Chapter 13

1. Chapter 13 Time-Dependent Perturbation Theory

2. 13.A The eigenproblem • Let us assume that we have a system with an (unperturbed) Hamiltonian H0, the eigenvalue problem for which is solved and the spectrum is discrete and non-degenerate: • The eigenstates form a complete orthonormal basis: • At t = 0, a small perturbation of the system is introduced so that the new Hamiltonian is: • At t < 0, the system is in the stationary state

3. 13.A The eigenproblem • At t > 0, the system evolves and can be found in a different state • What is the probability of finding the system at time t in another eigenstate of the unperturbed Hamiltonian ? • The evolution of the system is described by the Schrödinger equation: • Then:

4. 13.B.1 The approximate solution • Let us employ the following expansion: • Where: • Then:

5. 13.B.1 The approximate solution • Let us employ the following expansion: • Where: • Then:

6. 13.B.1 The approximate solution • Let us employ the following expansion: • Where: • Then:

7. 13.B.1 The approximate solution

8. 13.B.1 The approximate solution

9. 13.B.1 The approximate solution • If the perturbation is zero: • With a non-zero perturbation we can look for the solution in the form: • Then:

10. 13.B.1 The approximate solution • If the perturbation is zero: • With a non-zero perturbation we can look for the solution in the form: • Then:

11. 13.B.2 The approximate solution • This equation is equivalent to the Schrödinger equation • We will look for the solutions in the following form:

12. 13.B.2 The approximate solution • This equation is equivalent to the Schrödinger equation • We will look for the solutions in the following form:

13. 13.B.2 The approximate solution • For the 0th order: • For the higher orders: • From the 0th order solution we can recursively restore solutions for the higher orders

14. 13.B.3 First order solution • Let us recall that at t < 0, the system is in the stationary state • Therefore: • And: • Since: • Then:

15. 13.B.3 First order solution • Let us recall that at t < 0, the system is in the stationary state • Therefore: • And: • Since: • Then:

16. 13.B.3 First order solution • So, for this equation: • The solution is: • Let us recall the formula for the probability:

17. 13.B.3 First order solution • So, for this equation: • The solution is: • Let us recall the formula for the probability:

18. 13.B.3 First order solution • So, for this equation: • The solution is: • Let us recall the formula for the probability: • So, if i ≠ f, then:

19. 13.B.3 First order solution • Thereby, to the lowest power of λ, the probability we are looking for is: • It is nothing else but the square of the modulus of the Fourier transformation of the perturbation matrix element (coupling)

20. 13.C.1 Example: sinusoidal perturbation • Let’s assume that the perturbation is: • In this case: • Where: • Now we can calculate:

21. 13.C.1 Example: sinusoidal perturbation • Let’s assume that the perturbation is: • In this case: • Where: • Now we can calculate:

22. 13.C.1 Example: sinusoidal perturbation • Let’s assume that the perturbation is: • In this case: • Where: • Now we can calculate:

23. 13.C.1 Example: sinusoidal perturbation • So, if the perturbation is: • The probability is: • On the other hand, if the perturbation is: • Then the probability is:

24. 13.C.1 13.C.2 Example: sinusoidal perturbation • On the other hand, if the perturbation is: • Then the probability is:

25. 13.C.2 Example: sinusoidal perturbation • The probability of transition is greatest when the driving frequency is close to the “natural” frequency: resonance • The width of the resonance line is nothing else by the time-energy uncertainty relation • On the other hand, if the perturbation is: • Then the probability is:

26. 13.C.2 Example: sinusoidal perturbation • As a function of time, the probability oscillates sinusoidally • To increase the chances of transition to occur, the perturbation does not necessarily have to be kept on for a long time • On the other hand, if the perturbation is: • Then the probability is:

27. 13.C.1 Example: sinusoidal perturbation • For the special case: • The probability is: • On the other hand, if the perturbation is: • Then the probability is:

28. 13.C.1 Example: sinusoidal perturbation • For the special case: • The probability is:

29. 13.C.3 Coupling with the states of the continuous spectrum • So far we assumed that the final state belongs to a discrete part of the spectrum • How is the theory modified if the energy Ef belongs to a continuous part of the spectrum of H0? • First of all, we cannot measure a probability of finding the system in a well-defined final state • Instead, one has to employ integration over a certain group of final states

30. 13.C.3 Coupling with the states of the continuous spectrum • Let us assume that we have a system with an (unperturbed) Hamiltonian H0, the eigenvalue problem for which is solved and the spectrum is continuous: • The eigenstates form a complete orthonormal basis: • What is the probability of finding the system at time t in a given group of states in a domain Df?

31. 13.C.3 Coupling with the states of the continuous spectrum • Introducing the density of final statesρ: • Here β is the set of other parameters necessary to use if H0 is not a CSCO alone • Then the probability of finding the system at time t in a given group of states is: