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

Abstract We analyze a quantum system of two distinguishable spin <?CDATA $-\frac{1}{2}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo>−</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mrow> </mml:math> particles subjected to modified sinusoidal perturbation of the form <?CDATA $V(t)=({V}_{0}+{V}_{1}x+{V}_{2}y+{V}_{3}z){e}^{-\gamma t/\tau }\cos (\omega t){\overrightarrow{\sigma }}_{1}\cdot \hat{x};0\lt t\lt \tau $?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>V</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>V</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>V</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mi>x</mml:mi> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>V</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mi>y</mml:mi> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>V</mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mi>γ</mml:mi> <mml:mi>t</mml:mi> <mml:mo>/</mml:mo> <mml:mi>τ</mml:mi> </mml:mrow> </mml:msup> <mml:mi>cos</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>ω</mml:mi> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:msub> <mml:mover accent="true"> <mml:mi>σ</mml:mi> <mml:mo>→</mml:mo> </mml:mover> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>·</mml:mo> <mml:mover accent="true"> <mml:mi>x</mml:mi> <mml:mo stretchy="false">^</mml:mo> </mml:mover> <mml:mo>;</mml:mo> <mml:mn>0</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mi>t</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mi>τ</mml:mi> </mml:mrow> </mml:math> ; 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 ; <?CDATA ${\overrightarrow{\sigma }}_{1}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mover accent="true"> <mml:mi>σ</mml:mi> <mml:mo>→</mml:mo> </mml:mover> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:math> is the Pauli matrix for the first particle, and <?CDATA $\hat{x}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mover accent="true"> <mml:mi>x</mml:mi> <mml:mo stretchy="false">^</mml:mo> </mml:mover> </mml:math> , 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 <?CDATA $|000;00\rangle $?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo>|</mml:mo> <mml:mn>000</mml:mn> <mml:mo>;</mml:mo> <mml:mn>00</mml:mn> <mml:mo>〉</mml:mo> </mml:mrow> </mml:math> , and the final state of the form <?CDATA $|nlm;SM\rangle \equiv |nlm\rangle |\otimes SM\rangle $?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>n</mml:mi> <mml:mi>l</mml:mi> <mml:mi>m</mml:mi> <mml:mo>;</mml:mo> <mml:mi>S</mml:mi> <mml:mi>M</mml:mi> <mml:mo>〉</mml:mo> <mml:mo>≡</mml:mo> <mml:mo>|</mml:mo> <mml:mi>n</mml:mi> <mml:mi>l</mml:mi> <mml:mi>m</mml:mi> <mml:mo>〉</mml:mo> <mml:mo>|</mml:mo> <mml:mo>⊗</mml:mo> <mml:mi>S</mml:mi> <mml:mi>M</mml:mi> <mml:mo>〉</mml:mo> </mml:mrow> </mml:math> . The total spin state <?CDATA $|SM\rangle $?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>S</mml:mi> <mml:mi>M</mml:mi> <mml:mo>〉</mml:mo> </mml:mrow> </mml:math> actually contains the linear combination of the states of each particle. Next, we calculate the expected value of <?CDATA $\langle nlm;SM|V(t)|00;000\rangle 000\rangle $?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo>〈</mml:mo> <mml:mi>n</mml:mi> <mml:mi>l</mml:mi> <mml:mi>m</mml:mi> <mml:mo>;</mml:mo> <mml:mi>S</mml:mi> <mml:mi>M</mml:mi> <mml:mo>|</mml:mo> <mml:mi>V</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>|</mml:mo> <mml:mn>00</mml:mn> <mml:mo>;</mml:mo> <mml:mn>000</mml:mn> <mml:mo>〉</mml:mo> <mml:mn>000</mml:mn> <mml:mo>〉</mml:mo> </mml:mrow> </mml:math> is used in calculating the probability amplitude <?CDATA ${C}_{n}(\tau )$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>τ</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> , 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.