Simple Bra-Ket Manipulations
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| − | + | Here <math>\hat{O}</math> is an operator corresponding to some observable (like <math>\hat{H}</math> for energy), and its eigenvalues are <math>\lambda_{n}</math>. The corresponding eigenvectors are <math>\vert\phi_{n}\rangle</math>. <math>\vert\psi\rangle</math> is a generic state ket. <math>a_{n}</math> is the projection of <math>\vert\psi\rangle</math> onto <math>\vert\phi_{n}\rangle</math>. 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 <math>\vert\phi_{1}\rangle,\vert\phi_{2}\rangle,\vert\phi_{3}\rangle,\dots</math>, 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. | |
| − | + | ==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. | ||
| + | *<math>\langle\psi\vert=\vert\psi\rangle^{\dagger}</math> | ||
| + | *<math>\hat{A}\vert\psi\rangle=\vert\hat{A}\psi\rangle</math> | ||
| + | ===From Linear Algebra=== | ||
| + | ====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}\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==== | ||
| + | *<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.
