Time dependent Schrödinger equation

In this section, illustrative examples are shown of how the wave function evolves under certain scenarios. We present animations of the following scenarios:

• The free wave packet

• Quantum tunneling

• The simple, forced and 2D harmonic oscillators

• Examples of the wave-particle duality

We start by presenting the equation of movement that rules non-relativistic quantum systems, which is Schrödinger's time dependent equation:

$${\rm i} \hbar\frac{\partial \Psi(x,t)}{\partial t} = -\frac{\hbar^2}{2m}\nabla_{x}^2 \Psi(x,t) + U(x,t)\Psi(x,t)$$

which, unlike the wave function, describes the evolutiono of complex instead of a real function. However, one computational problem is that the scale at which quantum phenomena are appreciable is really small. For this reason, a change of variables is suggested. One change of dimensions that I came up with is the following:

$$\xi = \frac{x}{a_0},\tau = \frac{E_0}{\hbar}t,E_0 = \frac{\hbar^2}{ma_0^2}$$

where a_0 is a scale of length appropieate for the problem to be treated. For instance, if we are working with atomic scales, it could be one angstrom or Bohr's radius. These new variables are dimensionless and will be more apropiate for numerical calculations. Along with the axis escalation, the potential and wave functions must also change:

$$V(\xi,\tau) = \frac{U(x,t)}{E_0},\Psi(\xi,\tau) = \sqrt{a^n_0}\psi(x,t),$$

were n is the dimension in which the wave function is being solved. This is to ensure a dimensionless potential and a normalized wave function. Finally, whenever we want to calculate the momentum of the particle, we can use the dimensionless operator:

$$\hat{p_{\varepsilon}} = \frac{a_0}{\hbar}\hat{p} = -i\nabla_{\varepsilon}$$

The resulting dimensionless equation is the following:

$${\rm i} \frac{\partial \Psi(\xi,\tau)}{\partial \tau} = -\frac{1}{2}\nabla_{\xi}^2 \Psi(\xi,\tau) + V(\xi,\tau)\Psi(\xi,\tau)$$

However, to save some writing and have a more comfortable notation, we are going to simply use x and t again, remembering that that we are using code units:

$${\rm i}\frac{\partial \psi(x,t)}{\partial t} = -\frac{1}{2}\nabla^2 \psi(x,t) + V(x,t)\psi(x,t)$$

In the following chapters, we are going to solve this equation with different potentials and initial conditions.