1. Chapter 4
Two-Level Systems

2. 4.C.1
Two-level systems
Let us start with the simplest non-trivial state space, with only two dimensions
Despite its simplicity, such space is a good approximation of many physical quantum systems, where all other energy levels could be ignored
If the Hamiltonian of the system is H0, then eigenvalue problem can be written as:

3. Coupling in a two-level system
To account for either external perturbations or the neglected internal interactions of the two-level system, an additional (small) inter-level coupling term is introduced in the Hamiltonian:
In the original (unperturbed) basis the matrix of the perturbed Hamiltonian can be written as:
Let us assume that the coupling is time-independent
Since the coupling perturbation is observable

4. Coupling in a two-level system
What modifications of the two-level system will such coupling introduce?
Now, the eigenvalue problem is modified:
Thereby one has to find the following relationships:
In other words, the new (perturbed) eigen-problem has to be diagonalized

5. Coupling in a two-level system
The solution is:

6. Coupling in a two-level system
The solution is:

7. Coupling in a two-level system
The solution is:

8. Coupling in a two-level system
The solution is:

9. Evolution of the state vector
Let’s represent the state vector at instant t as a superposition of the two “uncoupled” eigenvectors:
Since
we get:

10. Evolution of the state vector
On the other hand:
Recall that if
then
Thus, assuming

11. Evolution of the state vector
On the other hand:
Recall that if
then
Thus, assuming
one gets

12. Evolution of the state vector
Let us choose a special case:
Recall that
Then
Since

13. Evolution of the state vector
The probability amplitude of finding the system at time t in state :

14. Evolution of the state vector
The probability amplitude of finding the system at time t in state :
Then the probability is
The system oscillates between two “unperturbed” states

15. Applications: quantum resonanceIf
then the unperturbed Hamiltonian is 2-fold degenerate
The inter-level coupling lifts this degeneracy giving rise to the ground state and an excited state
E.g., the benzene molecule
It has two equivalent electronic states

16. Applications: quantum resonance
However, there is coupling between the two states, so that the perturbed Hamiltonian matrix has non-diagonal elements
The two levels become separated
This makes the molecule more stable since the ground state energy is below Em while the ground state is a resonant superposition of the unperturbed states

17. Applications: quantum resonance
Another example: a singly ionized hydrogen molecule
There is coupling between two equivalent electronic states, yielding a lower ground state energy
This leads to the delocalization of the electron – the ground state is a resonant superposition of the unperturbed states, which is in essence a chemical bond

18. 4.A.2
Applications: spin
Let us label the elements of the perturbed Hamiltonian matrix as follows:

19. Applications: spin The perturbed Hamiltonian matrix
can be rewritten introducing a matrix corresponding to a spin ½ vector operator
Then

20. Magnetic Moment The vector is called the magnetic dipole
moment of the coil with current
Its magnitude is given by μ = IAN
The vector always points perpendicular to the plane of the coil
The potential energy of the system of a magnetic dipole in a magnetic field depends on the orientation of the dipole in the magnetic field and is equal to:

21. Some properties of spin operator
4.B.3
Some properties of spin operator
Let’s consider a uniform field B0 and chose the direction of the z axis along the direction of B0
Within the notation
So, the eigenvalue problem for H is:
The two states correspond to the spin vector parallel and antiparallel to the field

22. Some properties of spin operator
4.B.3
Some properties of spin operator
We can apply the formulas we derived for the two-level system
For example:
This is precession