please do parts a-c! **2. Consider the plane wave function of a free particle of mass \( m \) and characterized by a positive constant \( k_0 \)**\[\psi(x, t) = \exp \left[ i k_0 x - i \hbar^2 k_0^2 t / 2m \right]\]**(a) Find its momentum space wave function (Fourier transform) \(\Phi(k, t)\) in terms of a delta function.**\[ \left\{ \quad 2\pi \delta(a) = \int_{-\infty}^{+\infty} \exp \left[ i a b \right] db \quad \right\}\]**(b) Find the probability current \( J(x, t) = \left(i \hbar / 2m \right) \left[ \psi \left( \partial \psi^* / \partial x \right) - \psi^* \left( \partial \psi / \partial x \right) \right] \), simplifying the expression as much as possible.****(c) If the particle entered a region with constant potential \( V_0 > k_0^2 / 2m \), what would its wave function in this region become?**

2. Consider the plane wave function of a free particle of mass m and characterized by positive constant ko y(x, t)= exp [ i ko x-ih² ko²t/2m] (a) Find its momentum space wave function (Fourier transform) (k, t) in terms of a delta function. + ∞0 { 2 nd(a) = exp [ia b] db } - 00 (b) Find the probability current J(x, t) = (i ħ/2m) [y (@y*/əx) − y*(@y/dx)], simplifying the

2. Consider the plane wave function of a free particle of mass m and characterized by positive constant ko y(x, t)= exp [ i ko x-ih² ko²t/2m] (a) Find its momentum space wave function (Fourier transform) (k, t) in terms of a delta function. + ∞0 { 2 nd(a) = exp [ia b] db } - 00 (b) Find the probability current J(x, t) = (i ħ/2m) [y (@y/əx) − y(@y/dx)], simplifying the