- #1
bolbteppa
- 309
- 41
Two questions about the path integral:
a) Intuitively, how does one (as Landau does) start quantum mechanics from Heisenberg's uncertainty principle, which states there is no concept of the path of a particle, derive Schrodinger equation [itex] i \hbar \tfrac{\partial \psi}{\partial t} = H \psi[/itex] the operator solution [itex] \psi = Ae^{\tfrac{-i}{\hbar}Ht}\psi_0[/itex] and from this basis express [itex] Ae^{\tfrac{-i}{\hbar}Ht}[/itex] as an integral over all possible paths. In other words, starting from the assumption that no path exists we construct the wave function in terms of an integral over all paths (even though no true path exists) and get the right answer, mathematically how does that make sense?
b) Is there a way to derive the path integral from the quasi-classical approximation [itex] \psi = e^{\tfrac{i}{\hbar}S}[/itex]? I mean by using [itex] \psi = e^{\tfrac{i}{\hbar}S} \cdot 1 = e^{\tfrac{i}{\hbar}S} \cdot e^{\tfrac{i}{\hbar}S_1} \cdot e^{-\tfrac{i}{\hbar}S_1} [/itex] where [itex]S_1[/itex] is the action for a nearby path, then extending over all actions over all paths, or something?
a) Intuitively, how does one (as Landau does) start quantum mechanics from Heisenberg's uncertainty principle, which states there is no concept of the path of a particle, derive Schrodinger equation [itex] i \hbar \tfrac{\partial \psi}{\partial t} = H \psi[/itex] the operator solution [itex] \psi = Ae^{\tfrac{-i}{\hbar}Ht}\psi_0[/itex] and from this basis express [itex] Ae^{\tfrac{-i}{\hbar}Ht}[/itex] as an integral over all possible paths. In other words, starting from the assumption that no path exists we construct the wave function in terms of an integral over all paths (even though no true path exists) and get the right answer, mathematically how does that make sense?
b) Is there a way to derive the path integral from the quasi-classical approximation [itex] \psi = e^{\tfrac{i}{\hbar}S}[/itex]? I mean by using [itex] \psi = e^{\tfrac{i}{\hbar}S} \cdot 1 = e^{\tfrac{i}{\hbar}S} \cdot e^{\tfrac{i}{\hbar}S_1} \cdot e^{-\tfrac{i}{\hbar}S_1} [/itex] where [itex]S_1[/itex] is the action for a nearby path, then extending over all actions over all paths, or something?