The Schrödinger equation for a spin- 1/2 neutral particle with a magnetic dipole moment interacting with a helical periodic magnetic field of arbitrary strength and period is solved exactly. The solution is easily obtained by exploiting the symmetries of the Hamiltonian and it is expressed in terms of elementary functions. Several interesting physical aspects of the solution emerge. The behavior of the spin, the group velocity, and the effective mass tensor are obtained, yielding some novel and nontrivial results. Because of the mathematical simplicity, this problem is particularly suitable for an elementary graduate course in quantum mechanics.
The Schrödinger equation for a spin- 1/2 neutral particle with a magnetic dipole moment interacting with a helical periodic magnetic field of arbitrary strength and period is solved exactly. The solution is easily obtained by exploiting the symmetries of the Hamiltonian and it is expressed in terms of elementary functions. Several interesting physical aspects of the solution emerge. The behavior of the spin, the group velocity, and the effective mass tensor are obtained, yielding some novel and nontrivial results. Because of the mathematical simplicity, this problem is particularly suitable for an elementary graduate course in quantum mechanics.
The Pauli–Schrödinger equation for a spin 1/2, neutral particle with a nonvanishing magnetic moment μ0, interacting with an external scalar potential V and a static magnetic field B, both functions of only one of the coordinates, is solved exactly for four different choices of the potential and the field. By choosing in the examples the coordinate y, we present these solutions in the following cases: (i) V( y)=0, B( y)=(B0sin κy, 0, B0cos κy) where B0 and κ are two arbitrary constants. (ii) V( y)=λ′y2, B( y)=‖αy‖(cos 2κy, 0, sin 2κy) where 6* λ=ℏ2/2m and α,κ arbitrary constants and λ′=λ(μ0/2κλ)2. (iii) V( y)=λ′(c̄1tan αy+c̄2cot αy)2, B( y)=B( y) (sin (2κy+2δ(y)), 0, cos (2κy+2δ( y)) where B2( y)=(c̄1tan αy+c̄2cot αy)2 (+α2/4κ2)(c̄1sec2αy−c̄2csc2αy)2 and tan 2δ( y)=(2κ/λ)((c̄1tan αy+c̄2cot αy) /(c̄1sec2αy−c̄2csc2αy)); c̄1, c̄2, α, and κ are arbitrary constants. (iv) V( y)=λ′(ā tanh z+b̄)2 where z =[(y−cd)/d], B2( y)=(ā tanh z+b̄)2+(ā2/4κ2d2)sech2z, tan 2δ( y)=(2κd/ā)((ā tanh z+b̄)/sech2z); ā, b̄, c, d, and κ are arbitrary constants restricted only by (μ0/2κλ)ā2+ā/d>0. The functions B( y) and δ( y) define the vector B( y) as in (iii). Our method of solution is based on the familiar factorization technique for solving the Schrödinger eigenvalue equation. Several interesting physical results of these solutions are discussed.
We study the valence electron of an alkaline atom or a general charged particle with arbitrary spin and with magnetic moment moving in a rotating magnetic field. By using a time-dependent unitary transformation, the Schr\"odinger equation with the time-dependent Hamiltonian can be reduced to a Schr\"odinger-like equation with a time-independent effective Hamiltonian. Eigenstates of the effective Hamiltonian correspond to cyclic solutions of the original Schr\"odinger equation. The nonadiabatic geometric phase of a cyclic solution can be expressed in terms of the expectation value of the component of the total angular momentum along the rotating axis, regardless of whether the solution is explicitly available. For the alkaline atomic electron and a strong magnetic field, the eigenvalue problem of the effective Hamiltonian is completely solved, and the geometric phase turns out to be a linear combination of two solid angles. For a weak magnetic field, the same problem is solved partly. For a general charged particle, the problem is solved approximately in a slowly rotating magnetic field, and the geometric phases are also calculated.
We study a relativistic charged Dirac particle moving in a rotating magnetic field. By using a time-dependent unitary transformation, the Dirac equation with the time-dependent Hamiltonian can be reduced to a Dirac-like equation with a time-independent effective Hamiltonian. Eigenstates of the effective Hamiltonian correspond to cyclic solutions of the original Dirac equation. The nonadiabatic geometric phase of a cyclic solution can be expressed in terms of the expectation value of the component of the total angular momentum along the rotating axis, regardless of whether the solution is explictly available. For a slowly rotating magnetic field, the eigenvalue problem of the effective Hamiltonian is solved approximately and the geometric phases are calculated. The same problem for a charged or neutral Dirac particle with an anomalous magnetic moment is discussed briefly.
We study a relativistic charged Dirac particle moving in a rotating magnetic field. By using a time-dependent unitary transformation, the Dirac equation with the time-dependent Hamiltonian can be reduced to a Dirac-like equation with a time-independent effective Hamiltonian. Eigenstates of the effective Hamiltonian correspond to cyclic solutions of the original Dirac equation. The nonadiabatic geometric phase of a cyclic solution can be expressed in terms of the expectation value of the component of the total angular momentum along the rotating axis, regardless of whether the solution is explictly available. For a slowly rotating magnetic field, the eigenvalue problem of the effective Hamiltonian is solved approximately and the geometric phases are calculated. The same problem for a charged or neutral Dirac particle with an anomalous magnetic moment is discussed briefly.
For tutorial purposes, the simple problem of a classical magnetic moment interacting with various magnetic field configuration is reformulated in terms of coordinate-free, time-dependent rotation operators. In this approach, the corresponding equation of motion assumes the form of an operator equation, identical with the Schrödinger equation of a spin-1 interacting with the same field configurations. It is solved utilizing deliberately the concepts of a Hermitian Hamiltonian and a unitary time evolution operator to familiarize students early in their curriculum with these important ingredients of the quantum-mechanical formalism. It turns out that besides furnishing useful exercises for the manipulation of three-dimensional rotation operators, being an end in itself, the approach also provides simple but nontrivial examples for a number of theorems for which the students will have frequent use in their quantum-mechanics course.
An integrable dynamical problem of a classical neutral particle interacting with an inhomogeneous magnetic field via a magnetic dipole moment is considered; this has been studied quantum mechanically (1978). The equation for the centre-of-mass velocity is of second order resembling that of an anharmonic oscillator with velocity replacing position in the equation. The velocity undergoes curious oscillations, where the spin and centre-of-mass degrees of freedom exchange energy, although there is also, in general, a uniform component to the velocity. An application to the motion of a classical 'neutron' in a magnetic domain wall is discussed, and comparison with previous quantum treatments made.