# Schrödinger Equation

The Schrödinger equation is the backbone of quantum mechanics and is the foundation of computational quantum chemistry. It is an equation that describes the behavior of quantum systems, and it is used to predict the behavior of atoms and molecules.

We will only focus on time-independent Schrödinger equation, by which we mean the physical system does not evolve over time. It is used to calculate the energy levels and wave functions of a system that is in a steady state, such as an atom or a molecule that we live with in the normal life.

Understanding the Schrödinger equation is essential for anyone interested in AI for science. Let's kick-start!

Time-independent Schrödinger equation can be defined by:

$$
\hat H | \boldsymbol{\Psi} \rangle = E | \boldsymbol{\Psi} \rangle
$$

where $\hat H$ is the Hamiltonian operator, and $E$ is the eigenvalue of this operator, representing the energy level of the system.

What does this equation say? The equation says that the wavefunction $|\Psi\rangle$ is the eigenstate of Hamiltonian.

Let's recall the operator part of the previous Chapter: if the wavefunction is the eigenstate of Hamiltonian, then measure of the energy out of the system will produce a deterministic value, that is, the eigenvalue of the operator.

Intrisically, this corresponds to the meaning of time-independent system. The system is already in a steady state. Therefore, measuring the energy at any time will obtain deterministically the same value.

Solving the above equation is equivalent to solving the eigenvalue/eigenvector problem for the Hamiltonian operator. Among the eigenvalues, the minimum one is the ground-state energy of the system.

For different systems, the form of the Hamiltonian operator is different: each component should has its own physical meaning. We will see some examples below.

>**Example** (Hamiltonian for Hydrogen atom).
>There are three components of the Hamiltonian for Hydrogen atom: nuclear kinetic energy, electronic kinetic energy and the electron-nucleus attraction.
>Due to the Born-Oppenheimer approximation, the nucleus is much bigger and is assumed frozen (it is analogous to that in your daily life, you will assume the earth is not moving under your feet).
>Therefore, the nuclear kinetic energy is not considered. the Hamiltonian can be expressed as follows. The first part is the electronic kinetic energy, and the second part is the electron-nucleus attraction energy.

> $$\hat H = -\dfrac{1}{2}\nabla^2 - \dfrac{1}{|r-R|},$$

> Here, $\nabla^2$ is the Laplacian operator in physics that consists of the second-order derivative of wavefuncion, and the second term is anti-proportional to the distance between the electron and the nucleus, with $r$ and $R$ denoting the location of electron and nucleus respectively. In physics language, the second term is called the Coulomb potential operator.

> For example, in 1D case, the Laplacian operator is just the second-order derivative. When the Hamiltonian operator is fully expanded, the Schrödinger Equation looks like:


> It is interesting to note that you can see the wavefunction from 2 points of view. From the braket notation, the Schrödinger Equation is just saying that the solution is an eigenvector of Hamiltonian. If you concretize it by replacing the operator with its concrete form, the equation becomes a differential equation.

>**Example** (Hamiltonian for water molecule).

> The water molecule is more complicated than Hydrogen atom. There are in total 8 nuclei and 10 electrons.
> So apart from the energy considered above, we need to consider the interactions effect among the electrons and the nuclei. There will be additional terms to express the interactions, namely, electron-electron repulsion, and nucleus-nucleus repulsion. The Hamiltonian looks like:
>
>\begin{equation}
\hat H = \underbrace{-\sum_{i=1}^{10}\dfrac{1}{2}\nabla^2_{r_i}}_{\text{Kinetic energy} \\ \text{of electron i}} - \underbrace{\sum_{i=1}^{10} \dfrac{8}{|r_i - R_{O}|}}_{\text{Electron attraction} \\ \text{to the Oxygen atom}} - \underbrace{\sum_{k=1}^2\sum_{i=1}^{10} \dfrac{1}{r_i-R_{H_k}} }_{\text{Electron attraction to} \\ \text{two Hydrogen atoms}} + \underbrace{ \sum_{i=1}^{10} \sum_{j=1}^{10} \dfrac{1}{\vert r_i-r_j \vert}}_{\text{Electron-electron} \\ \text{repulsion}} + \underbrace{ \sum_{i=1}^{3} \sum_{j=1}^{3} \dfrac{e_ie_j}{\vert R_i-R_j \vert}}_{\text{Nucleus-nucleus} \\ \text{repulsion}}
\end{equation}

### Wave function for multi-particle system

When dealing with multiple particles (usually electrons here), the wave function is a function of the positions of all the particles in the system. For an $N$-particle wave funtion $\boldsymbol \Psi(\boldsymbol r_1, \boldsymbol r_2, \cdots, \boldsymbol r_N)$, if the particles are indistinguishable, the marginal density can be defined by

$$
n(\boldsymbol r_1) = \int | \boldsymbol \Psi(\boldsymbol r_1, \boldsymbol r_2, \cdots, \boldsymbol r_N) |^2 d\boldsymbol r_2 \cdots d\boldsymbol r_N.
$$

$n(\boldsymbol r_1)$ is the probability density of the particle at position $\boldsymbol r_1$.

The wave function must satisfy two specific conditions:

* **Normalization condition**:

$$
\int |\boldsymbol \Psi(\boldsymbol r)|^2 d\boldsymbol r = 1,
$$

since the interpretation of the wave function is that the square of it is a probability amplitude.

* **Anti-symmetry condition for fermions**:

$$\hat P_{12}\boldsymbol \Psi(\boldsymbol r_1,\boldsymbol r_2)= \boldsymbol \Psi(\boldsymbol r_2,\boldsymbol r_1) = -\boldsymbol \Psi(\boldsymbol r_1,\boldsymbol r_2),$$

where $\hat P_{12}$ is the permutation operator, which swaps the position of the two particles.

The second constraint is due to the nature of *fermions*. This is the first time this concept appears. Let's explain!

Fermions are a type of particle that obey the Pauli Exclusion Principle, which states that no two identical fermions can occupy the same quantum state simultaneously. In fact, many particles in our living world are fermions. Electrons, neutrons and protons are fermions.

To fully explain the second constraint, we should dive into the concept of *spin*, an intrinsic property of particle and it would be another long adventure. For now, let's just remember the conclusion it derives: If you swap the position of 2 fermions, the corresponding wavefunction should change the sign.