The Schrodinger Equation of Quantum Mechanics

The Schrodinger equation is the basic defining equation of non-relativistic quantum mechanics. It is a partial differential equation which determines the of the quantum wavefunction $\Psi$ as a function of time. The wavefunction parameterizes the probability density of observing a certain configuration of a quantum system in the sense that the probability density is the square magnitude of the wavefunction. This correspondence is known as the Born rule, or 2-norm probability formalism.

Introduction to the Schrodinger Equation

The Schrodinger equation is

\begin{equation} i \frac{\partial \Psi}{\partial t} = \hat{H} \Psi \end{equation}

where $\hat{H}$ is a linear operator known as the Hamiltonian. Typically, $\hat{H}$ is a second-order differential operator in terms of the spatial coordinates of a set of quantum particles. Here and in what follows, the units are chosen such that $\hbar=1$.

Single-particle Schrodinger Equation

Arguably the simplest example of the Schrodinger equation is the case of a single quantum particle moving in a in the presence of a potential $V(\mathbf{r})$. For this case, the Schrodinger equation takes the form:

\begin{equation} i \frac{\partial \Psi(\mathbf{r},t)}{\partial t} = \Big[\frac{-1}{2m}\nabla^2 + V(\mathbf{r})\Big] \Psi(\mathbf{r},t) \end{equation}

Having determined the wavefunction $\Psi(\mathbf{r},t)$, the probability to find the particle at the position $\mathbf{r}$ is given by $p(\mathbf{r},t) = |\Psi(\mathbf{r},t)|^2$.

Stationary Solutions and Time-Independent Equation

Of the many possible solutions of the Schrodinger equation, a special set of solutions are those known as stationary states or energy eigenstates. These are solutions of the form \begin{equation} \Psi(\mathbf{r},t) = \psi(\mathbf{r}) e^{-i E t} \end{equation} where $E$ is a constant interpreted as the energy of the quantum system. The reason such solutions are called stationary is that they correspond to a time-independent probability distribution $p(\mathbf{r})=|\psi(\mathbf{r}) e^{-i E t}|^2 = |\psi(\mathbf{r})|^2$.

For solutions of this form, the spatially-dependent part of the solution $\psi(\mathbf{r})$ is determined by the solution of the following eigenvalue equation, known as the time-independent Schrodinger equation: \begin{equation} \hat{H} \psi(\mathbf{r}) = E \psi(\mathbf{r}) \ . \end{equation} Thus the set of eigenvectors and eigenvalues of the Hamiltonian operator $\hat{H}$ determines the set of such solutions.

Many-Body Schrodinger Equation

In terms of modern research and applications the most important case of the Schrodinger equation is when it describes many interacting quantum particles. The particles could be electrons and protons, as in chemistry or materials science, or whole atoms in atomic gas systems or astrophysics. The many-body setting is where high-order tensors naturally arise, as we will see below.

The solution to the many-body Schrodinger equation is the many-body wavefunction \begin{equation} \Psi(\{\mathbf{r}\},t)= \Psi(\mathbf{r}_1,\mathbf{r}_2,\mathbf{r}_3,\ldots,\mathbf{r}_N,t) \end{equation} where $N$ is the number of particles and $\mathbf{r}_j$ is the location of particle $j$. The wavefunction can be interpreted as parameterizing the probability density of finding the particles at the positions $\{\mathbf{r}\}$ at time $t$ through the relation: \begin{equation} p(\{\mathbf{r}\},t)=|\Psi(\{\mathbf{r}\},t)|^2 \ . \end{equation}

Additional symmetry constraints on the solution $\Psi$ are also required to obtain correct results for a system of identical particles, such as electrons. If the solution is required to be symmetric under exchange of particle labels, the particles are known as bosons, and if antisymmetric, the particles are known as fermions. (Electrons and protons are fermions, whereas certain isotopes of neutral atoms are bosons when approximated as a single particle.)

The most common form of the many-body Schrodinger equation is: \begin{equation} i\frac{\partial}{\partial t} \Psi(\{\mathbf{r}\},t) = \Big[ \sum_{j=1}^N \left( -\frac{1}{2 m_j} \nabla^2 + v(\mathbf{r}_j) \right) + \frac{1}{2} \sum_{i=1}^N \sum_{j=1}^N u(\mathbf{r}_i,\mathbf{r}_j) \Big] \Psi(\{\mathbf{r}\},t) \end{equation} where $m_j$ is the mass of particle number $j$. The operator in square brackets on the right-hand side of the equation is the many-body Hamiltonian $\hat{H}$.

The first term in the Hamiltonian, involving a single sum over $j$, is known as the single-particle or non-interacting part of the Hamiltonian. The second term involving a double sum is the 2-particle or interaction term.

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