#pauli matrix

[peeterjoot]

Explicit form of the square root of p . sigma.

Explicit form of the square root of p . sigma.

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With the help of Mathematica, a fairly compact form was found for the root of \( p \cdot \sigma \) \begin{equation}\label{eqn:DiracUVmatricesExplicit:120} \sqrt{ p \cdot \sigma } = \inv{ \sqrt{ \omega_\Bp- \Norm{\Bp} } + \sqrt{ \omega_\Bp+ \Norm{\Bp} } } \begin{bmatrix} \omega_\Bp- p^3 + \sqrt{ \omega_\Bp^2 – \Bp^2 } & – p^1 + i p^2 \\ –…

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[peeterjoot]

Explicit expansion of the Dirac u,v matrices

Explicit expansion of the Dirac u,v matrices

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We found that the solution of the \( u(p), v(p) \) matrices were \begin{equation}\label{eqn:DiracUVmatricesExplicit:20} \begin{aligned} u(p) &= \begin{bmatrix} \sqrt{p \cdot \sigma} \zeta \\ \sqrt{p \cdot \overline{\sigma}} \zeta \\ \end{bmatrix} \\ v(p) &= \begin{bmatrix} \sqrt{p \cdot \sigma} \eta \\ -\sqrt{p \cdot \overline{\sigma}} \eta…

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[peeterjoot]

Reflection using Pauli matrices.

Reflection using Pauli matrices.

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In class yesterday (lecture 19, notes not yet posted) we used \( \Bsigma^\T = -\sigma_2 \Bsigma \sigma_2 \), which implicitly shows that \( (\Bsigma \cdot \Bx)^\T \) is a reflection about the y-axis. This form of reflection will be familiar to a student of geometric algebra (see [1] — a great book, one copy of which is in the physics…

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[peeterjoot]

Plane wave and spinor under time reversal

Plane wave and spinor under time reversal

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Q: [1] pr 4.7

(a) Find the time reversed form of a spinless plane wave state in three dimensions.

(b) For the eigenspinor of \( \Bsigma \cdot \ncap \) expressed in terms of polar and azimuthal angles \( \beta\) and \( \gamma \), show that \( -i \sigma_y \chi^\conj(\ncap) \) has the reversed spin direction.

A: part (a)

The Hamiltonian…

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[peeterjoot]

Determining the rotation angle and normal for a rotation through Euler angles

Determining the rotation angle and normal for a rotation through Euler angles

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[1] pr. 3.9 poses the problem to determine the total rotation angle for a set of Euler rotations given by

\begin{equation}\label{eqn:eulerAngleRotationAngleAndNormal:20} \mathcal{D}^{1/2}(\alpha, \beta, \gamma) = \begin{bmatrix} e^{-i(\alpha+\gamma)/2} \cos \frac{\beta}{2} & -e^{-i(\alpha-\gamma)/2} \sin \frac{\beta}{2} \\ e^{i(\alpha-\gamm…

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[peeterjoot]

An observation about the geometry of Pauli x,y matrices

An observation about the geometry of Pauli x,y matrices

[mathjax]

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Motivation

The conventional form for the Pauli matrices is

\begin{equation}\label{eqn:pauliMatrixXYgeometry:20} \begin{aligned} \sigma_x &= \begin{bmatrix} 0 & 1 \\ 1 & 0 \\ \end{bmatrix} \\ \sigma_y &= \begin{bmatrix} 0 & -i \\ i & 0 \\ \end{bmatrix} \\ \sigma_z &= \begin{bmatrix} 1 & 0 \\ 0 & -1 \\ \end{bmatrix} \end{aligned}. \…

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