Chapter 4 Two-Level Systems

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:

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

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

Coupling in a two-level system The solution is:

Coupling in a two-level system The solution is:

Coupling in a two-level system The solution is:

Coupling in a two-level system The solution is:

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:

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

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

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

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

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

Applications: quantum resonance If 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

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

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

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

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

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:

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

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