College Physics

Solution of spin state eigenvalue and eigenfunction in double electrons system

Wang Zhi-jun

Published Jan 1, 2010• 0 citations

Abstract

The eigenvalues and eigenfunctions of electron spin state in the double electrons system are solved by using the theory of the matrix direct product.The results are discussed and the consistency with usual teaching material is shown.

Related papers

Construction of S2 Eigenfunctions by the Method of Spin Operators II: Six-electron Systems

F Berencz

In the first part of this paper a spin operator was introduced which when operating on the eigenfunctions of the total S z operator related to maximal, minimal and zero projections of total spin creates each eigenfunction of the total S 2 corresponding to its different eigenvalues. In this second part a brief discussion of the proposed operator technique developed for the problem of six-electron system is presented.

Proceedings of the Physical SocietyFeb 1, 19584 citations

Electron Spin Dynamics in a Quantum Dot due to Hyperfine Coupling with Nuclei in the Dot - An Exact Analysis

A. K. Rajagopal

The time-dependent Schrodinger equation of a many particle spin system consisting of an electron in a quantum dot interacting with the spins of the nuclei (N) in the dot due to hyperfine interaction is solved exactly for a given arbitrary initial state. The electron spin dynamics is then expressed in terms of the reduced density matrix of the composite system by computing the marginal density matrix of the electron. This is accomplished by classifying states of the system by the total spin of the coupled electron and nuclear system that commutes with the system Hamiltonian. These states are used in enumerating and finding the exact solution of the time-dependent Schrodinger equation. In each sector of the total spin, the problem reduces to solving a linear simultaneous set of equations that are solved by matrix inversion. Such solutions enable one to make a reliable approximate scheme for purposes of numerical estimation of the various physical quantities. A methematical and physical discussion of this procedure with the density matrix approach to this problem is given here. To elucidate the procedure and its advantages, special cases of N=3,4 are given in some detail in an Appendix.

arXiv:quant-ph/0602092Feb 10, 20060 citations

Teaching Perturbation Theory through Spin Systems

L. E. Radford, J. S. Willis, P. W. Forbes

The spin system provides a relatively simple and compact vehicle for discussion of non-degenerate perturbation theory. In fact in many such systems the terms required to write the electron's energy correct to second order may be read directly from the spin-energy matrix. In this work a hypothetical spin system and three different physical spin systems are treated in order to demonstrate the conciseness of the spin-energy-matrix approach to some perturbation problems. The combination of a perturbation problem which is calculable and which may be related to experimental data provides an approach to perturbation theory which is particularly satisfying to the student.

American Journal of PhysicsMay 1, 19700 citations

Solutions of the three-magnon bound state equation. III. The physical eigenstate

Chanchal K. Majumdar, Indrani Bose

The physical eigenvalue and eigenfunction are found analytically by directly solving the three-magnon bound state equation in one dimension. The structure of the equation suggests a simple ansatz for the eigenfunction, and the calculation is reduced to the solution of a finite number of linear algebraic equations. The method is elementary, but the explicit solution is found after a long calculation. The connection of the wavefunction with previous results is also established.

Journal of Mathematical PhysicsOct 1, 19787 citations

One-Dimensional Many-Fermion System. I: Exact Solution in a Field Theoretical Framework

T. Kebukawa

The eigenvalue problem in a one-dimensional system of many fermions interacting via a delta-function potential is solved exactly in a field theoretical framework. Simultaneous eigenstates of spin and energy are represented in a systematic and compact form, that is, in such a form that the orbital and spin eigenfunctions are unified in harmony without the aid of Young’s tableau. The eigenstate and energy eigenvalue are specified by two sets of quantum numbers, particle momenta and spin momenta, besides two eigenvalues of a total spin angular momentum and its z-component. It is shown that for given particle momenta and spin momenta the energy eigenvalue is given by the solutions of coupled equations.

