Simple Bra-Ket Manipulations
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==Basics== | ==Basics== | ||
===Notation Definitions=== | ===Notation Definitions=== | ||
| + | Note that the ket symbols don't "do" anything except surround a normal column vector. However, the bra symbols do perform an operation: they represent taking the adjoint (conjugate transpose) of a column vector. | ||
*<math>\langle\psi\vert=\vert\psi\rangle^{\dagger}</math> | *<math>\langle\psi\vert=\vert\psi\rangle^{\dagger}</math> | ||
*<math>\hat{A}\vert\psi\rangle=\vert\hat{A}\psi\rangle</math> | *<math>\hat{A}\vert\psi\rangle=\vert\hat{A}\psi\rangle</math> | ||
===From Linear Algebra=== | ===From Linear Algebra=== | ||
====For All Vectors and Matrices==== | ====For All Vectors and Matrices==== | ||
| + | *<math>(\hat{A}\hat{B})\hat{C}=\hat{A}(\hat{B}\hat{C})</math> | ||
*<math>(\hat{A}^{\dagger})^{\dagger}=\hat{A}</math> | *<math>(\hat{A}^{\dagger})^{\dagger}=\hat{A}</math> | ||
*<math>(\hat{A}\hat{B})^{\dagger}=\hat{A}^{\dagger} \hat{B}^{\dagger}</math> | *<math>(\hat{A}\hat{B})^{\dagger}=\hat{A}^{\dagger} \hat{B}^{\dagger}</math> | ||
====For Hermitian Operators==== | ====For Hermitian Operators==== | ||
| − | *<math></math> | + | *<math>\hat{O}=\hat{O}^{\dagger}</math> |
| + | * | ||
Revision as of 21:11, 25 January 2007
Here 
is an operator corresponding to some observable (like 
for energy), and its eigenvalues are λn. The corresponding eigenvectors are 
. 
is a generic state ket. an is the projection of onto . One of the keys to what follows is that the eigenvectors form an orthonormal basis. That just means that any vector can be built from a unique combination of 
, just like any vector in 3-D space can be built from a unique combination of x, y, and z, or any complex number can be built from 1 and i.
Contents |
Basics
Notation Definitions
Note that the ket symbols don't "do" anything except surround a normal column vector. However, the bra symbols do perform an operation: they represent taking the adjoint (conjugate transpose) of a column vector.
