Off diagonal element of density matrix

[KFC]

For two level system , let denotes the ground state as 1 and exctied state as 2, for writing the office off-diagonal matrix element for the density operator, shall it be

$$\rho_{12} = |2\rangle\langle 1|$$

and

$$\rho_{21} = |1\rangle\langle 2|$$

?

 

[naima]

From a pure state f = a|0> + b|1> one gets the density opérator
(a|0> + b|1>)(a*<0| + b*<1|)
with a off diagonal part ab*|0><1|
the off diagonal matrix element is ab* = <f |0><1|f> (with |0> and |1> orthogonal)
With a mixture of pure state f1,f2 with probabilities p1, p2
we get p1<f1 |0><1|f1> + p2<f2 |0><1|f2>
I hope there is no error in my answer!

 

[Entropia]

Yes, that is correct. The off-diagonal elements of the density matrix represent the coherence between the two states of the system. In a two level system, the ground state is denoted as |1⟩ and the excited state as |2⟩. The off-diagonal element ρ12 represents the coherence from state |1⟩ to state |2⟩, while the off-diagonal element ρ21 represents the coherence from state |2⟩ to state |1⟩. These elements can also be written as the outer products of the two states, as shown in the equations above. These off-diagonal elements are important in understanding the dynamics and behavior of the system, as they represent the probability amplitude for the system to transition between the two states.