Q1. Match the eign functions in column B to their operators in column A .What is the eign values for each of the eign function.
| Column A | Column B |
| (1-x 2 )d2 /dx2 – x d/dx | 4x 4-12x2 +3 |
| d2/dx 2 | 5x4 |
| d2 /dx2 -2x d/dx | 3e3x +e-3x |
| x d2/dx2 (1-x) d/dx | 4x3 – 3x |
Q2. (a)Show that d/dt < ψ(t)|ψ(t)> =0
(b) Derive the continuity equation for the system having potential V(x) = V 1(x) + iV2 (x). Write the explicit relations for current density and number density .
Q3. (a) you are given the following operators :
O1 ψ (x) =x3 ψ(x) , O2 ψ(x) =x (d/dx) ψ (x) , O3 ψ(x) = ∫x-∞ dx’ (ψ(x’) x’)
Solve the eign value problem problem O3 ψ(x) = λ ψ(x) .Also calculate the commutator [O1,O2 ] , [ O3,O4 ].
(b) Let A and B be Hermition and give four operators eiA , eAB , eAB+BA , eAB-BA , . Show which of them are the unitary .