Simple Bra-Ket Manipulations
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*<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> | ||
+ | *<math>a* b*=(ab)*</math> | ||
+ | *A scalar can be moved around in a set of multiplications, but vectors and matrices often cannot. Inner products, of the form <math>\langle\psi_{1}\vert\psi_{2}\rangle</math>, are scalars. | ||
+ | <math>(\langle\psi_{1}\vert\psi_{2}\rangle)^{\dagger}=\langle\psi_{2}\vert\psi_{1}\rangle</math> | ||
+ | |||
====For Hermitian Operators==== | ====For Hermitian Operators==== | ||
*<math>\hat{O}=\hat{O}^{\dagger}</math> | *<math>\hat{O}=\hat{O}^{\dagger}</math> | ||
− | * | + | *<math>\langle\psi_{1}\vert\hat{O}\vert\psi_{2}\rangle=\langle\hat{O}\psi_{1}\vert\psi_{2}\rangle</math> |
Latest revision as of 02:09, 26 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.
From Linear Algebra
For All Vectors and Matrices
- a * b * = (ab) *
- A scalar can be moved around in a set of multiplications, but vectors and matrices often cannot. Inner products, of the form , are scalars.