Chapter 4 Two-Level Systems, Spin

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 H 0, then eigenvalue problem can be written as: 4.C.1

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.C.1 4.C.2

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 4.C.1 4.C.2

Coupling in a two-level system The solution is: 4.C.2

Coupling in a two-level system The solution is: 4.C.2

Coupling in a two-level system The solution is: 4.C.2

Coupling in a two-level system The solution is: 4.C.2

Evolution of the state vector Let at instant t the state vector is a superposition of the two “uncoupled” eigenvectors: Since we get: 4.C.3

Evolution of the state vector On the other hand: Recall that if then Thus, assuming 4.C.3

Evolution of the state vector On the other hand: Recall that if then Thus, assuming one gets 4.C.3

Evolution of the state vector Let us choose a special case: Recall that Then Since 4.C.3

Evolution of the state vector The probability amplitude of finding the system at time t in state : 4.C.3

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 4.C.3

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 4.C.2

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 E m while the ground state is a resonant superposition of the unperturbed states 4.C.2

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.C.2

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

Applications: spin The perturbed Hamiltonian matrix can be rewritten introducing a matrix corresponding to a vector operator This matrix is Hermitian, therefore it should correspond to some observable Let’s define this observable as the spin Then 4.A.2

Applications: spin It turns out that such observable indeed exists Moreover, it is one of the most fundamental properties of an electron and other elementary particles Spin can be measured experientially, and it gives rise to many macroscopic phenomena (such as, e.g., magnetism) 4.A.2

Some properties of spin operator Let’s consider a uniform field B 0 and chose the direction of the z axis along the direction of B 0 Within the notation So, the eigenvalue problem for H is: The two states correspond to the spin vector parallel and antiparallel to the field 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 4.B.3