13 Bra-ket notation
Bra-ket notation, introduced by Dirac, is a compact way of writing the linear algebra of quantum mechanics. A vector in a Hilbert space is written as a ket, \[ |\psi\rangle, \] and its Hermitian adjoint is a bra, \[ \langle\psi|. \]
The inner product of two vectors is therefore written \[ \langle\phi|\psi\rangle, \] and is a scalar. It satisfies \[ \langle\phi|\psi\rangle = \langle\psi|\phi\rangle^*. \] In particular, \[ \langle\psi|\psi\rangle = \|\psi\|^2, \] so a normalized state satisfies \(\langle\psi|\psi\rangle=1\).
The product in the opposite order, \[ |\phi\rangle\langle\psi|, \] is an outer product. It is an operator: acting on a ket \(|\chi\rangle\) it gives \[ \bigl(|\phi\rangle\langle\psi|\bigr)|\chi\rangle = |\phi\rangle\langle\psi|\chi\rangle. \]
A linear operator \(\hat A\) acts on kets from the left, \[ |\phi\rangle=\hat A|\psi\rangle. \] Taking the Hermitian adjoint reverses the order, \[ \langle\phi|=\langle\psi|\hat A^\dagger. \] For a Hermitian operator, \(\hat A^\dagger=\hat A\). Observables in quantum mechanics are represented by Hermitian operators, and the expectation value of \(\hat A\) in a normalized state \(|\psi\rangle\) is \[ \langle A\rangle=\langle\psi|\hat A|\psi\rangle. \]
For an orthonormal basis \(\{|i\rangle\}\), \[ \langle i|j\rangle=\delta_{ij}, \] and every vector can be expanded as \[ |\psi\rangle=\sum_i c_i|i\rangle,\qquad c_i=\langle i|\psi\rangle. \] Equivalently, the basis satisfies the completeness relation \[ \hat I=\sum_i|i\rangle\langle i|. \]
In quantum chemistry we also frequently use nonorthogonal basis functions, such as atomic orbitals. If the basis is \(\{|b_i\rangle\}\), its overlap matrix is \[ S_{ij}=\langle b_i|b_j\rangle. \] In this case the expansion coefficients cannot in general be obtained simply by projecting with \(\langle b_i|\); the overlap matrix must also be taken into account.
Exercises
Exercise 13.1 (Bra-ket notation and column vectors) Consider the orthonormal basis \(\{|1\rangle,|2\rangle,|3\rangle\}\). In this basis, the ket \[ |\psi\rangle=2|1\rangle-i|2\rangle+3|3\rangle \] can be represented by a column vector.
Write \(|1\rangle\), \(|2\rangle\), and \(|3\rangle\) as column vectors.
Write \(|\psi\rangle\) as a column vector.
Write the bra \(\langle\psi|\) in bra-ket notation.
Write \(\langle\psi|\) as a row vector. Remember that taking the adjoint involves both transposing and complex conjugating.
Calculate \(\langle 2|\psi\rangle\), first using bra-ket notation and then by multiplying a row vector by a column vector.
Calculate \(\langle\psi|\psi\rangle\) using the row and column vectors. What is the norm \(\|\psi\|\)?
Exercise 13.2 (Reading simple bra-ket expressions) Consider two orthonormal states \(|1\rangle\) and \(|2\rangle\) and the state \[ |\psi\rangle=c_1|1\rangle+c_2|2\rangle. \]
Write the corresponding bra \(\langle\psi|\).
Compute \(\langle1|\psi\rangle\).
Compute \(\langle2|\psi\rangle\).
Compute \(\langle\psi|\psi\rangle\).
What condition must \(c_1\) and \(c_2\) satisfy if \(|\psi\rangle\) is normalized?
