Question

4. Determine approximately the law of motion of a particle of mass m in the field Ucx) in the vicinity of turning point of motion E-Ua) where E is total energy. Proceed by expanding U(x) in a Taylor series about x-a. Consider the cases when (a) U(a)#0 and 5. Find the law according to which the period of motion T for a particle of mass nm moving in the field sketched below approaches infinity as ε=um-E goes to zero. The answer of the form T(e)fe will suffice. Tip: The particle spends most of its time in the vicinity of the lef turning point x,. Approximating Uc) as in part (b) of previous problem, and looking at the time the particle spends near x, provides an estimate for 6. Sketch the trajectory corresponding to E=um (see Figure to previous Problem) when the particle moves between xa and x = x2 clearly labeling the x-a and x =x2 locations.

Answer 1

"Determine approximately the law of motion of a particle of mass m in the field U(x) in one vicinity of turning point of motion E = U(a) where, E is total energy. Schrodinger equation: - strok^2/2m d^2 psi(x)/phi x^2 + v(x) psi(x) = E psi(2) . . . I Taylor series expansion of v(X) U(x) = U(a) + (x - a) U'(a) + (x - a)^2/2 U'(a) + 0(U^3) . . . U(a) = E total energy [Take only 1st two terms] U'(a) + 0 -hstrok^2/2m d^2 psi(x)/dx^2 + (x - a) U'(a) psi(x) = 0 d^2 psi(x)/dx^2 + (-2 m/hstrok^2) (x - a) U'(a) psi(x) = 0 d^2 psi(x)/dx^2 + (-2 m/hstrok^2) (x - a) U'(a) psi(x) = 0 d^2 psi(x)/dx^2 - 2m U'(a)/hstrok^2 (x - a) psi(x) = 0 U'(a) = 0 U""(a) notequalto 0 (take upto second term) -hstrok/2m d^2/dx^2 psi(x) + 1/2 (x - a)^2 U""(a) psi(x) = 0 d^2 psi(x)/dx^2 - k u""(a)/hstrok^2 (x - a)^2 psi(x) = 0 \r\n for particle of mass ""m"" in field U(x) force on particle "