Abstract 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 〈
Abstract 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 〈
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.
Abstract Protonium is a type of onium consisting of a proton (p) and an anti-proton ( <?CDATA $ \bar p $?> p ¯ ) which is called antiprotonic hydrogen. Since the proton and the anti-particle have the same mass, and each of them is a fermion, the Hamiltonian form of this system consists of the kinetic energy terms of each particle, the potential energy, and the spin interactions of both (hyperfine structure). To simplify the analysis, the Hamiltonian is then converted into a Hamiltonian which contains the relative motion terms of the two particles. This new Hamiltonian contains their reduced mass because the two particles orbit each other towards a certain center of mass. In the initial state, t < 0, the system is in the ground state and then at 0 < t < τ, the system is given a disturbance of the form <?CDATA $ V(t) = ({V_0} + {V_1}\,r\,\sin \theta \,{e^{i\phi }})\,\cos (\omega t)\,{\hat \sigma _p}.\hat x $?> V ( t ) = ( V 0 + V 1 r sin θ e i ϕ ) cos ( ω t ) σ ^ p . x ^ , where V 0 is a constant, V 1 is a constant and very small, <?CDATA $ {\hat \sigma _p} $?> σ ^ p is the Pauli matrix for proton, and <?CDATA $ \hat x $?> x ^ is the three-dimensional position operator in spherical coordinates. We analyze this system using a time-dependent perturbation theory that begins by defining the initial state, then calculating the transition amplitude for the new state when the perturbation is applied to the system. From the results of this calculation, we compute the system transition probability. We find that the transition probability of this system can only occur for the following quantum number terms, namely n even positive integer, l = 2,S = 1, and with the certain combinations of m and M.
A system consisting of two neutral spin-1/2 particles is analyzed for two magnetic field perturbations: (1) an inhomogeneous magnetic field over all space, and (2) external fields over a half space containing only one of the particles. The field is chosen to point from one particle to the other, which results in essentially a one-dimensional problem. A number of interesting features are revealed for the first case: the singlet, which has zero potential energy in the unperturbed case, remains unstable in the perturbing field. The spin-zero component of the triplet evolves into a bound state with a double well potential, with the possibility of tunneling. Superposition states can be constructed which oscillate between entangled and unentangled states. For the second case, we show that changes in the magnetic field around one particle affect measurements of the spin of the entangled particle not in the magnetic field nonlocally. By using protective measurements, we show it is possible in principle to establish a nonlocal interaction using the two particles, provided the dipole-dipole potential energy does not vanish and is comparable to the potential energy of the particle in the external field.
Consider a two-level atom represented by the spin operators σ ^ ± , σ ^ Z , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429092053/5e12ab0e-ba42-4caa-9bdf-6325e9117d3e/content/ineqn19_1.tif"/> interacting with a one-dimensional harmonic oscillator, represented by the creation and annihilation operators â † and â . The Hamiltonian of the system is given by ( 19.1 ) H ^ = 1 2 ℏ ω 0 σ ^ Z + ℏ w 0 ( a ^ † a ^ + 1 2 )− 1 2 i ℏ g + ( σ ^ + a ^ − σ ^ − a ^ † ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429092053/5e12ab0e-ba42-4caa-9bdf-6325e9117d3e/content/eqn19_1.tif"/>
Since a quantum system is inevitably influenced by some unpredictable perturbations,we thereby conclude that all the experimental realizations of Grover quantum search algorithm reported were,in fact,achieved in a three-dimensional complex subspace.We also prove that in a two-dimensional complex subspace,for any given initial superposition of basis states |γ0〉=cos β0|α〉+sin β0eiζ|β〉(β0 is a small positive real number,ζ is an arbitrary real number),there exists a set of solutions Fj={(θj,θj-1,…,θ1),(φj,φj-1,…,φ1)} such that a desired state can be found with certainty for some positive integer j≥2,where the phase rotation angles θl and φl are real numbers but not equal to 2k′π,1≤l≤j,k′ is an arbitrary integer.If it is only required that a desired state can be found with high success probability,then as the total number of the desired and undesired states in an unsorted database is sufficiently large the above set of solutions Fj can be written in the form Σji=1θl=Σji=1φl.for a relatively small positive integer j.