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Physica Scripta

On the quaternionic form of the Pauli–Schrödinger equation

Marc M. Cahay, D Morris

Published Sep 25, 2019• 8 citations
Publisher

Abstract

Abstract A thorough approach to quantum mechanics should rely on an algebraic operation of multiplication based on proper algebraic structures, i.e. division algebras. Quaternions, with their commutation rules isomorphic to SU (2), offer such a framework. In this work, a quaternion algebraic approach derived from the finite groups is used to solve the spatially invariant time-dependent Pauli–Schrödinger equation and derive the Rabi formula and the principle of nuclear magnetic resonance and its generalization. The spatial and time-dependent Pauli–Schrödinger equation is then written using left-chirality quaternions, i.e. by expressing the Pauli spinor, its time derivative, and the Hamiltonian as elements of the same division algebra. The quaternion formalism is used to derive the dispersion relation of a free electron and of an electron in a constant magnetic field. Finally, the formalism is applied to calculate the transmission probabilities through a one-dimensional delta scatterer and through a resonant tunneling structure composed of two one-dimensional quaternionic delta-scatterers in series. The results are compared to the transmission probabilities calculated using traditional (complex) quantum mechanics.

DOI
10.1088/1402-4896/ab47d2

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Abstract A thorough approach to quantum mechanics should rely on an algebraic operation of multiplication based on proper algebraic structures, i.e. division algebras. Quaternions, with their commutation rules isomorphic to SU (2), offer such a framework. In this work, a quaternion algebraic approach derived from the finite groups is used to solve the spatially invariant time-dependent Pauli–Schrödinger equation and derive the Rabi formula and the principle of nuclear magnetic resonance and its generalization. The spatial and time-dependent Pauli–Schrödinger equation is then written using left-chirality quaternions, i.e. by expressing the Pauli spinor, its time derivative, and the Hamiltonian as elements of the same division algebra. The quaternion formalism is used to derive the dispersion relation of a free electron and of an electron in a constant magnetic field. Finally, the formalism is applied to calculate the transmission probabilities through a one-dimensional delta scatterer and through a resonant tunneling structure composed of two one-dimensional quaternionic delta-scatterers in series. The results are compared to the transmission probabilities calculated using traditional (complex) quantum mechanics.
Abstract The solution to the time-dependent Pauli–Schrödinger equation is written using quaternion algebra, i.e. by expressing the unitary operator describing the time evolution of the Pauli spinor and the Pauli spinor itself as left-chirality quaternions. The quaternion approach is used to express the time evolution of a qubit on the Bloch sphere under the action of the commonly used quantum gates and to implement the two-qubit controlled-NOT (CNOT) gate and the three-qubit Tofolli gate. The quaternion formalism is also applied to describe: (1) a general two-qubit quaternionic gate, (2) the Larmor precession of an electron spin in an uniform magnetic field, and (3) the issue of measurements in quantum mechanics. Finally, the time-dependent Pauli–Schrödinger equation is also solved using right-chirality quaternions.
In this chapter, we will consider the general approach used to calculate the time evolution of the probability of a spinor reaching a particular location on the Bloch sphere from another location, when subjected to the simultaneous actions of two spatially uniform magnetic fields. One of these fields is static (time-independent) and directed along a specific axis, while the other is timedependent, has a different amplitude, and rotates in the plane perpendicular to the first magnetic field. This problem was originally considered by I. I. Rabi who calculated the probability of a “spin flip,” i.e., the probability for a spinor, originally located at the north pole on the Bloch sphere, to ultimately reach the south pole [1, 2, 3]. Rabi calculated the time evolution of the probability in the case of a constant (dc) magnetic field along the z-axis and a circularly polarized time-varying (ac) magnetic field of different amplitude, with a specific frequency, rotating in the (x-y) plane. This led to the derivation of what is known as the Rabi formula which is one of the key ingredients in Nuclear Magnetic Resonance (NMR). In most textbooks, the Rabi formula is derived by solving the time-dependent Pauli equation [Equation (2.31)] with a judicious choice of the reference frame to describe the time-evolution of the 2-component spinor wavefunction. Here, we use a different approach based on the concept of the Bloch sphere described in the previous chapter. This approach is more instructive since it directly relates to NMR concepts and is more synergistic with concepts used in spin-based quantum computing.
In this chapter, we will consider the general approach used to calculate the time evolution of the probability of a spinor reaching a particular location on the Bloch sphere from another location, when subjected to the simultaneous actions of two spatially uniform magnetic fields. One of these fields is static (time-independent) and directed along a specific axis, while the other is timedependent, has a different amplitude, and rotates in the plane perpendicular to the first magnetic field. This problem was originally considered by I. I. Rabi who calculated the probability of a “spin flip”, i.e., the probability for a spinor, originally located at the north pole on the Bloch sphere, to ultimately reach the south pole [1, 2, 3]. Rabi calculated the time evolution of the probability in the case of a constant (dc) magnetic field along the z-axis and a circularly polarized time-varying (ac) magnetic field of different amplitude, with a specific frequency, rotating in the (x-y) plane. This led to the derivation of what is known as the Rabi formula which is one of the key ingredients in Nuclear Magnetic Resonance (NMR). In most textbooks, the Rabi formula is derived by solving the time-dependent Pauli Equation [Equation (2.31)] with a judicious choice of the reference frame to describe the time-evolution of the 2-component spinor wavefunction. Here, we use a different approach based on the concept of the Bloch sphere described in the previous chapter. This approach is more instructive since it directly relates to NMR concepts and is more synergistic with concepts used in spin-based quantum computing.
In this paper the physical implications of quaternion quantum mechanics are further explored. In a quanternionic Hilbert space HQ, the lattice of subspaces has a symmetry group which is isomorphic to the group of all co-unitary transformations in HQ. In contrast to the complex space HC (ordinary Hilbert space), this group is connected, while for HC it consists of two disconnected pieces. The subgroup of transformations in HQ which associates with every quaternion q of magnitude 1, the correspondence ψ → qψq−1 for all ψ∈HQ (called Q conjugations), is isomorphic to the three-dimensional rotation group. We postulate the principle of Q covariance: The physical laws are invariant under Q conjugations. The full significance of this postulate is brought to light in localizable systems where it can be generalized to the principle of general Q covariance: Physical laws are invariant under general Q conjugations. Under the latter we understand conjugation transformations which vary continuously from point to point. The implementation of this principle forces us to construct a theory of parallel transport of quaternions. The notions of Q-covariant derivative and Q curvature are natural consequences thereof. There is a further new structure built into the quaternionic frame through the equations of motion. These equations single out a purely imaginary quaternion η(x) which may be a continuous function of the space—time coordinates. It corresponds to the i in the Schrödinger equation of ordinary quantum mechanics. We consider η(x) as a fundamental field, much like the tensor gμν in the general theory of relativity. We give here a classical theory of this field by assuming the simplest invariant Lagrangian which can be constructed out of η and the covariant Q connection. It is shown that this theory describes three vector fields, two of them with mass and charge, and one massless and neutral. The latter is identifiable with the classical electromagnetic field.