7.1 Let's first consider a free particle, whose wavefunction can be expressed as

\begin{equation*}
\psi(x, t) = \int_{-\infty}^{\infty} \frac{d p}{\sqrt{2 \pi \hbar}} g(p) e^{-i \frac{E_{p} t - p x}{\hbar}}
\end{equation*}


where \(g(p)\) is a complex-valued, \(L^{2}\)-normalized function of momentum \(p\) and \(E_{p}=p^{2} / 2 m\), the kinetic energy.

(a) Assume that this wavefunction is a coherent state at time \(t=0\) :

\[\begin{equation*}\hat{a}|\psiangle=\lambda|\psiangle, \tag{7.127}\end{equation*}\]

where \(\hat{a}\) is the lowering operator we introduced with the harmonic oscillator and \(\lambda\) is a complex number. What differential equation must \(g(p)\) satisfy for this wavefunction to be a coherent state? Using the results of Sec. 6.2.2, can you explicitly solve this differential equation?

(b) What is the speed of the center-of-probability of this initial coherent state, at time \(t=0\) ? What is the acceleration of the center-of-probability for any time \(t\) ? From these results, provide an interpretation of \(\lambda\).

(c) We had demonstrated that the eigenstates of \(\hat{a}^{\dagger}\) had problems in the harmonic oscillator. For the free particle, what are the states that are eigenstates of the raising operator, \(\hat{a}^{\dagger}\) ? Are they allowed in this case?