Can Quantum Mechanics Have 'Elegant' Trajectories in Hilbert Space?

[Logic Cloud]

State-space trajectories in classical mechanics can be used to nicely represent the time evolution of a given system. In the case of the harmonic oscillator, for instance, we get ellipses. How does this situation carry over to quantum mechanics? Can the time evolution of, say, the quantum harmonic oscillator be represented by some 'trajectory' in Hilbert space? The obtained wave functions do not seem to lend themselves for such 'elegant' representations.

 

[atyy]

If you choose a unit vector, say for spin 1/2, the trajectory in Hilbert space is just rotation of a unit vector. It's very elegant.

 

[andresB]

There is also the trajectories in the projective hilbert space and their hamiltonian dynamics and poisson brackets.

 

[bhobba]

State-space trajectories in classical mechanics can be used to nicely represent the time evolution of a given system. In the case of the harmonic oscillator, for instance, we get ellipses. How does this situation carry over to quantum mechanics? Can the time evolution of, say, the quantum harmonic oscillator be represented by some 'trajectory' in Hilbert space? The obtained wave functions do not seem to lend themselves for such 'elegant' representations.

If that sort of thing interests you here is the book to get:
https://www.amazon.com/dp/0387493859/?tag=pfamazon01-20

Be warned - its what mathematicians call decidedly non-trivial - meaning it's hard.

Said with a German accent - We have ways and means of stopping you asking certain questions

Thanks
Bill