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Show, from it's definition,
[tex] \psi(x,t) = \int dx' G(x,t;x',t_0) \psi(x',t_0)[/tex]
[tex] G(x,t;x',t_0)= \langle x | U(t,t_0) | x' \rangle [/tex]
that the propagator G(x,t;x',t') is the Green Function of the Time-Dependent Schrodinger Equation,
[tex] \left ( H_x - i \hbar \frac{\partial}{\partial t} \right ) G(x,t;x',t_0) = -i \hbar \delta(x-x') \delta(t-t') [/tex]
where H is the Hamiltonian expressed as a differential operator in the x representation. Calculate the propagator for a free particle by this method.Is the best way to do this to substitute that form of the propagator into the Schrodinger eqn and take the derivatives? If so, how are the delta functions defined?
[tex] \psi(x,t) = \int dx' G(x,t;x',t_0) \psi(x',t_0)[/tex]
[tex] G(x,t;x',t_0)= \langle x | U(t,t_0) | x' \rangle [/tex]
that the propagator G(x,t;x',t') is the Green Function of the Time-Dependent Schrodinger Equation,
[tex] \left ( H_x - i \hbar \frac{\partial}{\partial t} \right ) G(x,t;x',t_0) = -i \hbar \delta(x-x') \delta(t-t') [/tex]
where H is the Hamiltonian expressed as a differential operator in the x representation. Calculate the propagator for a free particle by this method.Is the best way to do this to substitute that form of the propagator into the Schrodinger eqn and take the derivatives? If so, how are the delta functions defined?
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