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Calculating the Pauli Matrix Equivalent for Spin-1 Particles and Further

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Calculating the Pauli Matrix Equivalent for Spin-1 Particles and Further Calculating the Pauli Matrix equivalent for Spin-1 Particles and further implementing it to calculate the Unitary Operators of the Harmonic Oscillator involving a Spin-1 System Rajdeep Tah To cite this version: Rajdeep Tah. Calculating the Pauli Matrix equivalent for Spin-1 Particles and further implementing it to calculate the Unitary Operators of the Harmonic Oscillator involving a Spin-1 System. 2020. hal-02909703 HAL Id: hal-02909703 https://hal.archives-ouvertes.fr/hal-02909703 Preprint submitted on 31 Jul 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution - NonCommercial - NoDerivatives| 4.0 International License Calculating the Pauli Matrix equivalent for Spin-1 Particles and further implementing it to calculate the Unitary Operators of the Harmonic Oscillator involving a Spin-1 System Rajdeep Tah1, ∗ 1School of Physical Sciences, National Institute of Science Education and Research Bhubaneswar, HBNI, Jatni, P.O.-752050, Odisha, India (Dated: July, 2020) Here, we derive the Pauli Matrix Equivalent for Spin-1 particles (mainly Z-Boson and W-Boson). 1 Pauli Matrices are generally associated with Spin- 2 particles and it is used for determining the 1 properties of many Spin- 2 particles. But in our case, we try to expand its domain and attempt to implement it for calculating the Unitary Operators of the Harmonic oscillator involving the Spin-1 system and study it. I. INTRODUCTION discuss its analytical solution along with its features. In Section IV, we involve with the transformation of Pauli Matrices are a set of three 2 × 2 complex matri- Unitary Operators and after that in Section V, we ces which are Hermitian and Unitary in nature and they involve the implementation of the Spin-1 system into occur in the Pauli Equation which takes into account the Quantum Harmonic Oscillator with relevant trans- the interaction of the spin of a particle with an exter- formations and generalization which ultimately leads to nal electromagnetic field. In Quantum Mechanics, each the derivation of the Hamiltonian of our system. Then Pauli matrix is related to an angular momentum opera- in Section VI, we derive the Unitary Operators of our tor that corresponds to an observable describing the spin system using the informations from the previous sections 1 and then in its following subsection we represent the of a Spin- 2 particle, in each of the three spatial direc- tions. But we seldom need to deal with particles which 16 × 16 Matrix of the different Unitary Operators. At 1 last, in Section VII, we discuss the Results of our project are having spin more than 2 i.e. Spin-1 Particles; Spin- 3 followed by the Conclusion and Acknowledgement. 2 Particles; Spin-2 Particles; etc. and for that we need to search for matrices which perform similar function to 1 that of Pauli Matrices (in case of Spin- 2 particles). In 1 II. MATHEMATICAL MODELLING Spin- 2 particles, the Pauli Matrices are in the form of: 0 1 Let us assume that we have a Spin-s system for which X = σ = (1) 2 x 1 0 the Eigenvalue S is given by: 2 2 S = s(s + 1)~ (4) 0 −i Or Y = σy = (2) i 0 p S = s(s + 1)~ The eigenvalues of S are written s , where s is al- 1 0 z z ~ z Z = σ = (3) lowed to take the values s; s−1; ··· ; −s+1; −s i.e. there z 0 −1 are 2s + 1 distinct allowed values of sz. We can repre- sent the state of the particle by (2s + 1) different Wave- Also sometimes the Identity Matrix, I is referred to 0 functions which are in-turn denoted as sz (x ). Here as the `Zeroth' Pauli Matrix and is denoted by σ0. 