giova7_89
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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:
<br /> <br /> i\hbar\frac{d}{dt}\psi = \hat{H}\psi<br /> <br />
and this is a evolution equation where \psi 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 |\vec{x}> basis, one gets the wave equation that everyone knows, where the hamiltonian on this basis acts on the state ket as
<br /> <br /> -\frac{\hbar^2}{2m}\nabla^2 + U(\vec{x})<br /> <br />
does.
In CM one has:
<br /> <br /> \frac{d}{dt}f = \hat{L}f<br /> <br />
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 t0 to time t do not depend on t0 explicitly, otherwise I should have added \partial_t f to that equation)
If one represents that equation in the |\vec{q},\vec{p}> basis one gets:
<br /> <br /> \frac{d}{dt}f (\vec{q},\vec{p}) = \{f(\vec{q},\vec{p}),H(\vec{q},\vec{p})\}<br /> <br />
and if I solve Hamilton equations and get the hamiltonian flow \Phi^H_{(t,t_0)}, I know that the solution to the equation with initial condition f0 is:
<br /> <br /> (e^{\hat{L}\Delta t}[f])(\vec{q},\vec{p}) = f(\Phi^H_{(t,t_0)}(\vec{q},\vec{p}))<br /> <br />
(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 \vec{X} whose associated flow (which i can find by solving \frac{d}{dt}\vec{x} = \vec{X}) one can use to evolve the initial state ket of QM?
<br /> <br /> i\hbar\frac{d}{dt}\psi = \hat{H}\psi<br /> <br />
and this is a evolution equation where \psi 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 |\vec{x}> basis, one gets the wave equation that everyone knows, where the hamiltonian on this basis acts on the state ket as
<br /> <br /> -\frac{\hbar^2}{2m}\nabla^2 + U(\vec{x})<br /> <br />
does.
In CM one has:
<br /> <br /> \frac{d}{dt}f = \hat{L}f<br /> <br />
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 t0 to time t do not depend on t0 explicitly, otherwise I should have added \partial_t f to that equation)
If one represents that equation in the |\vec{q},\vec{p}> basis one gets:
<br /> <br /> \frac{d}{dt}f (\vec{q},\vec{p}) = \{f(\vec{q},\vec{p}),H(\vec{q},\vec{p})\}<br /> <br />
and if I solve Hamilton equations and get the hamiltonian flow \Phi^H_{(t,t_0)}, I know that the solution to the equation with initial condition f0 is:
<br /> <br /> (e^{\hat{L}\Delta t}[f])(\vec{q},\vec{p}) = f(\Phi^H_{(t,t_0)}(\vec{q},\vec{p}))<br /> <br />
(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 \vec{X} whose associated flow (which i can find by solving \frac{d}{dt}\vec{x} = \vec{X}) one can use to evolve the initial state ket of QM?