Exercise 13.3 (From coordinate vectors to Hilbert-space vectors) Let \(B=\{|b_1\rangle,\ldots,|b_n\rangle\}\) be a basis for an \(n\)-dimensional Hilbert space \(V\), and let \(\{|i\rangle\}\) be the standard orthonormal basis of \(\mathbb C^n\). Define \[ \hat B=\sum_{i=1}^n|b_i\rangle\langle i|. \]
What are the domain and range of \(\hat B\)?
Let \[ |x\rangle=\sum_i x_i|i\rangle\in\mathbb C^n. \] Show that \[ \hat B|x\rangle=\sum_i x_i|b_i\rangle. \] Interpret the numbers \(x_i\).
Find \(\hat B^\dagger\).
Show that \[ \hat B^\dagger\hat B=\sum_{ij}|i\rangle\langle b_i|b_j\rangle\langle j|. \]
Explain why \(\hat B^\dagger\hat B\) is the overlap matrix of the basis \(B\), expressed as an operator on \(\mathbb C^n\).
Exercise 13.4 (Molecular orbitals in a nonorthogonal atomic-orbital basis) Suppose a molecular orbital is expanded in two normalized atomic orbitals, \[ |\psi\rangle=c_1|\chi_1\rangle+c_2|\chi_2\rangle, \] where \[ \langle\chi_1|\chi_1\rangle=\langle\chi_2|\chi_2\rangle=1,\qquad \langle\chi_1|\chi_2\rangle=\langle\chi_2|\chi_1\rangle=s, \] with \(s\) real.
Compute \(\langle\psi|\psi\rangle\).
Write the overlap matrix \(S\) in the basis \(\{|\chi_1\rangle,|\chi_2\rangle\}\).
Show that the normalization condition can be written \[ \mathbf c^\dagger S\mathbf c=1, \] where \(\mathbf c=(c_1,c_2)^T\).
Consider the symmetric combination \[ |\psi_+\rangle=N_+\bigl(|\chi_1\rangle+|\chi_2\rangle\bigr). \] Find the normalization constant \(N_+\).
Similarly, find the normalization constant \(N_-\) for \[ |\psi_-\rangle=N_-\bigl(|\chi_1\rangle-|\chi_2\rangle\bigr). \]
Show that \(\langle\psi_+|\psi_-\rangle=0\). Thus symmetric and antisymmetric combinations can be orthogonal even though the atomic orbitals from which they are constructed are not.
Exercise 13.5 (Nonorthogonal bases, dual vectors, and completeness) Let \(\{|b_i\rangle\}_{i=1}^n\) be a linearly independent, but not necessarily orthogonal, basis of an \(n\)-dimensional Hilbert space. Define the overlap matrix \[ S_{ij}=\langle b_i|b_j\rangle. \] Assume that \(S\) is invertible.
Define vectors \(|b^i\rangle\) by \[ |b^i\rangle=\sum_j|b_j\rangle(S^{-1})_{ji}. \] Show that \[ \langle b_k|b^i\rangle=\delta_k^{\,i}. \] The set \(\{|b^i\rangle\}\) is called the dual basis.
Show that \[ \hat I=\sum_i|b^i\rangle\langle b_i| \] and hence also \[ \hat I=\sum_{ij}|b_i\rangle(S^{-1})_{ij}\langle b_j|. \]
Let \[ |\psi\rangle=\sum_i c_i|b_i\rangle. \] Show that the expansion coefficients are \[ c_i=\langle b^i|\psi\rangle. \]
Define the projections \[ p_i=\langle b_i|\psi\rangle. \] Show that \[ p_i=\sum_jS_{ij}c_j \] and therefore, in matrix notation, \[ \mathbf c=S^{-1}\mathbf p. \]
Explain why the familiar orthonormal-basis formula \(c_i=\langle b_i|\psi\rangle\) is recovered when \(S=I\).
In quantum chemistry, atomic-orbital basis functions are generally nonorthogonal. Explain briefly why confusing the projections \(\langle b_i|\psi\rangle\) with the expansion coefficients \(c_i\) can therefore lead to incorrect molecular-orbital coefficients.