Here's my question: as soon as I learned Quantum Mechanics and Schrodinger equation, I saw a "similarity" with the equation one gets in classical mechanics for the evolution of a function in phase space. In QM one has:(adsbygoogle = window.adsbygoogle || []).push({});

[itex]

i\hbar\frac{d}{dt}\psi = \hat{H}\psi

[/itex]

and this is a evolution equation where [itex] \psi [/itex] is the element which evolves and it is an element of a space of functions.

If one represents this equation (considering one spinless particle) in the [itex] |\vec{x}> [/itex] basis, one gets the wave equation that everyone knows, where the hamiltonian on this basis acts on the state ket as

[itex]

-\frac{\hbar^2}{2m}\nabla^2 + U(\vec{x})

[/itex]

does.

In CM one has:

[itex]

\frac{d}{dt}f = \hat{L}f

[/itex]

where f is the element that evolves and it is an element of a space on functions, too. (here I assumed that the functions I want to evolve from time t_{0}to time t do not depend on t_{0}explicitly, otherwise I should have added [itex] \partial_t f [/itex] to that equation)

If one represents that equation in the [itex] |\vec{q},\vec{p}> [/itex] basis one gets:

[itex]

\frac{d}{dt}f (\vec{q},\vec{p}) = \{f(\vec{q},\vec{p}),H(\vec{q},\vec{p})\}

[/itex]

and if I solve Hamilton equations and get the hamiltonian flow [itex] \Phi^H_{(t,t_0)} [/itex], I know that the solution to the equation with initial condition f_{0}is:

[itex]

(e^{\hat{L}\Delta t}[f])(\vec{q},\vec{p}) = f(\Phi^H_{(t,t_0)}(\vec{q},\vec{p}))

[/itex]

(I assumed that H does not depend on time).

Then my question is: in CM i can solve the evolution equation for f (a PDE) by solving ODEs. Can a similar thing be done in QM with Schrodinger equation? Is there any vector field [itex] \vec{X} [/itex] whose associated flow (which i can find by solving [itex] \frac{d}{dt}\vec{x} = \vec{X} [/itex]) one can use to evolve the initial state ket of QM?

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# Solving PDE by means of ODE

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