3. Quantum Mechanics and its Postulates#
While the energy levels of the hydrogen atom played an important role in the historical development of quantum mechanics, it became almost immediately clear that the true challenge is in applying the laws of quantum mechanics to systems with many interacting particles or fields. Note that the formalism of quantum mechanics, and in particular its postulates, are generically valid and not restricted to the description of a single particle. Quantum field theory also follows these postulates and is thus not a generalisation of quantum mechanics, but rather a specific case of it. These postulates characterise the mathematical model by which quantum mechanics describes physical systems, and more specifically how it represents states, observables, measurements and dynamics. We briefly reiterate these postulates and base our discussion on the wonderfull lecture notes “Quantum Information and Computation” by John Preskill.
3.1. Postulate 1: States#
The state of an isolated quantum system is associated to a ray of vectors in a complex Hilbert space \(\mathbb{H}\).
A Hilbert space is a metric complete inner product space. Let us unpack this definition:
\(\mathbb{H}\) is a vector space in this case over the complex numbers. We will denote elements of this vector space with Dirac’s ket notation \(\ket{\psi}\). In particular, we can build linear combinations
\[\ket{\psi} = a \ket{\psi_1} + b \ket{\psi_2}\]for all \(a, b \in \mathbb{C}\) and all \(\ket{\psi_1}, \ket{\psi_2} \in \mathbb{H}\).
\(\mathbb{H}\) has an inner product, which maps two vectors \(\ket{\psi}\) and \(\ket{\varphi}\) onto a scalar \(\braket{\varphi|\psi} \in \mathbb{C}\) with the properties of
Linearity: \(\bra{\varphi} ( a \ket{\psi_1} + b \ket{\psi_2}) = a \braket{ \varphi | \psi_1} + b \braket{ \varphi | \psi_2}\)
Skew-symmetry: \(\braket{ \varphi | \psi} = \braket{ \psi | \varphi}^\ast\)
Positivity: \(\braket{ \psi | \psi} \geq 0\) with equality only if \(\ket{\psi} = 0\).
This last property enables us to define a norm \(\lVert \psi \rVert = \lVert \ket{\psi} \rVert = \sqrt{\braket{\psi|\psi}}\), which satisfies known properties such as \(\lVert \psi \rVert = 0 \Leftrightarrow \ket{\psi} = 0\) \(\lVert a \ket{\psi} \rVert = \vert a\vert \lVert \psi \rVert\) and the triangle inequality \(\lVert \ket{\varphi} + \ket{\psi} \rVert \leq \lVert \varphi \rVert + \lVert \psi \rVert\).
The final property of metric completeness is a technical requirement that is only relevant in infinite-dimensional Hilbert spaces. Firstly, a metric is a notation of distance between the elements in \(\mathbb{H}\), which is provided by the norm of the difference, i.e. \(d(\varphi, \psi) = \lVert \varphi - \psi \rVert\).
Completeness of the metric is a specific property that guarantees that certain sequences of vectors are guaranteed to have a limit value that also exists in \(\mathbb{H}\). This is necessary to make sense of e.g. Fourier series.
The state of a quantum system is associated to a ray of vectors, which is the one-dimensional space \(\{ a \ket{\psi} , \forall a \in \mathbb{C}\}\) spanned by a single (nonzero) vector \(\ket{\psi} \in \mathbb{H}\). We will describe the state of the system using a single representative \(\ket{\psi}\) of this ray, which we typically choose such that \(\braket{ \psi | \psi} = 1\). However, this does not fix the representative completely, as we can still add arbitrary phases \(\exp(\mathrm{i}\alpha)\), i.e. \(\ket{\psi}\) and \(\mathrm{e}^{\mathrm{i}\alpha} \ket{\psi}\) describe the same state.
The best known Hilbert space from your courses on single-particle quantum mechanics is probably \(L^2(\mathbb{R}^n)\), the Hilbert space for a single quantum particle moving in the \(n\)-dimensional coordinate space \(\mathbb{R}^n\) (typically \(n=1,2,3\)). This Hilbert space corresponds to the space of all square-integrable functions \(\psi:\mathbb{R}^d \to \mathbb{C}: x \mapsto \psi(x)\) and the inner product is given by
However, this is already a complicated Hilbert space from a technical perspective. Hilbert spaces can also be finite-dimensional, i.e. \(\mathbb{C}^d\), the space of column vectors of length \(d\), with the standard Euclidean inner product
These Hilbert spaces will be very important in our discussion. The simplest nontrivial case corresponds to \(d=2\) and the associated quantum system is known under various names. It is often referred to as a qubit in the context of quantum information theory. There are various ways in which qubits can be physically realised. Another common example of a two-dimensional Hilbert space is for describing the spin degree of freedom of an electron, or another particle with spin quantum number 1/2. Such a reduced description (forgetting about the position) is possible if the electron is localised in space, for example when it is strongly bound to an atom.
