The appearance of the classical Lagrangian and action in the path integral might be surprising given how we had constructed quantum mechanics throughout this book, but it was actually hidden in plain sight from the beginning. In this problem, we will show that we've seen the classical action many times before.

(a) A momentum eigenstate in position space can of course be expressed as

\[\begin{equation*}\langle x \mid pangle=e^{\frac{i}{\hbar} p x} \tag{11.149}\end{equation*}\]

for momentum \(p\) and position \(x\). Show that the classical action for a free particle with momentum \(p\) and energy \(E\) that starts at position 0 at time \(t=0\) and ends at position \(x\) at time \(t=T\) is

\[\begin{equation*}S_{\text {class }}=p x-E T \tag{11.150}\end{equation*}\]

(b) For motion in two spatial dimensions, an eigenstate of the angular momentum operator in "angle" space can be expressed as

\[\begin{equation*}\langle\theta \mid Langle=e^{\frac{i}{\hbar} L \theta} \tag{11.151}\end{equation*}\]

for angular momentum \(L\) and angle \(\theta\). Show that the classical action for a free particle with conserved angular momentum \(L\) and energy \(E\) that starts at angle 0 at time \(t=0\) and ends at angle \(\theta\) at time \(t=T\) is
\[\begin{equation*}S_{\text {class }}=L \theta-E T \text {. } \tag{11.152}\end{equation*}\]
(c) In Sec. 10.4, we introduced the WKB approximation to construct an estimate wavefunction by exploiting properties of the unitary translation operator when momentum is not conserved. We had demonstrated that the wavefunction \(\psi(x)\) can be approximated from its known value at position \(x_{0}\) as
\[\begin{equation*}\psi(x) \approx \exp \left[\frac{i}{\hbar} \int_{x_{0}}^{x} d x^{\prime} \sqrt{2 m\left(E-V\left(x^{\prime}\right)\right)}\right] \psi\left(x_{0}\right) \tag{11.153}\end{equation*}\]
for some potential \(V(x)\) and total energy \(E\). Give this expression for the wavefunction time dependence, can you relate the factor in the exponent to the classical action for the particle in the potential? How is this expression related to the wavefunction constructed from the path integral, Eq. (11.56)?