As emphasized in this chapter, calculating the path integral for even very simple quantum systems is highly non-trivial. In Example 11.2, we calculated the freeparticle path integral one way, and here we will calculate it in two different ways.

(a) A zero-frequency harmonic oscillator is just a free particle. Take the \(\omega \rightarrow 0\) limit of the harmonic oscillator path integral, Eq. (11.137), and show that it agrees with the result of Example 11.2.

(b) We can calculate this free-particle path integral in a different way. For a free particle, momentum eigenstates are also energy eigenstates, so we can also calculate the path integral using a procedure similar to that used in constructing Eq. (11.79). Generalize this equation for the case of continuous energy or momentum eigenstates and use it to evaluate the path integral for the free particle. Does it agree with your result in part (a)?