* Partial Differential Equations and Fourier Transform

The Schrödinger equation for a free particle,

$$i\hbar\frac{\partial}{\partial t} \Psi(x,t) =-\frac{\hbar^2\partial_x^2}{2m} \Psi(x,t), \quad \Psi(x,t=0)=\Psi_0(x)$$ (53)

can be solved by Fourier transformation:

$$i\hbar\frac{\partial}{\partial t} \int_{-\infty}^{\infty}\frac{dk}{2\pi} e^{ikx} \tilde{\Psi}(k,t) = - \frac{\hbar^2\partial_x^2}{2m} \int_{-\infty}^{\infty}\frac{dk}... ...hbar^2}{2m} \int_{-\infty}^{\infty}\frac{dk}{2\pi} k^2e^{ikx} \tilde{\Psi}(k,t)$$

$$\leadsto 0= \int_{-\infty}^{\infty}\frac{dk}{2\pi} e^{ikx} \left[ i\hbar\frac... ...\partial t} \tilde{\Psi}(k,t) - \frac{\hbar^2 k^2}{2m}\tilde{\Psi}(k,t) \right]$$

$$\leadsto 0= i\hbar\frac{\partial}{\partial t} \tilde{\Psi}(k,t) - \frac{\hbar^2 k^2}{2m}\tilde{\Psi}(k,t).$$ (54)

The last equation is an ordinary differential equation in $t$ that can be solved easily:

$$0= i\hbar\frac{\partial}{\partial t} \tilde{\Psi}(k,t) - \frac{\hbar^2 k^2}{2m}\tilde{\Psi}(k,t)$$

$$\leadsto \tilde{\Psi}(k,t)=\tilde{\Psi}(k,t=0)e^{-i\omega(k)t},\quad \omega(k):=\frac{\hbar k^2}{2m}.$$ (55)

Here, the initial value $\tilde{\Psi}(k,t=0)$ appears; it is given by our definition for the Fourier transform

$$\tilde{\Psi}(k,t=0) = \int_{-\infty}^{\infty}dx \Psi(x,t=0) e^{-ikx}= \int_{-\infty}^{\infty}dx \Psi_0(x) e^{-ikx}=\tilde{\Psi}_0(k)$$ (56)

Therefore, if we know the initial wave function $\Psi_0(x)$, we know its Fourier transform $\tilde{\Psi}(k,t=0)$ and can calculate the solution $\Psi(x,t)$ at a later time $t>0$ by Fourier back transformation,

$${\Psi(x,t)} = \frac{1}{2\pi}\int_{-\infty}^{\infty}dk \tilde{\Psi}(k,t)e^{ikx} ... ...int_{-\infty}^{\infty}dk e^{ikx} e^{-i\frac{\hbar k^2t}{2m}} \tilde{\Psi}_0(k)}$$

$$= \frac{1}{2\pi}\int_{-\infty}^{\infty}dk e^{ikx} e^{-i\omega(k) t} \int_{-\infty}^{\infty}dx' \Psi(x',t=0) e^{-ikx'}$$

$$= \int_{-\infty}^{\infty}dx' \underline{ \frac{1}{2\pi}\int_{-\infty}^{\infty}dk e^{ik(x-x')-i\omega(k) t} } \Psi(x',t=0)$$

$$= \int_{-\infty}^{\infty}dx' \underline{G(x,x';t,t=0)} \Psi(x',t=0),$$ (57)

where we defined the propagator of the particle which propagates the wave function from its initial form at $t=0$ to its form at a later time $t>0$.