- #1
Rajat
- 1
- 0
Just out of pure curiosity, can anyone here give me any advice on the problem of solving the following differential equation
\left\{\sum_i \vec{\alpha}_i \cdot \nabla_i + \sum_{i neq j}\beta_{ij}\delta(x_i -x_j)\delta(\tau_i - \tau_j) \right\}\vec{\psi} = K\frac{\partial}{\partial t}\vec{\psi} = iKm\vec{\psi}.
where,
[\alpha_{ix} , \alpha_{jx}] \equiv \alpha_{ix} \alpha_{jx} + \alpha_{jx}\alpha_{ix} = \delta_{ij}
[\alpha_{i\tau} , \alpha_{j\tau}] = \delta_{ij}
[\beta_{ij} , \beta_{kl}] = \delta_{ik}\delta_{jl}
[\alpha_{ix}, \beta_{kl}]=[\alpha_{i\tau}, \beta_{kl}] = 0 \text{ where } \delta_{ij} = <br /> \left\{\begin{array}{ c c }<br /> 0, & \text{ if } i \neq j \\<br /> 1, & \text{ if } i=j<br /> \end{array} \right.<br />
Any advice on approaching this problem would be greatly appreciated.
Thank you very much!
\left\{\sum_i \vec{\alpha}_i \cdot \nabla_i + \sum_{i neq j}\beta_{ij}\delta(x_i -x_j)\delta(\tau_i - \tau_j) \right\}\vec{\psi} = K\frac{\partial}{\partial t}\vec{\psi} = iKm\vec{\psi}.
where,
[\alpha_{ix} , \alpha_{jx}] \equiv \alpha_{ix} \alpha_{jx} + \alpha_{jx}\alpha_{ix} = \delta_{ij}
[\alpha_{i\tau} , \alpha_{j\tau}] = \delta_{ij}
[\beta_{ij} , \beta_{kl}] = \delta_{ik}\delta_{jl}
[\alpha_{ix}, \beta_{kl}]=[\alpha_{i\tau}, \beta_{kl}] = 0 \text{ where } \delta_{ij} = <br /> \left\{\begin{array}{ c c }<br /> 0, & \text{ if } i \neq j \\<br /> 1, & \text{ if } i=j<br /> \end{array} \right.<br />
Any advice on approaching this problem would be greatly appreciated.
Thank you very much!
