0.5 Geometry

The inner product brings geometry: the length, or norm, of |v\rangle is given by \|v\|=\sqrt{\langle v|v\rangle}, and we say that |u\rangle and |v\rangle are orthogonal if \langle u|v\rangle=0. Any maximal set of pairwise orthogonal vectors of unit length6 forms an orthonormal basis, and so any vector can be expressed as a linear combination of the basis vectors: \begin{gathered} |v\rangle =\sum_i v_i|e_i\rangle \\\text{where $v_i=\langle e_i|v\rangle$}. \end{gathered} Then the bras \langle e_i| form the dual basis \begin{gathered} \langle v| =\sum_i v_i^\star\langle e_i| \\\text{where $v_i^\star=\langle v|e_i\rangle$}. \end{gathered}

To make the notation a bit less cumbersome, we will sometimes label the basis kets as |i\rangle rather than |e_i\rangle, and write \begin{aligned} |v\rangle &= \sum_i |i\rangle\langle i|v\rangle \\\langle v| &= \sum_j \langle v|i\rangle\langle i|. \end{aligned} But do not confuse |0\rangle with the zero vector! We never write the zero vector as |0\rangle, but only ever as 0, without any bra or ket decorations (so e.g. |v\rangle+0=|v\rangle).

  • With any isolated quantum system, which can be prepared in n perfectly distinguishable states, we can associate a Hilbert space \mathcal{H} of dimension n such that each vector |v\rangle\in\mathcal{H} of unit length (i.e. \langle v|v\rangle =1) represents a quantum state of the system.

  • The overall phase of the vector has no physical significance: |v\rangle and e^{i\varphi}|v\rangle (for any real \varphi) both describe the same state.

  • The inner product \langle u|v\rangle is the probability amplitude that a quantum system prepared in state |v\rangle will be found in state |u\rangle upon measurement.

  • States corresponding to orthogonal vectors (i.e. \langle u|v\rangle=0) are perfectly distinguishable, since, if we prepare the system in state |v\rangle, then it will never be found in state |u\rangle, and vice versa. In particular, states forming orthonormal bases are always perfectly distinguishable from each other. Choosing such states, as we shall see in a moment, is equivalent to choosing a particular quantum measurement.


  1. That is, consider sets of vectors |e_i\rangle such that \langle e_i|e_j\rangle=\delta_{ij} (where the Kronecker delta \delta_{ij} is 0 if i\neq j, and 1 if i=j.), and then pick any of the largest such sets (which must exist, since we assume our vector spaces to be finite dimensional).↩︎