[Physics] How to simulate an atom

atomic-physicshamiltonianschroedinger equationsimulationswavefunction

Let us assume I wish to simulate a Helium atom, since there does not exist a closed-form solution.

However, I presume I would need to simulate the time-dependent Schrodinger wave equation. I would like to know what the time-dependent hamiltonian looks like for this.

I think I would need the terms for electromagnetic, strong, and weak nuclear forces — I do not think I need relativistic corrections, since I'm only interested in local information.

I wish to simulate an equation on simulation gives me the evolving probability distributions of the electrons.

I understand that this might be computationally infeasible, but I still wish to know what the exact PDE is that I need to solve — I'm not looking for approximations!

Best Answer

The Hamiltonian for the He atom is: $$H = -\frac{\hbar^2}{2m_e}(\nabla_1^2 + \nabla_2^2) - \frac{2e^2}{4\pi\epsilon_0 r_1} - \frac{2e^2}{4\pi\epsilon_0 r_2} + \frac{e^2}{4\pi\epsilon_0 r_{12}}$$ where the electrons are denoted 1 and 2, and $r_i$ is the distance to the nucleus at the origin and $r_{12}$ the distance between the electrons. Since the electrons have 3D position vectors $\mathbf{r_i}$ this corresponds to 6 degrees of freedom.

One can reformulate the equation to only act in $r_1,r_2,r_{12}$ coordinates for the spherically symmetric $^1S$ state. With the right approximation methods one can get energy eigenvalues with absurd precision. But I suspect the aim of the question is rather to let loose an electrong wave packet and see how it sloshes around.

Simulating the nucleus requires quite a lot more elaboration. Now you have 4 particles, making a 12D state space (18 in total). The hamiltonian is about the same but with more interaction terms (see section 3.1), corresponding to nucleon-nucleon potentials that generally look somewhat messy.

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Newtonian Mechanics – How to Simulate Rotational Instability

$\vec\omega = I^{-1} \vec L$, and $\vec L$ is constant in the absence of external forces. The bit that I think you're missing is that $I$ rotates with the rigid body, so it is not constant in general and neither is $\vec\omega$.

I played with your online example, and the angular velocity does seem to always remain constant when I'm not poking the block, which is consistent with an inertia tensor that's a scalar multiple of the identity. I've never used Unity, but it seems to support arbitrary tensors of inertia via Rigidbody.inertiaTensor and Rigidbody.inertiaTensorRotation, so maybe you just need to set those properly.

If you end up having to roll your own physics, here's some naive, untested, and potentially wrong pseudocode:

const double time_step = ...;
const Quaternion L = {0, Lx, Ly, Lz};  // angular momentum
const double K1 = 1/I1, K2 = 1/I2, K3 = 1/I3;  // reciprocals of principal-axis moments of inertia; no idea if there's a standard letter for this
Quaternion orientation = {1, 0, 0, 0};
while (1) {
    Quaternion transformed_L = conjugate(orientation) * L * orientation;
    Quaternion transformed_omega = {0, K1 * transformed_L.i, K2 * transformed_L.j, K3 * transformed_L.k};
    Quaternion omega = orientation * transformed_omega * conjugate(orientation);
    orientation = quaternion_exp(time_step * omega) * orientation;
    // ... or orientation = normalize((1 + time_step * omega) * orientation);
    // ... or orientation = normalize((1 + time_step * omega + 0.5 * (time_step * omega)**2) * orientation);
    output(orientation);
}

Best Answer

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Newtonian Mechanics – How to Simulate Rotational Instability

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