Editing De Broglie–Bohm theory (section)

=== Spin ===
To incorporate [[Spin (physics)|spin]], the wavefunction becomes complex-vector-valued. The value space is called spin space; for a [[spin-1/2]] particle, spin space can be taken to be <math>\Complex^2</math>. The guiding equation is modified by taking [[inner product]]s in spin space to reduce the complex vectors to complex numbers. The Schrödinger equation is modified by adding a [[Pauli equation|Pauli spin term]]:

<math display="block">\begin{align}
\frac{d\mathbf{Q}_k}{dt}(t) &= \frac{\hbar}{m_k} \operatorname{Im}\left(\frac{(\psi,D_k \psi)}{(\psi,\psi)}\right)(\mathbf{Q}_1, \ldots, \mathbf{Q}_N, t), \\
  i\hbar\frac{\partial}{\partial t}\psi &= \left(-\sum_{k=1}^{N}\frac{\hbar^2}{2m_k}D_k^2 + V - \sum_{k=1}^{N} \mu_k \frac{\mathbf{S}_k}{\hbar s_k} \cdot \mathbf{B}(\mathbf{q}_k)\right) \psi,
\end{align}</math>

where
* <math>m_k, e_k,\mu_k</math> — the mass, charge and [[magnetic moment]] of the <math>k</math>–th particle
* <math>\mathbf{S}_k</math> — the appropriate [[Spin (physics)#Operator|spin operator]] acting in the <math>k</math>–th particle's spin space
* <math>s_k</math> — [[spin quantum number]] of the <math>k</math>–th particle (<math>s_k = 1/2</math> for electron)
* <math>\mathbf{A}</math> is [[vector potential]] in <math>\R^{3}</math>
* <math>\mathbf{B}=\nabla\times\mathbf{A}</math> is the [[magnetic field]] in <math>\R^{3}</math>
* <math display="inline">D_k = \nabla_k - \frac{ie_k}{\hbar}\mathbf{A}(\mathbf{q}_k)</math> is the covariant derivative, involving the vector potential, ascribed to the coordinates of <math>k</math>–th particle (in [[International System of Units|SI units]])
* <math>\psi</math> — the wavefunction defined on the multidimensional configuration space; e.g. a system consisting of two spin-1/2 particles and one spin-1 particle has a wavefunction of the form <math display="block">\psi: \R^9 \times \R \to \Complex^2 \otimes \Complex^2 \otimes \Complex^3,</math> where <math>\otimes</math> is a [[tensor product]], so this spin space is 12-dimensional
* <math>(\cdot,\cdot)</math> is the [[inner product]] in spin space <math>\Complex^d</math>: <math display="block">(\phi, \psi) = \sum_{s=1}^d \phi_s^* \psi_s.</math>