Question: 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}
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?
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71 a The lowering operator can be expressed in the momentum basis as beginequationhatafracisqrt2 m hbar omega hatpsqrtfracm omega2 hbarhatxfracisqrt2 ... View full answer
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