Q1. An electron is described by the wave function
ψ (x) =0 x<0
ψ(x) =C e-x (1-e-x ) x>0
where x is in nm C is a constant
(a) Determine the value of C that normalize ψ(x).
(b) Where is the electron mass likely to be found ? That is,for what value of x is the probability of finding the electron the largest?
(c) Calculate the average postion >x< for the electron .Compare this result with the most likely position ,and comment on difference.
Q2 . If ψ(x) = A e-(ma/h) x2+it Calculate < ∇ P >
Q3. Simple harmonic oscillator is always an approximation . Real problems always have potential of the form V(x)=1/2 k x2 +b x3 +c x4 +…….The contribution beyond 1/2 k x2 are called anharmonic terms .
(a) Show that to leading order the nth energy eign values of oscillator is changed b2 /h w (h /2mw)3 (30n2+ 30n +1) if all terms are ignored except bx3 .
(b) Calculate the coresponding energy eign value if only cx4 survived .
(c) Calculate the time evalution of creation and annihilation operator i.e <a(t)> and <at (t)>.
