Quantum Trajectory


Since most of our understanding of quantum mechanics comes from analogies with classical theory, it doesn’t seem fair that we have thrown the idea of classical trajectory completely out of the window.

The problem is that we define our quantum states using purely spatial/spin coordinates and completely ignore momentum (which propels spatial translation). Under such parametrization, it is impossible to construct any spatial trajectory. The obvious remedy is to label quantum states using both spatial and momentum coordinates. Unfortunately, any attempt to label a state with determined spatial and momentum is destined for failure owing to the uncertainty principle. One may, however, label a state with some distribution of spatial coordinates and momentum.

Schrodinger Molecular Dynamics (SMD)

The most obvious route to quantum dynamics is to express a quantum state in some given time-independent basis wave functions and let the expansion coefficients carry time dependence. The coefficient can be evolved according to time-dependent Schrodinger equation and quantum dynamics is achieved. The problem with this approach is that the dimensionality of a quantum state grows exponentially with system size. A naive implementation of SMD will require the number of basis functions to grow exponentially with system size.

Centroid Molecular Dynamics (CMD)

One method that bares striking resemblance to Classical MD is the centroid molecular dynamics. This method is inspired by the path integral Monte Carlo (PIMC) method.