4.1 Hilbert spaces, briefly

A formal mathematical setting for a quantum system is that of a Hilbert space $\mathcal{H}$, which is (for us⁹⁰) just a vector space along with an inner product.

As mentioned in Section 0.3, we only work with finite dimensional vector spaces, and it is a very convenient fact that any finite dimensional inner product space is automatically a Hilbert space.↩︎

Given a Hilbert space corresponding to our system, the result of any preparation of the system is then represented by some unit vector $|\psi\rangle\in \mathcal{H}$, and any test is represented by some other unit vector $|e\rangle\in \mathcal{H}$. 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 $|e\rangle$. As always, 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 $|e\rangle$. That is, if we immediately measure the object again, we will find it to still be in state $|e\rangle$ with probability $1$. This is the mysterious quantum collapse, which we will further 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.

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90. As mentioned in Section 0.3, we only work with finite dimensional vector spaces, and it is a very convenient fact that any finite dimensional inner product space is automatically a Hilbert space.↩︎