Summary: The authors study the bound-state solutions of vanishing angular momentum in a quaternionic spherical square-well potential of finite depth for the “quaternionic Schrödinger equation”. In other to see the novelty involved in this paper we firstly need to see what is that “quaternionic Schrödinger equation”, which was defined by the authors as \[ \hbar\partial_t\Phi= I{\hbar^2\over 2m} \nabla^2\Phi- (IV_1- JV_2- KV_3)\Phi,\tag{\(*\)} \] where \(I\), \(J\), \(K\) are the quaternionic units, \(\Phi= \phi_0+ \phi_1I+ \phi_2J+ \phi_3K\) with \(\phi_i: \mathbb{R}^4\to \mathbb{R}\) \((i=0,1, 2,3)\) is the “quaternionic wave function” and the other symbols have their usual meanings. But it is well-known that the quaternionic units can be represent using Pauli matrices \(\sigma_1\), \(\sigma_2\) and \(\sigma_3\) as \[ I= i\sigma_3= \begin{pmatrix} i & 0\\ 0 & -i\end{pmatrix},\quad J= i\sigma_2= \begin{pmatrix} 0 & 1\\ -1 & 0\end{pmatrix},\quad K= i\sigma_1= \begin{pmatrix} 0 & i\\ i & 0\end{pmatrix}, \] where \(i\) is the complex imaginary unit. The matrix form of the “quaternionic Schrödinger equation” is \[ \begin{pmatrix} \hbar\partial_t\alpha &-\hbar\partial_t\beta^*\\ \hbar\partial_t\beta &\hbar\partial_t \alpha^*\end{pmatrix}= \begin{pmatrix} i{\hbar^2\over 2m} \nabla^2+ iV_1 & V_2+ iV_3\\ -V_2+ iV_3 &-i{\hbar^2\over 2m} \nabla^2- iV_1\end{pmatrix} \begin{pmatrix} \alpha & -\beta^*\\ \beta & \alpha^*\end{pmatrix}, \] where \(\alpha= \phi_0+ i\phi_1\) and \(\beta= -\phi_2+ i\phi_3\). There are only two independent equations coupling the functions \(\alpha\) and \(\beta\), \[ i\hbar\partial_t\alpha= -{\hbar^2\over 2m} \nabla^2\alpha+ V_1\alpha+ (V_3- iV_2)\beta,\tag{1a} \]
\[ i\hbar\partial_t\beta= {\hbar^2\over 2m} \nabla^2\beta- V_1\beta+ (V_3+ iV_2)\alpha.\tag{1b} \] On the other hand, the Pauli-Schrödinger equation, which is the nonrelativistic wave equation for spin \(1/2\) particles, is \[ i\hbar\partial_t\varphi= \Biggl({1\over 2m} (-i\hbar\nabla+ e\vec A)^2- eU+ {e\hbar\over 2m} \vec\sigma\cdot \vec B+ V\Biggr)\varphi, \] where \(U\) and \(\vec A\) are, respectively, the scalar and vector electromagnetic potentials, \(\vec B= \nabla\times\vec A\) is the external magnetic field, \(V\) is an additional external potential, and \(\varphi\) a Pauli spinor field. In the absence of electromagnetic interactions and in the particular case where \(V= \vec V'\cdot\vec\sigma\) we have \[ i\hbar\partial_t \varphi= -{\hbar^2\over 2m} \nabla^2\varphi+ (V_1'\sigma_1+ V_2'\sigma_2+ V_3'\sigma_3)\varphi. \] Writing \(\varphi= {\alpha\choose\beta}^T\) we get the equations \[ i\hbar\partial_t\alpha= -{\hbar^2\over 2m} \nabla^2\alpha+ V_3'\alpha+ (V_1'- iV_2')\beta,\;i\hbar\partial_t\beta= -{\hbar^2\over 2m} \nabla^2\beta- V_3'\beta+ (V_1'+ iV_2')\alpha. \] Since \(IV_1+ JV_2+ KV_3= i(V_1\sigma_3+ V_2\sigma_2+ V_3\sigma_1)\), let us take \(V_1'= V_3\), \(V_2'= V_2\) and \(V_3'= V_1\). In this case, the Pauli-Schrödinger equation for the components of the spinor field are \[ i\hbar\partial_t\alpha= -{\hbar^2\over 2m} \nabla^2\alpha+ V_1\alpha+ (V_3- iV_2)\beta,\tag{2a} \]
\[ i\hbar\partial_t\beta= -{\hbar^2\over 2m} \nabla^2\beta- V_1\beta+ (V_3+ iV_2)\alpha.\tag{2b} \] Now if we compare (1–2) we see that (1a) is identical to (2a) and (2a) differs from (2b) by the sign of the kinetic energy term. It is easy that the (2a) and (2b) can be written in a quaternion-fashioned way as \[ \hbar\partial_t\Phi= {\hbar^2\over 2m} \nabla^2\Phi I- (IV_1- JV_2- KV_3)\Phi.\tag{\(**\)} \] The difference between \((*)\) and \((**)\) is, therefore, the positioning of the quaternion unity \(I\) in the kinetic energy term as left multiplication or right multiplication.
Since the study of solutions of Pauli-Schrödinger equation for certain types of potentials are taken as elementary exercises in quantum mechanics courses, we can say that the novelty in this paper is to solve those kind of exercises with a misplaced \(I\).
Finally, we have to mention that there is a difference between the usual Pauli-Schrödinger equation written in terms of Pauli spinor fields and its quaternionic version \((**)\) when we consider the transformation properties of these fields under the action of the \(\text{SU}(2)\) group. The Pauli spinor field is transformed according to \(\varphi\mapsto U\varphi\), where \(U\in\text{SU}(2)\), while the quaternion field can be transformed either as \(\Phi\mapsto U\Phi\) or as \(\Phi\mapsto U\Phi U^†\). The first possibility implies that \(\Phi\) changes sing after a \(2\pi\) rotation while the second possibility implies that \(\Phi\) does not changes sign. This interesting fact was not mentioned by the authors. However, the study of this problem using the \((*)\) may lead to some unusual facts since as we see from (1) the spin-up and spin-down components of a free particle satisfy different equations, namely the Schrödinger and the time-reversed Schrödinger equations, respectively.