1. Time-Dependent Perturbation Theory
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. The approximate solution
13.B.1
The approximate solution
Let us employ the following expansion:
Where:
Then:

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

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

7. The approximate solution
13.B.1
The approximate solution

8. The approximate solution
13.B.1
The approximate solution

9. 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:

10. 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:

11. 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:

12. 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:

13. 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

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. 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:

17. 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:

18. 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:

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

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

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

23. 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:

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

25. 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:

26. 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:

27. 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:

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

29. 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. 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. 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: