Path Integral for Motion as the Harmonic Oscillator

In the same way as done for free motion, see solution (4.95), worth rewritten the actual classical solution (4.106) in terms of relations (4.107) and (4.108), for instance as 

On the other side the classical action of the Lagrangian (4.104) looks like  

in order having classical action in terms of only space-time coordinate of the ending points, one has to replace the end-point velocities in Eq. (4.111) with the aid of relations (4.109) and (4.110) in which the current time is taken as the t = and t = t, respectively thus we firstly get  

Note that the correctness of Eq. (4.116) may be also checked by imposing the limit ft) 0 in which case the previous free motion has to be recovered indeed by employing the consecrated limit 

Quantum Nanochemistry-Volume I Quantum Theory and Observability 

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