4.1 Hilbert spaces, briefly

A formal mathematical setting for a quantum system is that of a Hilbert space \mathcal{H}, i.e. a vector space with an inner product. The result of any preparation of a system is represented by some unit vector |\psi\rangle\in \mathcal{H}, and any test is represented by some other unit vector |e\rangle\in \mathcal{H}.62 The inner product of these two vectors, \langle e|\psi\rangle, gives the probability amplitude that an object prepared in state |\psi\rangle will pass a test for being in state |e\rangle. Probabilities are obtained by squaring absolute values of probability amplitudes: |\langle e|\psi\rangle|^2 = \langle\psi|e\rangle\langle e|\psi\rangle. After the test, in which the object was found to be in state |e\rangle, say, the object forgets about its previous state |\psi\rangle and is, indeed, actually now in state |e\rangle. This is the mysterious quantum collapse which we will briefly discuss later on.

A more complete test involves multiple states e_k that form an orthonormal basis \{|e_1\rangle,\ldots,|e_n\rangle\}. These states are perfectly distinguishable from each other: the condition \langle e_k|e_l\rangle = \delta_{kl} implies that a quantum system prepared in state |e_l\rangle will never be found in state |e_k\rangle (unless k=l). The probability amplitude that the system in state |\psi\rangle will be found in state |e_k\rangle is \langle e_k|\psi\rangle and, given that the vectors |e_k\rangle span the whole vector space, the system will be always found in one of the basis states, whence \sum_k |\langle e_k|\psi\rangle|^2 = 1. As a result:

A complete measurement in quantum theory is determined by the choice of an orthonormal basis \{|e_i\rangle\} in \mathcal{H}, and every such basis (in principle) represents a possible complete measurement.

  1. The term “Hilbert space” used to be reserved for an infinite-dimensional inner product space that is complete, i.e. such that every Cauchy sequence in the space converges to an element in the space. Nowadays, as in these notes, the term includes finite-dimensional spaces, which automatically satisfy the condition of completeness.↩︎