🫐 Hamiltonian eigenstates

What are you looking at?

This demo visualizes the stationary eigenstates of a 2D wave function:

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

Where:

• $\psi(x, y, t)$ is the wave function of the system.
• $\partial^2/\partial x^2 + \partial^2/\partial y^2 = \nabla^2$ is the 2D Laplacian operator.
• $\hbar$ is the reduced Planck’s constant, set to $1$ in the code.
• $m$ is the mass of the particle, set to $1$ in the code.
• $V(x, y)$ is the potential energy function, e.g. an isotropic harmonic oscillator potential:

$$V(x,y)=\dfrac{1}{2}k(x^2+y^2)$$

Demo

Select different eigenstates to explore how their amplitude and phase form distinct wave-like patterns, including the characteristic lobes and nodes of quantum wavefunctions.

Interpretation

For the isotropic harmonic oscillator, the energies are given by $\sqrt{k}(n_x +n_y+1)$:

$E$	$n_x$	$n_y$
0.2236	0	0
0.4472	1	0
0.4472	0	1
0.6708	0	2
0.6708	1	1
0.6708	2	0
0.8944	0	3
0.8944	1	2
0.8944	2	1
0.8944	3	0
1.1180	0	4
1.1180	1	3
1.1180	2	2
1.1180	3	1
1.1180	4	0