The document presents an overview of time-dependent perturbation theory in quantum mechanics, focusing on evaluating the probability of finding a system in a particular state over time when a time-dependent perturbation is applied. It discusses the assumptions made regarding the Hamiltonian and the derivation of probability amplitudes using Schrödinger's equation, highlighting the need for approximations when transition probabilities are small. The intended audience is physics students with prior knowledge of Dirac braket notation.

1. Time-Dependent Perturbation Theory Prepared by: James Salveo L. Olarve Graduate Student January 28, 2010

2. Introduction The presentation is about how to evaluate the probability of finding the system in any particular state at any later time when the simple Hamiltonian was added by time dependent perturbation. S o now the wave function will have perturbation-induced time dependence. The intended reader of this presentation were physics students. The author already assumed that the reader knows dirac braket notation. This presentation was made to facilitate learning in quantum mechanics.

3. Time-Dependent Perturb Hamiltonian We look at a Hamiltonian is some time-dependent perturbation in which we assumed to be small compared to the time-independent part. here unperturbed eigenvalue equation Note: We label here not since for a time-dependent Hamiltonian, the energy will not be conserved. Therefore energy corrections are futile to solve Now, even for V=0, the wave functions have the usual time dependence with as constant

4. On introducing the acquire time dependence Time-Dependent Perturb Hamiltonian This time dependence can be determined by Schödinger’s Equation with

5. Time-Dependent Perturb Hamiltonian Now, Taking the inner product with the bra and introducing

6. Time-Dependent Perturb Hamiltonian This is a matrix differential equation for the and solving this set of coupled equations will give us the , and hence the probability of finding the system in any particular state at any later time.

7. If the system is in initial state at t = 0, the probability amplitude for it being in state at time t is to leading order in the perturbation Time-Dependent Perturb Hamiltonian The probability that the system is in fact in state at time t is therefore Obviously, this is only going to be a good approximation if it predicts that the probability of transition is small—otherwise we need to go to higher order, using the Interaction Representation

8. Reference: Retrieved from http://galileo.phys.virginia.edu/classes/752.mf1i.spring03/Time_Ind_PT.htm , January 19 2010, Michael Fowler. Introduction to Quantum Mechanics. David J. Griffiths. 1994