I'm not quite satisfied by the derivation I've found in Sakurai (Modern Quantum Mechanics) and was trying to 'derive' it myself. I'd like some help to seal the deal. I've described below what I've done. Please tell me where to go from there.(adsbygoogle = window.adsbygoogle || []).push({});

I know the solution to the Schrodinger equation can be given in terms of a unitary operator,

[tex] \left| \psi(t) \right\rangle = U(t,t_{0})\left| \psi(t_{0}) \right\rangle [/tex]

[tex] \left\langle x \left| \psi(t) \right\rangle = \left\langle x \right| U(t,t_{0})\left| \psi(t_{0}) \rangle [/tex]

which describes time evolution.

I know that I can introduce intermediary time intervals, thus splitting the above term

[tex] \langle x \left| \psi(t) \right\rangle = \int dx_{0} \left\langle x \right| U(t,t_{0}) \left| x_{0} \rangle \left\langle x_{0} \right| \psi(t_{0}) \rangle[/tex]

I here consider

[tex] \left\langle x \right| U(t,t_{0}) \right| x_{0} \rangle = \left\langle x,t \right| x_{0},t_{0} \rangle [/tex]

as the Green's function of this operation.

Now, splitting the interval between x and x_{0} by introducing n points [tex]x_{1}, \dots , x_{n} [/tex] such that

[tex] \left\langle x,t \right| x_{0},t_{0} \rangle = \int dx_{1} \int dx_{2} \dots \int dx_{n} \left\langle x_{1},t_{1} \right| x_{0},t_{0} \rangle \left\langle x_{2},t_{2} \right| x_{1},t_{1} \rangle \dots \left\langle x,t \right| x_{n},t_{n} \rangle [/tex]

So far so good. Now comes the crunch. How does everyone seem to get

[tex]exp(\frac{-iS}{\hbar})[/tex]

as the integrand as they let

[tex]n \rightarrow \infty [/tex].

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# Path integral formulation

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