Two-state system

schematic representation of a two-state system with absorption and emission of an energy quantum

A two-state system or two-level system in quantum mechanics is a simple but important model system that can be used to describe many situations. The system can only be in one of two possible states , and named, or in a superposition of these two states ( Bra-Ket notation ). These two states usually have different energies and . An example is an electron bound to an atom that can occupy one of two levels of the atomic spectrum (ground state, excited state, see figure on the right). Often the model system of a quantum mechanical spin -1/2 ( angular momentum ) is used, which can only be in two settings. There is a transition between the levels (e.g. an optical transition that can be excited by visible light ). Once the system is in one of the two states, it stays there forever, at least as long as you don't disturb the system. If a fault is switched on in the system, one can observe that the states can merge into one another: ${\ displaystyle \ left | 1 \ right \ rangle}$ ${\ displaystyle \ left | 2 \ right \ rangle}$ ${\ displaystyle E_ {1}}$ ${\ displaystyle E_ {2}}$

Is z. If , for example, an electron is in the state (which is energetically lower than ), it can change into the state through a resonant laser pulse . An electron in the state can fall back into the state by emitting a photon that carries the difference energy between the states . The adjacent figure shows this schematically. ${\ displaystyle \ left | 1 \ right \ rangle}$ ${\ displaystyle \ left | 2 \ right \ rangle}$ ${\ displaystyle \ left | 2 \ right \ rangle}$ ${\ displaystyle \ left | 2 \ right \ rangle}$ ${\ displaystyle \ Delta E = E_ {2} -E_ {1} = \ hbar \ omega}$ ${\ displaystyle \ left | 1 \ right \ rangle}$

If the disturbance is present for a longer period of time, the probability of finding the atom in one of the states oscillates. After half an oscillation period there is a high probability of finding the atom in the excited state, after a whole period it is most likely again in the ground state, etc. This phenomenon corresponds to the Rabi oscillations .

Mathematical description in the context of quantum mechanics

Static treatment

A Hamilton operator belongs to the given system . The states are eigen-states of this Hamiltonian for the eigenvalues : ${\ displaystyle {\ hat {H}} _ {0}}$ ${\ displaystyle \ left | 1 \ right \ rangle, \ left | 2 \ right \ rangle}$ ${\ displaystyle E_ {1}, E_ {2}}$

{\ displaystyle {\ hat {H}} _ {0} \ left | n \ right \ rangle = E_ {n} \ cdot \ left | n \ right \ rangle, \ \ \ \ \ \ n \ in \ left \ { 1,2 \ right \}}

If a Hermitian perturbation is switched on in addition , the eigenstates of the new Hamiltonians are no longer present . The new eigenstates are denoted with and the new eigenenergies with . In the obtained - base following representation for : ${\ displaystyle {\ hat {H}} _ {0}}$ ${\ displaystyle {\ hat {W}}}$ ${\ displaystyle \ left | 1 \ right \ rangle, \ left | 2 \ right \ rangle}$ ${\ displaystyle {\ hat {H}} = {\ hat {H}} _ {0} + {\ hat {W}}}$ ${\ displaystyle \ left | + \ right \ rangle, \ left | - \ right \ rangle}$ ${\ displaystyle E _ {+}, E _ {-}}$ ${\ displaystyle \ {\ left | 1 \ right \ rangle, \ left | 2 \ right \ rangle \}}$ ${\ displaystyle {\ hat {H}}}$

{\ displaystyle ({\ hat {H}}) = {\ begin {pmatrix} E_ {1} + W_ {11} & W_ {12} \\ W_ {21} & E_ {2} + W_ {22} \ end { pmatrix}}, \ \ \ \ \ W_ {12} = W_ {21} ^ {*}}

Is , then only the energy eigenvalues shift; the eigen-states remain the same. The following then applies: ${\ displaystyle W_ {12} = W_ {21} = 0}$

{\ displaystyle E _ {-} = E_ {1} + W_ {11}, \ \ \ \ \ E _ {+} = E_ {2} + W_ {22}}

{\ displaystyle \ left | - \ right \ rangle = \ left | 1 \ right \ rangle, \ \ \ \ \ \ left | + \ right \ rangle = \ left | 2 \ right \ rangle}

Energy shift in the disturbed two-state system

In the case we neglect the diagonal elements (i.e.:) and thus get: ${\ displaystyle W_ {12} = W_ {21} ^ {*} \ neq 0}$ ${\ displaystyle W_ {11} = W_ {22} = 0}$

{\ displaystyle E _ {\ pm} = E_ {m} \ pm {\ sqrt {\ Delta ^ {2} + | W_ {12} | ^ {2}}}}

{\ displaystyle \ left | + \ right \ rangle = \ cos {\ frac {\ theta} {2}} \ cdot e ^ {- i \ varphi / 2} \ cdot \ left | 1 \ right \ rangle + \ sin {\ frac {\ theta} {2}} \ cdot e ^ {i \ varphi / 2} \ cdot \ left | 2 \ right \ rangle}

{\ displaystyle \ left | - \ right \ rangle = - \ sin {\ frac {\ theta} {2}} \ cdot e ^ {- i \ varphi / 2} \ cdot \ left | 1 \ right \ rangle + \ cos {\ frac {\ theta} {2}} \ cdot e ^ {i \ varphi / 2} \ cdot \ left | 2 \ right \ rangle}

The following definitions were used:

{\ displaystyle E_ {m} = {\ frac {E_ {1} + E_ {2}} {2}}; \ \ \ \ \ \ \ Delta = {\ frac {E_ {1} -E_ {2}} { 2}}; \ \ \ \ \ \ tan \ theta = {\ frac {| W_ {12} |} {\ Delta}}; \ \ \ \ \ H_ {12} = | H_ {12} | \ cdot e ^ {-i \ varphi}}

You can see that in this case the energy eigenvalues are shifted in such a way that their distance increases:

{\ displaystyle E _ {+} - E _ {-}> E_ {2} -E_ {1}}

This phenomenon is also called avoided crossing or avoided level crossing , sometimes also level repulsion , repulsion of the energy levels, since the energy levels without the disturbance are represented by two crossing lines, while in the disturbed system the levels converge , but no longer cross.

Oscillations in the QM two-state system

Time development

If the system is prepared in its own state at the time , it remains in this state for all times. If, however, the disturbance (with non-vanishing secondary diagonal elements) is switched on, the probability of finding the system in the state at time t is no longer 0. This is essentially due to the fact that the states and are no longer eigen-states of the system . From the somewhat extensive calculation you get: ${\ displaystyle t = 0}$ ${\ displaystyle \ left | 1 \ right \ rangle}$ ${\ displaystyle {\ hat {W}}}$ ${\ displaystyle P_ {12} (t)}$ ${\ displaystyle \ left | 2 \ right \ rangle}$ ${\ displaystyle \ left | 1 \ right \ rangle}$ ${\ displaystyle \ left | 2 \ right \ rangle}$