Transition (quantum mechanics)

The quantum mechanical transition generally describes the change in the state of a system from an initial state to another. An alternative, outdated term is the quantum leap , which is rarely used in technical terminology. The transitions occurring in quantum mechanics were seen as instantaneous, random phenomena, but it has been shown experimentally that this is not correct.

Transition using the example of the two-state system

In order to clarify the mathematical modeling of transitions in quantum mechanical systems, a two-level system can be considered as an example. The energy levels of a system are given by the eigenvectors of the Hamilton operator , hereinafter referred to as. An energy is assigned to the respective states . For the sake of simplicity, it is assumed for this example that the energies are not the same, i. H. ; the states are therefore not degenerate . The matrix representation of the Hamilton operator (in the energy eigenbase) is given by ${\ displaystyle {\ hat {H}} _ {0}}$ ${\ displaystyle \ {| a \ rangle, | b \ rangle \}}$ ${\ displaystyle E_ {a / b}}$ ${\ displaystyle E_ {a} \ neq E_ {b}}$

{\ displaystyle {\ hat {H}} _ {0} = {\ begin {pmatrix} E_ {a} & 0 \\ 0 & E_ {b} \ end {pmatrix}}.}

If the system is in one of the two eigenstates or , no transitions between the two states are possible with this Hamiltonian. In other words, transitions can take place if and only if the matrix representation of the Hamilton operator has at least one non-diagonal element not equal to zero. ${\ displaystyle | a \ rangle}$ ${\ displaystyle | b \ rangle}$

In a system with a Hamilton operator of the form , where the term is replaced by the expression ${\ displaystyle {\ hat {H}} = {\ hat {H}} _ {0} + {\ hat {V}}}$ ${\ displaystyle {\ hat {V}}}$

{\ displaystyle {\ hat {V}} = {\ begin {pmatrix} 0 & g \\ g ^ {*} & 0 \ end {pmatrix}}}

is given, the Hamilton operator has non-diagonal elements. Transitions are thus possible. In practice, such a term would come about through an interaction of the system with its environment (e.g. interaction with an electromagnetic field).

One can show that the transition probability of a system in the initial state in the state itself increases ${\ displaystyle {\ mathcal {P}} _ {a \ rightarrow b}}$ ${\ displaystyle | a \ rangle}$ ${\ displaystyle | b \ rangle}$

{\ displaystyle {\ mathcal {P}} _ {a \ rightarrow b} (t) = | \ langle b | \ varphi (t) \ rangle | ^ {2} = {\ frac {| g | ^ {2} } {| g | ^ {2} + \ Delta ^ {2}}} \ sin ^ {2} \ left ({\ sqrt {| g | ^ {2} + \ Delta ^ {2}}} {\ frac {t} {2 \ hbar}} \ right)}

results. It can detect a sinusoidal oscillation of probability, known as the Rabi oscillation . The validity of the previous statement that a diagonal Hamilton operator shows no transitions can easily be determined here, since it follows. ${\ displaystyle {\ mathcal {P}} _ {a \ rightarrow b} \ equiv 0}$ ${\ displaystyle g = 0}$

Individual evidence

^ ZK Minev u. a .: To catch and reverse a quantum jump mid-flight . In: Nature . tape 570 , no. 7760 , June 2019, p. 200-204 , doi : 10.1038 / s41586-019-1287-z .
^ Claude Cohen-Tannoudji, Bernard Diu, Frank Laloë: Quantum Mechanics . 3rd, through and verb. Edition volume 1 . De Gruyter, Berlin 2007, ISBN 978-3-11-019324-4 .

[1] ZK Minev u. a .: To catch and reverse a quantum jump mid-flight . In: Nature . tape 570 , no. 7760 , June 2019, p. 200-204 , doi : 10.1038 / s41586-019-1287-z .

[2] Claude Cohen-Tannoudji, Bernard Diu, Frank Laloë: Quantum Mechanics . 3rd, through and verb. Edition volume 1 . De Gruyter, Berlin 2007, ISBN 978-3-11-019324-4 .