A modified sinusoidal perturbation on a two distinguishable spin −12 particles system

We analyze a quantum system of two distinguishable spin − 1 2 particles subjected to modified sinusoidal perturbation of the form V ( t ) = ( V 0 + V 1 x + V 2 y + V 3 z ) e − γ t / τ cos ( ω t ) σ → 1 · x ^ ; 0 < t < τ ; where γ is a real positive number; V 0 , V 1 , V 2 , V 3 are real constants and have very small values; x = r sin θ cos ϕ ; y = r sin θ sin ϕ ; = z = r cos θ ; σ → 1 is the Pauli matrix for the first particle, and x ^ , is the three-dimensional position operator in spherical coordinates; V ( t ) is zero when t ≤ 0 and t ≥ τ . We use the time-dependent perturbation theory to solve this problem. Initially, we define the initial system state before being subjected to perturbation, namely | 000 ; 00 〉 , and the final state of the form | n l m ; S M 〉 ≡ | n l m 〉 | ⊗ S M 〉 . The total spin state | S M 〉 actually contains the linear combination of the states of each particle. Next, we calculate the expected value of 〈 n l m ; S M | V ( t ) | 00 ; 000 〉 000 〉 is used in calculating the probability amplitude C n ( τ ) , and then this form is used to analyze the conditions required for the transition to occur. From the results of the analysis, we present various conditions which include the conditions for n , l , m , M , S ; constants V 1 , V 2 and V 3 ; and the relationship between τ and γ ; so that the transition process can occur. We have also presented the energy emission and absorption processes that this type of perturbation phenomenon might produce in the system under study.