# 3.3: The Schrödinger Equation is an Eigenvalue Problem

Skills to Develop

- Classical-Mechanical Quantities Are Represented by Linear Operators in Quantum Mechanics

As per the definition, an operator acting on a function gives another function, however a special case occurs when the generated function is proportional to the original

\[\hat{A}\psi \propto \psi\label{3.3.1a}\]

This case can be expressed in terms of a equality by introducing a proportionality constant \(k\)

\[\hat{A}\psi = k \psi\label{3.3.1b}\]

The \(k\) constant that is corresponds to the specific \(\psi\) that solves this equation since not all equations will do so. In this special case, \(\psi\) is known as an **eigenfunction** (typically one of many function that satisfy Equation \(\ref{3.3.1b}\)) and the constant \(k\) is called an **eigenvalue** (these terms are hybrids with German, the purely English equivalents being characteristic function' and `characteristic value'). Solving eigenvalue problems is a key objective from linear algebra courses. To every dynamical variable \(a\) in quantum mechanics, there corresponds an eigenvalue equation, usually written

\[\hat{A}\psi=a\psi\label{3.3.2}\]

The \(a\) eigenvalues represents the possible measured values of the \(A\) operator. Classically, \(a\) would be allowed to vary continuously, but in quantum mechanics, \(a\) typically has only a sub-set of allowed values (hence the quantum aspect). Both time-dependent and time-independent Schrödinger equations are the best known instances of an eigenvalue equations in quantum mechanics, with its eigenvalues corresponding to the allowed energy levels of the quantum system.

\[ { \left[-\dfrac{\hbar^2}{2m}\nabla^2+V(\vec{r})\right]\psi(\vec{r})=E\psi(\vec{r})} \label{3.3.3}\]

The object on the left that acts on \(\psi (x)\) is an example of an *operator*.

\[ \left[-\dfrac{\hbar^2}{2m}\nabla^2+V(\vec{r})\right] \label{3.3.4}\]

In effect, what is says to do is "take the second derivative of \(\psi (x)\), multiply the result by \(-(\hbar^2 /2m)\) and then add \(V(x)\psi (x)\) to the result of that." Quantum mechanics involves many different types of operators. This one, however, plays a special role because it appears on the left side of the Schrödinger equation. It is called the *Hamiltonian operator* and is denoted as

\[\hat{H}=-\dfrac{\hbar^2}{2m}\nabla^2+V(\vec{r}) \label{3.3.5}\]

Therefore, the time-dependent Schrödinger equation can be (and it more commonly) written as

\[\hat{H} \psi (x,t) = i \hbar \dfrac{\partial}{\partial t} \psi(x,t) \label{3.3.6a}\]

and the time-independent Schrödinger equation

\[\hat{H}\psi (x)=E \psi (x) \label{3.3.6b}\]

Correspondence Principle

Note that \(\hat{H}\) is derived from the classical energy \(p^2 /2m+V(x)\) simply by replacing \(p \rightarrow -i\hbar(d/dx)\). This is an example of the *Correspondence Principle* initially proposed by Niels Bohr that states that the behavior of systems described by quantum theory reproduces classical physics in the limit of large quantum numbers.

### Commonly Used Operators

Although we could theoretically come up with an infinite number of operators, in practice there are a few which are much more important than any others.

**Linear Momentum:**

The linear momentum operator of a particle moving in one dimension (the \(x\)-direction) is

\[\hat p_x = -i \hbar \dfrac{\partial}{\partial x} \label{3.3.7}\]

and can be generalized in three dimensions:

**Kinetic Energy**

Classically, the kinetic energy of a particle moving in one dimension (the \(x\)-direction), in terms of momentum, is

\[KE_{classical}= \dfrac{p_x^2}{2m} \label{3.3.9}\]

Quantum mechanically, the corresponding kinetic energy operator is

\[ \hat {KE}_{quantum}= -\dfrac{\hbar^2}{2m} \dfrac{\partial^2}{\partial x^2}\label{3.3.10}\]

and can be generalized in three dimensions:

\[ \hat {KE}_{quantum}= -\dfrac{\hbar^2}{2m} \nabla^2 \label{3.3.11}\]

**Angular Momentum:**

Angular momentum requires a more complex discussion, but is the cross product of the position operator \(\hat{\vec{r}}\) and the momentum operator \(\hat p\)

\[ \hat {\vec{L}} = -i \hbar ( \vec{r} \times \nabla) \label{3.3.12}\]

**Hamiltonian:**

The Hamiltonian operator corresponds to the total energy of the system

and it represents the total energy of the particle of mass \(m\) in the potential \(V(x)\). The Hamiltonian in three dimensions is

\[\hat{H}=-\dfrac{\hbar^2}{2m}\nabla^2+V(\vec{r}) \label{3.3.5a}\]

**Energy:**

The energy operator from the time-dependent Schrödinger equation

The right hand side of Equation \(\ref{3.3.5}\) is the Hamiltonian Operator. In addition determining system energies, the Hamiltonian operator dictates the time evolution of the wavefunction

\[ \hat {H} \Psi(x,t) = i \hbar \dfrac{\partial \Psi(x,t)}{\partial t} \label{3.3.15}\]

This aspect will be discussed in more detail elsewhere.

Eigenstate, Eigenvalues, Wavefunctions, and Measurables

Often in discussions of quantum mechanics, the terms *eigenstate *and *wavefunction *are used interchangeably. The term eigenvalue is used to disignate the value of measurable quantity associated with the wavefunction. For example, when discussing the eigenstates of the Hamiltonian (\hat{H}\), the associated eigenvalues represent energies and within the context of the momentum operators, the associated eigenstate refer to the momentum of the particle.

### Contributors

Seymour Blinder (Professor Emeritus of Chemistry and Physics at the University of Michigan, Ann Arbor)

Adapted from "Quantum States of Atoms and Molecules" by David M. Hanson, Erica Harvey, Robert Sweeney, Theresa Julia Zielinski