Introduction: Two Ways to Combine States
Dirac notation elegantly distinguishes between two fundamental operations: the inner product \(\langle\phi|\psi\rangle\) (bra times ket = scalar) and the outer product \(|\psi\rangle\langle\phi|\) (ket times bra = operator). Both are essential in quantum mechanics.
The Inner Product (Bracket)
The inner product combines a bra and a ket to produce a complex number:
\[\langle\phi|\psi\rangle = \sum_i \bar{\phi}_i \psi_i\]Properties:
- \(\langle\phi|\psi\rangle = \overline{\langle\psi|\phi\rangle}\)
- \(\langle\psi|\psi\rangle \geq 0\) (equals 0 iff \(|\psi\rangle = 0\))
- \(\langle\phi|c\psi\rangle = c\langle\phi|\psi\rangle\)
- \(\langle c\phi|\psi\rangle = \bar{c}\langle\phi|\psi\rangle\)
The Outer Product (Projector)
The outer product combines a ket and a bra to produce an operator (matrix):
\[|\psi\rangle\langle\phi| = \begin{pmatrix} \psi_1 \\ \psi_2 \\ \vdots \end{pmatrix} (\bar{\phi}_1, \bar{\phi}_2, \ldots) = \begin{pmatrix} \psi_1\bar{\phi}_1 & \psi_1\bar{\phi}_2 & \cdots \\ \psi_2\bar{\phi}_1 & \psi_2\bar{\phi}_2 & \cdots \\ \vdots & \vdots & \ddots \end{pmatrix}\]If \(|\psi\rangle\) is normalized, \(|\psi\rangle\langle\psi|\) is a projection operator onto the state \(|\psi\rangle\).
Worked Examples
Example 1: Inner Product Calculation
Let \(|\phi\rangle = \begin{pmatrix} 1 \\ i \end{pmatrix}\) and \(|\psi\rangle = \begin{pmatrix} 2 \\ 1 \end{pmatrix}\):
\[\langle\phi|\psi\rangle = (1)(-i) \cdot \begin{pmatrix} 2 \\ 1 \end{pmatrix} = 1 \cdot 2 + (-i) \cdot 1 = 2 - i\]Example 2: Outer Product as Matrix
\[|0\rangle\langle 0| = \begin{pmatrix} 1 \\ 0 \end{pmatrix}(1, 0) = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}\]This projects any state onto \(|0\rangle\).
Example 3: Completeness Relation
For an orthonormal basis \(\{|n\rangle\}\):
\[\sum_n |n\rangle\langle n| = I\]In \(\mathbb{C}^2\): \(|0\rangle\langle 0| + |1\rangle\langle 1| = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} + \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\)
The Quantum Connection
The inner product \(\langle\phi|\psi\rangle\) is the probability amplitude; its squared modulus \(|\langle\phi|\psi\rangle|^2\) is the probability of finding \(|\psi\rangle\) in state \(|\phi\rangle\). The outer product \(|n\rangle\langle n|\) is the measurement projector for outcome \(n\). After measuring and getting outcome \(n\), the state becomes \(|n\rangle\langle n|\psi\rangle\) (up to normalization).