Pauli matries and Quaternions

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.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.

Physica ScriptaApr 2, 201919 citations

Pauli matrices

Giuseppe Mussardo

Springer eBooksJan 1, 20250 citations

On the quaternionic form of the Pauli–Schrödinger equation

Marc M. Cahay, D Morris

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 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.

Physica ScriptaSep 25, 20198 citations

The ubiquitous Pauli matrices

Steven B. Berger

American Journal of PhysicsJan 1, 19853 citations

Tsirelson Bound from Bloch Sphere Geometry and SU(2) Rotations — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: The CHSH inequality's quantum bound (Tsirelson bound = 2√2) emerges from the geometry of the qubit manifold (Bloch sphere), where optimal measurement angles are fixed by SU(2) rotations and Pauli observable non-commutativity — not by arbitrary choice. | MATH: CHSH operator $S = \langle AB \rangle + \langle AB' \rangle + \langle A'B \rangle - \langle A'B' \rangle$ . Classical bound: $|S| \leq 2$ . Quantum maximum: $S_{\text{max}} = 2\sqrt{2} \approx 2.828$ . Optimal angles: for Pauli observables $A = \vec{a}\cdot\vec{\sigma}, B = \vec{b}\cdot\vec{\sigma}$ , maximal violation occurs when $\angle(\vec{a},\vec{a}') = \angle(\vec{b},\vec{b}') = 90^\circ$ and $\angle(\vec{a},\vec{b}) = 45^\circ$ , $\angle(\vec{a},\vec{b}') = 135^\circ$ . This yields $\cos^2(45^\circ) = 1/2$ per term, summing to $2\sqrt{2}$ . Quaternion correspondence: $SU(2) \cong \mathbb{H}_1$ (unit quaternions), with Pauli matrices $\sigma_i$ mapping to imaginary quaternions $ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: The CHSH inequality's quantum bound (Tsirelson bound = 2√2) emerges from the geometry of the qubit manifold (Bloch sphere), where optimal measurement angles are fixed by SU(2) rotations and Pauli observable non-commutativity — not by arbitrary choice. | MATH: CHSH operator \( S = \langle AB \rangle + \langle AB' \rangle + \langle A'B \rangle - \langle A'B' \rangle $. Classical bound: $ |S| \leq 2 $. Quantum maximum: $ S_{\text{max}} = 2\sqrt{2} \approx 2.828 $. Optimal angles: for Pauli observables $ A = \vec{a}\cdot\vec{\sigma}, B = \vec{b}\cdot\vec{\sigma} $, maximal violation occurs when $ \angle(\vec{a},\vec{a}') = \angle(\vec{b},\vec{b}') = 90^\circ $ and $ \angle(\vec{a},\vec{b}) = 45^\circ $, $ \angle(\vec{a},\vec{b}') = 135^\circ $. This yields $ \cos^2(45^\circ) = 1/2 $ per term, summing to $ 2\sqrt{2} $. Quaternion correspondence: $ SU(2) \cong \mathbb{H}_1 $ (unit quaternions), with Pauli matrices $ \sigma_i $ mapping to imaginary quaternions \( Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Sep 28, 20260 citations

S = \langle AB \rangle + \langle AB' \rangle + \langle A'B \rangle - \langle A'B' \rangle

. Classical bound:

|S| \leq 2

. Quantum maximum:

S_{\text{max}} = 2\sqrt{2} \approx 2.828

. Optimal angles: for Pauli observables

A = \vec{a}\cdot\vec{\sigma}, B = \vec{b}\cdot\vec{\sigma}

, maximal violation occurs when

\angle(\vec{a},\vec{a}') = \angle(\vec{b},\vec{b}') = 90^\circ

and

\angle(\vec{a},\vec{b}) = 45^\circ

\angle(\vec{a},\vec{b}') = 135^\circ

. This yields

\cos^2(45^\circ) = 1/2

per term, summing to

2\sqrt{2}

. Quaternion correspondence:

SU(2) \cong \mathbb{H}_1

(unit quaternions), with Pauli matrices

\sigma_i

mapping to imaginary quaternions \( Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com