

E_(0)

and

\\\\Gamma

are real and show that if Eqn. 1.20 is to hold for all

t,\\\\Gamma

must vanish.\ b. The time-independent wave function

u(x)

can always be taken to be real unlike

\\\\Psi (x,t)

which is necessarily complex. If

u(x)

satisfies Eqn. 2.5, for a given

E

, so too does its complex conjugate, and hence also the real linear combinations

(u+u^(**))

and

i(u-u^(**))

.\ c. If

V(x)

is an even function, i.e.,

V(-x)=V(x)

, then

u(x)

can always be taken to be either even or odd. Note that if

u(x)

satisfies Eqn. 2.5, for a given

E

, so does

u(-x)

, and thus also the even and odd linear cominations

u(x)+-u(-x)

.

Prove that the following are true: a. For a normalizable solution, the separation constant E must be real. Do this by writing E in Eqn. 2.7 as E0+i where E0 and are real and show that if Eqn. 1.20 is to hold for all t, must vanish. b. The time-independent wave function u(x) can always be taken to be real unlike (x,t) which is necessarily complex. If u(x) satisfies Eqn. 2.5, for a given E, so too does its complex conjugate, and hence also the real linear combinations (u+u) and i(uu). c. If V(x) is an even function, i.e., V(x)=V(x), then u(x) can always be taken to be either even or odd. Note that if u(x) satisfies Eqn. 2.5, for a given E, so does u(x), and thus also the even and odd linear cominations u(x)u(x)