Greens functions

1. Greens functions dr. Imalie GamalathDept. of PhysUniversity of Colombo (Sri Lanka) http://www.cmb.ac.lk/academic/Science/Departments/Physics/

2. Response functions and Greens functions • When an external stimulationF(t) is applied to a physical system, if the measured response is A(t), • Linear Response • If A(t) is doubled when F(t) is doubled. Then is independent of F(t). • This is generally true if F(t) is not too large.

3. Non Linear Response • If A(t) is different for different choices of F(t). • Response of a system to a unit impulse • is the response at time t, to a unit impulse delivered at time t0. This is called the Green’s function or influence function and denoted by

4. The Green’s function satisfies the equation for a linear operator

5. Assuming that the integral falls off faster than 1/r2, can simplify the problem by taking the volume so large, that the surface integralvanishes.

6. Example: Consider an harmonic oscillator with external driving forceF(t).

8. To solve this, generalize ω into complex plane and use contour integration. Contour can be closed only in the upper half plane.

9. -R R This is consistent with the causality condition. No responsebefore the stimulation is applied.

12. Fourier Transform Space

13. Dissipative Response function

15. Rate of Energy Loss

18. Green’s function can be used to solve differential equations. • Time independent Schrodinger equation in one dimension

19. Applying the boundary conditions

22. 3.One dimensional heat conduction equation

25. This is consistent with the causality condition. No response before the stimulation is applied. • Contour can be closed only in the lower half plane.

28. Time independent Schrodinger equation in Three dimension

30. In spherical polar coordinates

31. Consider the integration • Contour can be closed only in the upper half plane

32. Consider the integration • Contour can be closed only in the lower half plane

34. Eigenfunction expansion

36. Find the Greens function for a free particle inside a rectangular box with box planes defined as • Schrödinger equation for a free particle

42. Example: Consider an infinite stretched string subject to an external harmonic force per unit length

48. Green’s function for one particle Schrodinger equation • Green’s function are particularly useful in dealing with problems in perturbation theory. • If the eigenstateH0 is known and V is a perturbation

49. Schrodinger equation • Solution can be written as an integral equation

50. The solution can be iterated • By comparison with (b) • Green function can be derived from Dyson’s equation