1.7: Dirac Notation - Analogies with vectors and matrices

You probably remember Dirac notation as a shorthand for integrals, for example the overlap between two wavefunctions can be written as:

\[\langle \chi |\phi \rangle \quad \text{ instead of } \quad \int \int \int \chi^* ({\bf r})\phi ({\bf r}) d^3 {\bf r}. \nonumber\]

(Where \(d^3 {\bf r}\) is the scalar volume element, sometimes called \(r^2 \sin \theta d \theta d\phi dr\), dxdydz, dV or d\(\tau\))

But also if we have a complete set of orthonormal basis states \(i\), the overlap is also the sum of the overlaps between each \(i\) and \(\chi\) and \(\phi\)

\[\langle \chi |\phi \rangle = \sum_i \langle \chi |i \rangle \langle i |\phi \rangle \nonumber\]

Warning: A summation convention is also sometimes used, such that when a state symbol appears twice, first as a ket, then as a bra, it is assumed to be summed over a complete set of orthonormal basis states. The expression above is then further abbreviated to \(\langle \chi |i \rangle \langle i |\phi \rangle\). This convention can be confusing and will not be used in these notes.

Compare this with the vector dot product formula

\[{\bf b.a} = b_xa_x + b_ya_y + b_za_z = \sum_i ({\bf b.e_i})({\bf e_i .a}) \nonumber\]

where \({\bf e_i}\) are the unit vectors in x, y and z directions. Just as any vector can be expressed as a linear combination of \({\bf e_i}\), so any quantum state can be expressed as a linear combination of basis states \(i\). There are certain conditions on the basis states, e.g. they must be ‘orthonormal’ \(\langle j | i \rangle = \delta_{ij}\) just as \({\bf e_i.e_j} = \delta_{ij}\). Just as the three Cartesian vectors span a three dimensional space, so the many basis states span a many-dimensional space. In some cases (e.g. Fourier expansions, hydrogen wavefunctions) there are an infinite number of basis states which are therefore related to spanning an infinite-dimensional space. Mathematicians call these ‘Hilbert spaces’. Any state \(\phi\) can thus be viewed as a vector in a multi-dimensional space, where each dimension corresponds to one of the basis functions. It is thus common to use the words eigenstate and eigenvector interchangeably to refer to \(|\phi \rangle\) Even before the discovery of quantum mechanics, mathematicians had solved many of the problems in this area.

In Dirac notation we have two quantities, the bra and the ket, whereas in vector algebra we have only one, this is because there is not an exact analogy to commutation for Dirac brackets: \(\langle \chi | \phi \rangle = \langle \phi | \chi \rangle^* \) includes taking a complex conjugate. Consider manipulating the bras and kets. We can write a vector in terms of its components thus

\[{\bf A} = \sum_i {\bf e_i} ( {\bf e_i.A} ) \nonumber\]

where \((e_i .A)\) is the amount of \({\bf A}\) along the \({\bf e_i}\) axis; the components. The quantities on either side of the equation are not numbers but vectors. We can generate a whole algebra based on vectors.

Likewise we can write a state thus: \(|\phi \rangle = \sum _i |i\rangle \langle i|\phi \rangle\)

where \(\langle i|\phi \rangle\) is the amount of \(\phi\) along the \(i\) basis state; the components or expansion coefficients. The quantities on each side of this equation are not numbers but functions. \(\phi\) is a normalised wavefunction iff \(\sum _i |\langle {\bf i}|\phi \rangle |^2 = 1\). We can then generate a whole algebra based on bras and kets.

For any different complete sets of basis states \(i\) and \(j\), we can write: \(|\phi \rangle = \sum _j |j\rangle \langle j|\phi \rangle\), and \(|\phi \rangle = \sum _i |i \rangle \langle i | \phi \rangle\). Expansions in \(i\) and \(j\) are called different representations of \(\phi\). This is very similar to using different coordinate systems: the bases \(i\) and \(j\) are analogous to two sets of axes rotated with respect to one another. We might choose complete set of wavefunctions as a representation which includes \(\phi\), just as we sometimes choose axes such that some special vector points along the \(z\)-axis.

Going even further, the expansion in a basis can be done for any \(|\phi \rangle\), so we can dispense with \(|\phi \rangle\) and write:

\[| = \sum_i |i\rangle \langle i|, \text{ the unit operator} \nonumber\]

All this means is that in any equation you can always proceed by breaking the states down into a complete, orthonormal set of basis functions. This may be useful when dealing with a Hamiltonian for which the eigenstates \(i\) with eigenenergies \(E_i\) are already known. A general mixed state \(|\phi \rangle\) has energy:

\[\langle \phi | H_0 | \phi \rangle = \sum_i \sum_j \langle \phi | i \rangle \langle i | H_0 | j \rangle \langle j | \phi \rangle = \sum_i | \langle \phi | i \rangle |^2E_i \text{ since for } i \neq j \langle j | E_i | i \rangle, \langle j| i \rangle = 0 \nonumber\]

So we could use the solution to an easier problem (the eigenvalue problem, which we need solve only once per Hamiltonian) so that we never need to apply the complicated Hamiltonian to the complicated mixed state! This is a very useful trick - reformulating a problem so that we can make use of some work that has already been done. In this case the single, hard, problem of finding the energy of a mixed state is changed to the many, easier, problems of finding the energy of the eigenstates and the amount of each eigenstate in the mixed state.