The wave packet

Let us come back to de Broglie's idea to describe a particle as a wave or better as a superposition of waves. We assume that a particle with energy $E=p^2/2m$ can be described by a function that is a superposition of plane waves,

$$\Psi(x,t) = \int_{-\infty}^{\infty}dk a(k) e^{i(kx-\omega(k) t)},\quad \hbar\omega(k)=E=\hbar^2k^2/(2m).$$ (17)

We have used the relation between momentum and wave vector, $p=\hbar k$, and the relation between energy and angular frequency, $E=\hbar \omega$. As with waves, the angular frequency $\omega$ in general depends on the wave length and therefore $\omega=\omega(k)$. For simplicity, we adopted a one-dimensional version. Note that the time evolution of a single plane wave $e^{i(kx -\omega t)}$ goes with the minus sign. We would like to know the time evolution of the function $\Psi(x,t)$, i.e. to find its equation of motion. Equations of motion often represent fundamental laws in physics, like Newton's $F=ma$, which is a second order differential equation $\ddot{x}=(1/m)F(x)$. We therefore differentiate (1.17) with respect to time (we write $\partial_t$ for $\partial/\partial t$ etc.

$$i\hbar\partial_t \Psi(x,t) = \int dk a(k) \hbar \omega(k)e^{i(kx-\omega(k) t)} =\int dk a(k) E(k) e^{i(kx-\omega(k) t)}=$$

$$= \int dk a(k) \frac{\hbar^2 k^2}{2m} e^{i(kx-\omega(k) t)} =-\frac{\hbar^2\partial_x^2}{2m}\int dk a(k) e^{i(kx-\omega(k) t)}=$$

$$= -\frac{\hbar^2\partial_x^2}{2m} \Psi(x,t).$$ (18)

So far we have only considered a particle that only has kinetic energy $E=p^2/2m=E(k)=(\hbar k)^2/2m$. In general, a particle can have both kinetic energy and potential energy $V(x)$.

Example: A harmonic oscillator with angular frequency $\omega$ and mass $m$ in one dimension has the potential energy $V(x)=(1/2)m\omega^2x^2$.

We now postulate that the above equation for a free particle (zero potential energy),

$$i\hbar\partial_t \Psi(x,t) =\int dk a(k) E(k) e^{i(kx-\omega(k) t)},$$

has to be generalized by replacing $E(k)$ with the total energy $E(k)+V(x)$ for a particle in a non-zero potential. Then, the equation of motion becomes

$$i\hbar\partial_t \Psi(x,t) = \int dk a(k) [E(k)+V(x)] e^{i(kx-\omega(k) t)}=$$

$$= [-\frac{\hbar^2\partial_x^2}{2m} + V(x) ]\Psi(x,t).$$ (19)

The equation

$$i\hbar\frac{\partial}{\partial t} \Psi(x,t) =\left[-\frac{\hbar^2\partial_x^2}{2m} + V(x) \right]\Psi(x,t)$$ (20)

is called Schrödinger equation and is one of the most important equations of physics at all. We only have given the one-dimensional version of it so far, the generalization to two or three dimensions is not difficult: the variables $x$ and $k$ become vectors ${\bf x}$ and ${\bf k}$. Instead of the differential operator $\partial_x^2$, one has $\partial_x^2+\partial_y^2$ in two or $\partial_x^2+\partial_y^2+\partial_z^2$ in three dimensions. This is nothing else but the Laplace operator $\Delta=\partial_x^2+\partial_y^2+...$. The Schrödinger equation reads

$$i\hbar\frac{\partial}{\partial t} \Psi({\bf x},t) =\left[-\frac{\hbar^2\Delta}{2m} + V({\bf x}) \right]\Psi({\bf x},t).$$ (21)