1. Lecture 15: Time-Dependent Perturbation
Source: D. Griffiths, Introduction to Quantum Mechanics (Prentice Hall, 2004)
R. Scherrer, Quantum Mechanics An Accessible Introduction (Pearson Int’l Ed., 2006)
R. Eisberg & R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (Wiley, 1974)

2. Two-Level Systems and are eigenstates of an unperturbed Hamiltonian
These states are orthonormal.
Any state can be written as:
Time-dependent expression in the absence of perturbation:
Probability that a measurement of energy would yield the value Ea.
Normalization of the function:

3. Perturbed System Time-Dependent Perturbations
Time-Dependent Schroedinger Equation

4. Perturbed System
Take inner product with Ψa, and use orthogonality of Ψa and Ψb.
Multiply through by

5. Perturbed System

6. Example: Problem 9.1

7. Example: Problem 9.1

8. Time-Dependent Perturbation Theory
If H’ is small, the above equations can be solved by successful approximations.
Suppose
Particle starts from lower state
Zeroth order:
First order:
Second Order:

9. Interaction of Light with Atoms
Absorption
Stimulated Emission.
Laser: Light amplification by stimulated emission of radiation.
Population Inversion: Majority of atoms in upper state.
Spontaneous Emission: An atom in the excited state makes a transition downwards, release a photon without any applied EM field to initiate process. 9

10. Sinusoidal Perturbation
Assume:
Transition Probability: The probability that a particle which started out in state Ya will be found at time t, in the state Yb.

11. Sinusoidal Perturbation
Transition probability as a function of driving frequency. The probability of transition is greatest at wo.

12. Emission and Absorption of Radiation: EM Waves
Therefore, the diagonal matrix elements of H’ vanish.

13. Probability of Transition
If atom starts out in lower state Ya, a polarized monochromatic is shone on it, the probability of transition to upper level (absorption):
The probability of transition to lower level (emission):
The probabilities are equal, it just switching from a to b , and b to a.

14. Problem 1

15. Problem 2