0 1 sz (x ) is the Probability density for observing the parti- The above denoted Matrices are useful for Spin- 2 0 particles like Fermions (Proton, Neutrons, Electrons, cle at position x with spin angular momentum sz~ in the Quarks, etc.) and not for other particles. So, in this z- direction. Now, by using the extended Pauli scheme, paper we will see how to further calculate the equiv- we can easily find out the Momentum operators and the alent Pauli Matrix for Spin-1 particles and implement Spin operators. The Spin Operator comes out in the it to calculate the Unitary Operators of the Quan- form: tum Harmonic Oscillator involving a Spin-1 system. hs; jjSkjs; li (σk)jl = (5) s~ Structure: Where, j and l are integers and j; l 2 (−s; +s). In Section II, we do the Mathematical Modelling Now, to make our calculations easier, we continue with for the Equivalent Matrices and discuss the results for σ matrix. We know that: higher spin particles. Then, in Section III we revisit the z harmonic oscillator in-context of our quantum world and Szjs; ji = j~js; ji (6) 2 ) We can write: oscillator in general can be written as: hs; jjS js; li j (σ ) = k = δ (7) 3 jl s s ij p^2 1 p^2 1 ~ H^ = + m!2x^2 = + kx^2 (16) 2m 2 2m 2 Here we have used the Orthonormality property of js; ji. Thus, σz is the suitably normalized diagonal matrix of where H^ is the Hamiltonian of the System, m is the eigenvalues of Sz. The elements of σx and σy are most easily obtained by considering the ladder operators: the mass of the particle, k is the bond stiffness (which is analogous to spring constant in clas- S± = Sx ± iSy (8) sical mechanics),x ^ is the position operator and Now, according to Eq.(5)-(8), we can write9: 1=2 @ S+ js; ji = [s (s + 1) − j (j + 1)] ~ js; j + 1i (9) p^ = −i (17) ~@x and is the momentum operator where ~ − 1=2 is the reduced Plank's constant. S js; ji = [s (s + 1) − j (j − 1)] ~ js; j − 1i: (10) Now, by combining all the conditions and equations, we The analytical solution of the Schrodinger 1 have: wave equation is given by Ref. : 1=2 [s (s + 1) − j (j − 1)] 1 1 (σ ) = δ 1=2 2 2 1 j l j l+1 X X 1 m! ζ β 2 s − 2 − 2 (11) Ψ = n e e Hnx (ζ)Hny (β)U(t) [s (s + 1) − j (j + 1)]1=2 2 n! π~ + δ nx=0 ny =0 2 s j l−1 (18) and Where; r r [s (s + 1) − j (j − 1)]1=2 m! m! (σ2)j l = δj l+1 ζ = x and β = y (19) 2 i s (12) ~ ~ [s (s + 1) − j (j + 1)]1=2 − δj l−1 Here Hn is the nth order Hermite polynomial. U(t) is the 2 i s unitary operator of the system showing its time evolution and is given by: ) According to Eq.(7), Eq.(11) and Eq.(12), we have: −itE −itEn 20 1 03 n 1 U(t) = exp( ) = e ~ (20) σ1 = p 41 0 15 (13) ~ 2 0 1 0 where En are the allowed energy eigenvalues of the par- ticle and are given by: 20 −i 0 3 1 1 1 En = (nx + )~! + (ny + )~! = (nx + ny + 1)~! (21) σ2 = p 4i 0 −i5 (14) 2 2 2 0 i 0 And the states corresponding to the various energy eigenvalues are orthogonal to each other and satisfy: 21 0 0 3 Z +1 σ = 0 0 0 (15) 3 4 5 j xdxi = 0 : 8 xi (22) 0 0 −1 −∞ A much simpler approach to the harmonic oscil- Where, σ1; σ2; σ3 are the Pauli Matrix equivalents for Spin-1 Particles. lator problem lies in the use of ladder opera- tor method where we make use of ladder oper- ators i.e. the creation and annihilation opera- III. HARMONIC OSCILLATOR IN BRIEF tors (^a; a^y), to find the solution of the problem. The most common and familiar version of ** Herea ^y denotes the `Creation' operator anda ^ the Hamiltonian of the quantum harmonic denotes the `Annihilation' operator in Spin-1 system. 3 We can also the Hamiltonian in terms of the creation V. IMPLEMENTATION ON A SPIN-1 SYSTEM and annihilation operators (^a; a^y)7: 4 1 1 The Hamiltonian of the full system is given by : H^ = !(^aa^ y − ) = !(^aa^ y + ) ~ 2 ~ 2 H^ = H^field + H^atom + H^int Now the Hamiltonian for \a discrete quantum harmonic oscillator" is given by: where H^field is the free Hamiltonian, H^atom is the atomic excitation Hamiltonian and H^ is the interaction Hamil- (^pd)2 (^xd)2 + (^yd)2 int H^ = + (23) tonian. 2 2 wherep ^d is the discrete momentum operator andx ^d and y^d are the discrete position operators in in x and y spatial A. MODEL dimension respectively. Alsop ^d can be expressed as: We have modeled our system using Rabi Hamiltonian5. d d −1 d d p^ = (F ) · x^ · (F ) (24) However, in our case we will be using somewhat modified 6 where F d is the standard discrete Quantum Fourier version of Rabi Hamiltonian : 3 d Transform matrix .
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