Time-Dependent Perturbation Theory Chapter 13 Time-Dependent Perturbation Theory

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

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:

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

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

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

The approximate solution 13.B.1 The approximate solution

The approximate solution 13.B.1 The approximate solution

The approximate solution 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:

The approximate solution 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:

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

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

The approximate solution 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

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

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

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

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

First order solution So, for this equation: The solution is: 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:

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)

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

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

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

Example: sinusoidal perturbation 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:

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

Example: sinusoidal perturbation 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:

Example: sinusoidal perturbation 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:

Example: sinusoidal perturbation 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:

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

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

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?

Coupling with the states of the continuous spectrum Introducing the density of final states ρ: 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: