1. Nonlinear Optics Lab. Hanyang Univ. Chapter 6. Time-Dependent Schrodinger Equation 6.1 Introduction Energy can be imparted or taken from a quantum system only if the system can jump from one energy E m to another energy E n. A change from one orbit to another can occur if an external time-dependent force F ext acts on the quantum system. We can associate this force with a new potential energy :, and the system’s total Hamiltonian can be given by (6.1.1) The Schrodinger equation becomes (6.1.2)

2. Nonlinear Optics Lab. Hanyang Univ. 6.2 Time-Dependent Solutions Time-independent Schrodinger equation ; : Complete & Orthonormal => Any function can be expressed by the * Following Dirac, the exact time-dependent wave function can be expressed by a sum of ; (6.2.1) (6.1.2) =>(6.2.2) (6.2.3)

3. Nonlinear Optics Lab. Hanyang Univ. where, : time-dependent Schrodinger equation Probability that the quantum system is in its m-th orbit. (6.2.7)

4. Nonlinear Optics Lab. Hanyang Univ. 6.3 Two-State Quantum Systems and Sinusoidal External Forces Time-dependent potential for the interaction between an EM field and an electron ; : dipole approximation For a monochromatic wave, Put, For a two-state system, (6.2.8) 0 (6.3.1) (6.3.4)

5. Nonlinear Optics Lab. Hanyang Univ. Normalization condition ; (6.2.7), (6.3.1) => where, Define, Set, (6.3.4) => : Rabi frequency (field-atom interaction energy in freq. unit) (6.3.11)

6. Nonlinear Optics Lab. Hanyang Univ. i) ( radiation field=0) ii) ( nearly resonant radiation field) trial solution, (6.3.11) => Neglected by rotating-wave approximation where, : detuning : Rabi frequency

7. Nonlinear Optics Lab. Hanyang Univ. Solution) initial condition ; where,: Generalized Rabi frequency Probability ;

8. Nonlinear Optics Lab. Hanyang Univ. 6.4 Quantum Mechanics and the Lorentz Model - Lorentz (classical) model can’t give the oscillator stength, - Why the classical model offers good explanation for a wide variety of phenomena ? Basic dynamic variable for an atomic electron : Displaceement, in classical model, Corresponding quantum displacement : expectation value, For the two-state atom, where,

9. Nonlinear Optics Lab. Hanyang Univ. For a case of linear polarization, Since where,

10. Nonlinear Optics Lab. Hanyang Univ. If we assume, Suppose the E-field points in the z-direction, example) Let atomic state 1 and 2 be the 100 and 210 (1S and 2P)

11. Nonlinear Optics Lab. Hanyang Univ. Table 6.1, 6.2, where, : Bohr radius cf) in classical model Homework : Appendix 5.A !

12. Nonlinear Optics Lab. Hanyang Univ. Classic Quantum mechanics (3.7.5) Oscillator Strength : example) Hydrogen n=1 => n=2,

13. Nonlinear Optics Lab. Hanyang Univ. 6.5 Density Matrix and (Collisional) Relaxation Two level system, time-dependent Schrodinger equation, (6.3.2) (6.3.12) (6.3.14) Via (6.4.3),, the combination variable are more useful than either

14. Nonlinear Optics Lab. Hanyang Univ. Define, similarly,

15. Nonlinear Optics Lab. Hanyang Univ. The equations are not yet in their most useful form, since they do not reflect the existence of relaxation such as collision. (6.3.14) =>

16. Nonlinear Optics Lab. Hanyang Univ. Average value This result can also be reached by a simple modification of the original equation of motion ; Similarly,

17. Nonlinear Optics Lab. Hanyang Univ. 1 2 A 21 22 11 (6.5.2) =>

18. Nonlinear Optics Lab. Hanyang Univ. (6.5.2) => where, : total relaxation rate

19. Nonlinear Optics Lab. Hanyang Univ. No dynamic information ! So, we can pay attention solely to the differences,

20. Nonlinear Optics Lab. Hanyang Univ. (Chapter 8 : Bloch equation) The notation used for : density matrix

21. Nonlinear Optics Lab. Hanyang Univ.