Dirac notation

Brackets: Bra’s and Kets

• Inner product in matrix notation (separate “vectors”)

\[ \langle \alpha | \beta \rangle = \pmatrix{a^*_1 & a^*_2 & \dots & a^*_n} \pmatrix{b_1 \\ b_2 \\ \vdots \\ b_n} = a^*_1 b_1 + a^*_2 b_2 + \dots a^*_n b_n \]

• “bra” acts on the ket by row vector multiplication
  
• “bra” vector is separate from the “ket” vector: bra sits in a dual vector space
• Now with possible infinite basis:

\[ \langle\alpha| = \sum_j a_j^* (\dots)_j \quad\longrightarrow\quad \langle \alpha | = \int \alpha^* (\dots) dx \]

Brackets: Bra’s and Kets

• Kets are vectors in vector space
• Bra’s are vectors in dual space
• In finite dimensions:
  • kets are column vectors
  • bra’s are complex conjugate row vectors

\[ \begin{aligned} \langle \textrm{bra} | &= \langle \alpha | = \pmatrix{a^*_1 & a^*_2 & \dots & a^*_n}\\ & \\ | \textrm{ket} \rangle &= | \beta \rangle = \pmatrix{b_1 \\ b_2 \\ \vdots \\ b_n} \end{aligned} \]

Dual space and Hermitian conjugates

• Converting a \(|\textrm{ket}\rangle\) to a \(\langle bra|\) and vice versa:

\[ \langle \alpha | = | \alpha \rangle^\dagger \]

• An operator acting on a \(\langle bra|\):

\[ \langle \alpha | \hat{Q}^\dagger = \langle \hat{Q} \alpha | = \left(\hat{Q} | \alpha \rangle\right)^\dagger \]

\(\longrightarrow\) operators can act to the left as this is allowed by associativity

• Why is this? See definition of Hermitian conjugate of operators:

\[ \langle \hat{Q}^\dagger \alpha | \beta \rangle = \langle \alpha | \hat{Q} \beta \rangle \]

In finite dimensions: matrix-formalism

• Example in two dimensions, an operator acting on a \(| \alpha \rangle = \pmatrix{a_1 \\ a_2}\):

\[ \hat{Q} | \alpha \rangle = Q {\bf a} = \pmatrix{Q_{11} & Q_{12} \\ Q_{21} & Q_{22}} \pmatrix{a_1 \\ a_2} = \pmatrix{Q_{11} a_1 + Q_{12} a_2 \\ Q_{21} a_1 + Q_{22} a_2} \]

The Hermitian conjugate gives

\[ \langle \alpha | \hat{Q}^\dagger = {\bf a}^\dagger Q^\dagger = \pmatrix{a^*_1 & a^*_2} \pmatrix{Q^*_{11} & Q^*_{21} \\ Q^*_{12} & Q^*_{22}} = \pmatrix{Q^*_{11} a^*_1 + Q^*_{12} a^*_2 & Q^*_{21} a^*_1 + Q^*_{22} a_2} \]

For this example we indeed see that:

\[ \langle \alpha | \hat{Q}^\dagger = \left(\hat{Q} | \alpha \rangle\right)^\dagger \]

The projection operator

• The projection operator defined for a normalized \(|\alpha\rangle\):

\[ \hat{P}_\alpha = | \alpha \rangle \langle \alpha | \]

\(\longrightarrow\) Projects any other vector \(|\beta\rangle\) onto the direction of \(|\alpha\rangle\):

\[ \hat{P}_\alpha|\beta\rangle = (\langle \alpha | \beta \rangle) \, | \alpha \rangle \]

The projection operator: example

Example: projection in two dimensions

\[ | \alpha \rangle = \frac{1}{\sqrt{5}}\pmatrix{1 \\ 2 i}, \quad | \beta\rangle = \pmatrix{2 \\ 1} \]

\[ \hat{P}_\alpha |\beta\rangle = | \alpha \rangle \langle \alpha | \beta \rangle = \frac{1}{5}\pmatrix{1 \\ 2i}\pmatrix{1 & -2i} \pmatrix{2 \\ 1} = \frac{2}{5} (1 - i) \pmatrix{1 \\ 2 i} \]

The operator itself is an outer product:

\[ \hat{P}_\alpha = | \alpha \rangle \langle \alpha | = \frac{1}{5}\pmatrix{1 \\ 2i}\pmatrix{1 & -2i} = \frac{1}{5}\pmatrix{1 & -2i \\ 2i & 4} \]

Two-dimensional vector spaces are actually useful: Spin, the two-level atom approximation, etc.

Identity operators

• If we have a complete basis \(\{ \,|e_n\rangle \}\)
• Projection operator:

\[ \hat{P}_n = | e_n \rangle \langle e_n | \]

Then the identity operator can be written as:

\[ \boxed{\sum_n \hat{P}_n = \sum_n | e_n \rangle \langle e_n | = \hat{1}} \]

Or for a continuous spectrum and eigenfunction basis:

\[ \langle e_z | e_z'\rangle = \delta(z - z') \qquad \boxed{\int | e_z \rangle \langle e_z | \,dz = \hat{1}} \]

Functions of operators: power series

• Sums and products of operators, order is important:

\[ (\hat{Q} + c\hat{R}) |\alpha\rangle = \hat{Q}|\alpha\rangle + c \hat{R} |\alpha\rangle \qquad \hat{Q}\hat{R} |\alpha\rangle = \hat{Q} \left(\hat{R} |\alpha\rangle\right) \]

• Functions of operators are represented by their power series
• Likewise with matrices (also operators in our case):

\[ \begin{aligned} e^{\hat{Q}} = 1 + \hat{Q} + \frac{1}{2} \, \hat{Q}^2 + \frac{1}{3!} \, \hat{Q}^3 + \dots\\ \frac{1}{1-\hat{Q}} = 1 + \hat{Q} + \hat{Q}^2 + \hat{Q}^3 + \hat{Q}^4 + \dots\\ \ln(1 + \hat{Q}) = \hat{Q} - \frac{1}{2} \, \hat{Q}^2 + \frac{1}{3} \, \hat{Q}^3 - \frac{1}{4} \, \hat{Q}^4 \dots\\ \end{aligned} \]

The wave function in Hilbert space

• The wave function of a quantum state \(|\Psi(t)\rangle\)

\[ \Psi(x,t) = \langle x |\Psi(t)\rangle, \qquad \hat{x} | x \rangle = x_0 | x \rangle \]

\(\longrightarrow \quad x_0\) are eigenvalues of position operator \(\hat{x}\)

\[ \langle x_0 |\Psi(t)\rangle = \int_{-\infty}^\infty \delta(x-x0) \psi(x) dx = \psi(x_0) \]

Momentum eigenvectors?

Momentum eigenvalue equation:

\[ \hat{p} |\Psi\rangle = p |\Psi\rangle \]

• Filling in momentum operator \(\hat{p} = -i \hbar \frac{d}{dx}\):

\[ \frac{d \psi_p(x)}{dx} = \frac{i p}{\hbar} \psi_p(x) \]

This differential equation has solution:

\[ \psi_p(x) = A e^{ipx/\hbar} = \frac{1}{\sqrt{2\pi}} e^{ipx/\hbar} \]