Progress of Theoretical PhysicsMay 1, 19853 citations

MATRIX MECHANICS OF ANGULAR MOMENTUM

Mircea S. Rogalski, Stuart B. Palmer

Ψ=Ψ Iˆ In view of the expansion (29.28), which holds for an arbitrary state function in the form Ψ = Ψ Ψ Ψ Ψ = Ψ Ψ Ψ∑ ∑ we may define the unit or identity operator as i iI ΨΨ=∑ =1 ˆ (33.4) Conversely, for every linear operator we can form a square matrix, in terms of a complete set of functions by multiplying the left-hand side of Eq.(33.3) by and integrating, which gives fˆ ,iΨ ∗Ψi ΨΨ=ΨΨ=ΦΨ=ΦΨ∫ ∗ Iffd iiii ˆˆˆV ΨΨ=ΨΨΨΨ= n ˆˆ Since ii b=ΦΨ this result is identical with that given by Eq.(33.1), provided we set Vˆˆ dfff kikiik ΨΨ=ΨΨ= ∫ ∗ (33.5) which are called the matrix elements of the linear operator in terms of the particular complete set of functions It is a straightforward matter to show that the rules of matrix algebra for addition, multiplication, associativity and distributivity (see Appendix II) apply to the matrix elements (33.5), following from their definition. Two properties of fˆ .iΨ matrix elements that are needed in quantum mechanics can be readily derived from Eq.(33.5). Using Eq.(29.19) we obtain ∗∗ =ΨΨ=ΨΨ= kiikkiik ffff ˆˆ (33.6) which means that if is a Hermitian operator, then its matrix representation in a complete set of functions is Hermitian. Hence, the elements in the principal diagonal of a Hermitian matrix must be real. When all the elements are real, the matrix must be symmetric. From Eq.(29.20) we have fˆ ikiikiik f δλλ =ΨΨ=ΨΨ ˆ (33.7) that is: if the functions are the eigenfunctions of the Hermitian operator the matrix representation is diagonal, and the matrix elements are the eigenvalues of the operator. A quantum mechanical problem can be stated in a differential equation form or in a matrix form. In both cases the solution consists of finding the eigenvalues and the eigenfunctions of an operator associated with a physical observable. In view of a matrix representation in any complete set of functions, the eigenvalue equation (29.20) must be formulated as iΨ ,fˆ (33.8) Ψ=Ψ λfˆ where the arbitrary eigenfunction Ψ can be expanded, as given by Eq.(29.4), in terms of a complete set of functions If we multiply Eq.(33.8) by and integrate, we obtain .iΨ ∗Ψi ˆ ˆ ˆV V ori i if d d f Iλ λ∗ ∗Ψ Ψ = Ψ Ψ Ψ Ψ = Ψ Ψ∫ ∫ i or also, in view of Eq.(33.4) k kki f λ ˆ This can be rewritten as = n 1 λ which reads (33.9) ⎟⎟ ⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜⎜ ⎜ ⎝ ⎛ = ⎟⎟ ⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜⎜ ⎜ ⎝ ⎛ ⎟⎟ ⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜⎜ ⎜ ⎝ ⎛ a a a a a a fff fff fff …… … ……… … … λ Equation (33.6) is called the matrix eigenvalue equation and consists of n linear homogenous equations for the given by ,ka ( ) 0... 1212111 =+++− nnafafaf λ ( ) 0... 2222121 =++−+ nnafafaf λ (33.10) ……… ( ) 0...2211 =−+++ nnnnn afafaf λ This system of simultaneous equations has a nontrivial solution for the if and only if the characteristic determinant vanishes ( ) ( ) ( ) ( ) 01ˆˆdet = ⎟⎟ ⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜⎜ ⎜ ⎝ ⎛ − − − =− λ λ λ λ nnnn fff fff fff f … ……… … … (33.11) If expanded, the determinant gives a polynominal of nth degree called the characteristic polynominal of fˆ with n roots which are the desired eigenvalues. For each eigenvalue Eqs.(33.10) can be solved for the giving the corresponding eigenfunction that is a column matrix.

CRC Press eBooksJan 8, 20140 citations