Tensor C*-algebra on Two Qubit Spin-1/2 System
Rivani Adistia Dewi, Imam Nugraha Albania, Rizky Rosjanuardi +1 more
Spin-1/2 system is one of the most important phenomenon in quantum mechanics.Such a system is known to be expressed as Pauli matrices, that isand 𝜎 3 = ( 1 0 0 -1 ) .By inserting the identity matrix 𝐼, we can construct a C*-algebra 𝐶 * ({𝐼, 𝜎 1 , 𝜎 2 , 𝜎 3 }) with |{𝐼, 𝜎 1 , 𝜎 2 , 𝜎 3 }⨂{𝐼, 𝜎 1 , 𝜎 2 , 𝜎 3 }| = 16 = |{𝐸 𝑖𝑗 : 𝑖, 𝑗 = 1,2,3,4} |, where 𝐸 𝑖𝑗 is the canonical basis of C*-algebra 𝑀 4 (ℂ).In this study, by using the linear independence concept, we got 𝐶 * ({𝐼, 𝜎 1 , 𝜎 2 , 𝜎 3 })⨂𝐶 * ({𝐼, 𝜎 1 , 𝜎 2 , 𝜎 3 }) ≅ 𝑀 4 (ℂ).This means, that for any observable in the composite of two spin-1/2 systems, by self-adjointness of Pauli matrices, it can be expressed by the field reduction from ℂ to ℝ in 𝑠𝑝𝑎𝑛{𝐼, 𝜎 1 , 𝜎 2 , 𝜎 3 }⨂{𝐼, 𝜎 1 , 𝜎 2 , 𝜎 3 } where the structure itself is nonassociative algebra.Furthermore, given a Hamiltonian 𝐻(𝑡) = (𝜎 1 ⨂𝜎 1 ) + 𝒥 𝑠𝑖𝑛(𝜔𝑡) (𝜎 3 ⨂𝐼) + 𝒥 𝑠𝑖𝑛 (𝜔𝑡 + 𝜋 2 ) (𝐼⨂𝜎 3 ) and the initial state which is a linear combination of Bell basis, by using the integration factor method and the commutation fact of the exponential operator, we got the following timedependent state |𝜓(𝑡)⟩ = 1 √2 (𝑒 ( 𝑖𝑡 ℏ )( 𝑖𝒥 ℏ𝜔 𝑐𝑜𝑠(𝜔𝑡))( 𝑖𝒥 ℏ𝜔 𝑠𝑖𝑛(𝜔𝑡))𝜈 𝑖(5-𝑖) ) 𝑖=1 4 with 𝜈 𝑖𝑗 = 𝛿 (𝑖+𝑗)5 (-1) 𝑚𝑎𝑘𝑠{𝑖,𝑗} , where 𝛿 𝑖𝑗 is Kronecker delta.
Advances in Social Science, Education and Humanities Research/Advances in social science, education and humanities research•Jan 1, 2024•0 citations