Bra-Ket Notation
Used in Quantum Mechanics.
Notation
\(|\psi\rangle\) - ket vector \(\in\mathcal{E}\).
\(\langle\psi|\) - bra vector \(\in\mathcal{E}^*\) dual space.
\(\langle\psi|\psi\rangle\) - bra-ket.
Projections
Position
\(\psi(\vec{r},t)=\langle\vec{r},t|\psi\rangle\).
Momentum
\(\psi(\vec{p},t)=\langle\vec{p},t|\psi\rangle\).
Examples
\(\langle\varphi|\psi\rangle = (\varphi,\psi)\) is the notation for an inner product.
For position/ momentum space representation:
\(\langle\varphi|\psi\rangle = \int\varphi^*(\vec{r},t)\psi(\vec{r},t)d^3\vec{r} = \int\varphi_P^*(\vec{p},t)\psi_P(\vec{p},t)d^3\vec{p}\).
Properties
- \((|\psi\rangle)^*=\langle\psi|\)
- \((a|\psi\rangle)^*=a\langle\psi|\)
- \(|a\psi\rangle=a|\psi\rangle\)
- \(\langle a\psi|=a^*\langle\psi|\)
- \(\langle\varphi|\psi = \overline{\langle\psi|\varphi\rangle}\) since the complex conjugate transposes the swapped bra-kets
- \(\langle\psi|\psi\rangle\) is real, positive
- Normalization of Wavefunction
- Schwarz Inequality
- Triangle Inequality
- Orthogonality of States
- Orthonormality of States
- \(|\varphi\rangle|\psi\rangle\) is allowed (and is shorthand for a tensor product of the spaces) only if \(|\varphi\rangle\) and \(|\psi\rangle\) are from different spaces (e.g. Positon vector and spin vector)
- \(\langle\varphi|\psi\rangle\) is a projection, or checks the overlap, of \(\psi\) on \(\varphi\)
Examples of Bra-Ket Algebra
- \(|\psi\rangle=3i|\varphi_1\rangle-7i|\varphi_2\rangle\)
- \(|\chi\rangle=-|\varphi_1\rangle+2i|\varphi_2\rangle\)
- \(\langle\psi|=-3i\langle\varphi_1|+7i\langle\varphi_2|\)
- \(|\psi+\chi\rangle=(-3i-1)|\varphi_1\rangle-5i|\varphi_2\rangle\).