Simple Bra-Ket Manipulations

Here $\hat{O}$ is an operator corresponding to some observable (like $\hat{H}$ for energy), and its eigenvalues are λ_n. The corresponding eigenvectors are $\vert\phi_{n}\rangle$. $\vert\psi\rangle$ is a generic state ket. a_n is the projection of $\vert\psi\rangle$ onto $\vert\phi_{n}\rangle$. 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 $\vert\phi_{1}\rangle,\vert\phi_{2}\rangle,\vert\phi_{3}\rangle,\dots$, 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.

• $\langle\psi\vert=\vert\psi\rangle^{\dagger}$
• $\hat{A}\vert\psi\rangle=\vert\hat{A}\psi\rangle$

From Linear Algebra

For All Vectors and Matrices

• $(\hat{A}\hat{B})\hat{C}=\hat{A}(\hat{B}\hat{C})$
• $(\hat{A}^{\dagger})^{\dagger}=\hat{A}$
• $(\hat{A}\hat{B})^{\dagger}=\hat{A}^{\dagger} \hat{B}^{\dagger}$
• 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 $\langle\psi_{1}\vert\psi_{2}\rangle$, are scalars.

$(\langle\psi_{1}\vert\psi_{2}\rangle)^{\dagger}=\langle\psi_{2}\vert\psi_{1}\rangle$

For Hermitian Operators

• $\hat{O}=\hat{O}^{\dagger}$
• $\langle\psi_{1}\vert\hat{O}\vert\psi_{2}\rangle=\langle\hat{O}\psi_{1}\vert\psi_{2}\rangle$