If you like us, please share us on social media.

The latest UCD Hyperlibrary newsletter is now complete, check it out.

ChemWiki: The Dynamic Chemistry E-textbook > Physical Chemistry > Quantum Mechanics > Quantum States of Atoms and Molecules > 3. The SchrÃ¶dinger Equation > Operators, Eigenfunctions, Eigenvalues, and Eigenstates

MindTouch

Copyright (c) 2006-2014 MindTouch Inc.

http://mindtouch.com

This file and accompanying files are licensed under the MindTouch Master Subscription Agreement (MSA).

At any time, you shall not, directly or indirectly: (i) sublicense,
resell, rent, lease, distribute, market, commercialize or otherwise
transfer rights or usage to: (a) the Software, (b) any modified version
or derivative work of the Software created by you or for you, or (c)
MindTouch Open Source (which includes all non-supported versions of
MindTouch-developed software), for any purpose including timesharing or
service bureau purposes; (ii) remove or alter any copyright, trademark
or proprietary notice in the Software; (iii) transfer, use or export the
Software in violation of any applicable laws or regulations of any
government or governmental agency; (iv) use or run on any of your
hardware, or have deployed for use, any production version of MindTouch
Open Source; (v) use any of the Support Services, Error corrections,
Updates or Upgrades, for the MindTouch Open Source software or for any
Server for which Support Services are not then purchased as provided
hereunder; or (vi) reverse engineer, decompile or modify any encrypted
or encoded portion of the Software.

A complete copy of the MSA is available at http://www.mindtouch.com/msa

- 1. Contributors

The Laplacian operator is called an operator because it does something to the function that follows: namely, it produces or generates the sum of the three second-derivatives of the function. Of course, this is not done automatically; you must do the work, or remember to use this operator properly in algebraic manipulations. Symbols for operators are denoted by a caret ^ over the symbol, unless the symbol is used exclusively for an operator, e.g. \(\nabla\) (del/nabla), or does not involve differentiation, e.g.\(r\) for position.

From Equation (3-19), we can identify the total energy operator, which is called the Hamiltonian operator, Ĥ, as consisting of the kinetic energy operator plus the potential energy operator.

\[\hat {H} = - \frac {\hbar ^2}{2m} \nabla ^2 + \hat {V} (x, y , z ) \tag {3-22}\]

Using this notation we write the Schrödinger Equation (Equation (3-19)) as

\[ \hat {H} \psi (x , y , z ) = E \psi ( x , y , z ) \tag {3-23}\]

The term Hamiltonian, named after the Irish mathematician Hamilton, comes from the formulation of Classical Mechanics that is based on the total energy, H = T + V, rather than Newton's second law, f = ma. Equation (3-23) says that the Hamiltonian operator operates on the wavefunction to produce the energy, which is a number, (a quantity of Joules), times the wavefunction. Such an equation, where the operator, operating on a function, produces a constant times the function, is called an eigenvalue equation. The function is called an eigenfunction, and the resulting numerical value is called the eigenvalue. Eigen here is the German word meaning self or own.

It is a general principle of Quantum Mechanics that there is an operator for every physical observable. A physical observable is anything that can be measured. If the wavefunction that describes a system is an eigenfunction of an operator, then the value of the associated observable is extracted from the eigenfunction by operating on the eigenfunction with the appropriate operator. The value of the observable for the system is the eigenvalue, and the system is said to be in an eigenstate. Equation (3-23) states this principle mathematically for the case of energy as the observable.

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

Last Modified

12:46, 20 May 2014

**Analytical Chemistry**

**Biological Chemistry**

**Inorganic Chemistry**

**Organic Chemistry**

**Physical Chemistry**

**Theoretical Chemistry**

**Cal Poly Pomona**

**Diablo Valley College**

**Florida State U**

**Hope College**

**Howard University**

**Purdue**

**Sacramento City College**

**UC Davis**

**UC Irvine**

**Zumdahl 9 ^{ed}**

An NSF funded Project

- © Copyright 2014 Chemwiki

Unless otherwise noted, content in the UC Davis ChemWiki is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License. Permissions beyond the scope of this license may be available at copyright@ucdavis.edu. Questions and concerns can be directed toward Prof. Delmar Larsen (dlarsen@ucdavis.edu), Founder and Director. Terms of Use