If we do want to describe a particle that moves in space, we might also consider it to exist only at discrete positions in space, i.e. on a lattice. For example, on a one-dimensional lattice (a.k.a. a chain) with \(L\) sites, the Hilbert space would also correspond to \(\mathbb{H} = \mathbb{C}^L\) and the standard basis vectors \(\vert j \rangle, j=1,\dots,L\) correspond to the state of the system if the particle is exactly localised on site \(j\). We can also consider infinitely large lattices, e.g. the one-dimensional chain where there is a site associated with every \(j \in \mathbb{Z}\) (or the n-dimensional hypercubic lattice \(\mathbb{Z}^n\)). The resulting Hilbert space is then spanned by the states \(\vert j \rangle\) for all \(j \in \mathbb{Z}\), and is thus infinite-dimensional but with a straightforward countably infinite basis.
Of course, our goal is to find the Hilbert space of a many body system. We return to this question below and devote a complete section to it.
3.2. Postulate 2: Observables#
Physical observables of the system correspond to self-adjoint (a.k.a. Hermitian) linear operators on the Hilbert space \(\mathbb{H}\).
An operator \(\hat{A}\) on \(\mathbb{H}\) is a linear map \(\hat{A}:\mathbb{H} \to \mathbb{H}\), i.e. a map from vectors to vectors that satisfies
The adjoint of an operator \(\hat{A}\) is a new operator \(\hat{A}^\dagger\) that is constructed such that
for all \(\ket{\varphi}, \ket{\psi} \in \mathbb{H}\) and where \(\vert \hat{A}\psi \rangle = \hat{A} \ket{\psi}\). This definition requires that \((a_1 \hat{A}_1 + a_2 \hat{A}_2)^\dagger = a_1^\ast \hat{A}_1^\dagger + a_2^\ast \hat{A}_2^\dagger\) and \((\hat{A}_1 \hat{A}_2)^\dagger = \hat{A}_2^\dagger \hat{A}_1^\dagger\).
A self-adjoint operator is an operator such that \(\hat{A}^\dagger = \hat{A}\) or thus
for all \(\ket{\varphi}, \ket{\psi} \in \mathbb{H}\). Linear combinations of self-adjoint operators with real coefficients are self-adjoint. The composition of two self-adjoint linear operators \(\hat{A}_1 \hat{A}_2\) is self-adjoint if and only if
i.e. if the operators also commute. Self-adjoint operators have real eigenvalues, and eigenvectors associated to distinct eigenvalues are orthogonal. In a finite-dimensional Hilbert space, self-adjoint operators admit a spectral decomposition
where \(\hat{P}_n\) is the spectral projector onto the eigenspace associated with \(\lambda_n\). The spectral projectors satisfy \(\hat{P}_n \hat{P}_m = \delta_{n,m} \hat{P}_n\), \(\hat{P}_n^\dagger = \hat{P}_n\) and \(\sum_{n} \hat{P}_n = \mathbb{1}\), the identity operator. If \(\lambda_n\) has one-dimensional eigenspace spanned by the eigenvector \(\vert\phi_n\rangle\), then
where the denominator can be omitted if the eigenvector is normalised.
In the language of matrices, these properties can be rephrased as follows: With respect to an orthonormal basis choice, self-adjoint operators are represented as hermitian matrices. Such matrices can be diagonalised by a unitary transformation, or thus, we can construct a complete basis consisting of eigenvectors. With respect to this basis, the self-adjoint operator is represented by a diagonal matrix with real values on the diagonal.
3.3. Postulate 3: Measurements, Expectation Values and Collapse#
Given an observable to which we associate the operator \(\hat{A}\), we now need to prescribe the result of measuring this observable with respect to a system that is in a state \(\ket{\psi}\). The most compact way of describing the result is by stating that, the expectation value \(\braket{\hat{A}}\) (= the mean value of the measurement when averaging over an ensemble of identical copies of the system) is given by
By exploiting the fact that this also prescribes the expectation value of all higher moments \(\braket{\hat{A}^k}\), this determines the full probability distribution of the measurement outcome, and yields the more familiar result: The only possible measurement outcomes are given by the eigenvalues \(\lambda_n\) of \(\hat{A}\), and for a system in state \(\ket{\psi}\) (now assumed normalized), the probability of obtaining \(\lambda_n\) is given by \(p_n = \braket{\psi \vert \hat{P}_n \vert \psi}\) with \(\hat{P}_n\) the spectral projector from above. In the case that \(\lambda_n\) has a single (linearly independent) eigenvector \(\ket{\phi_n}\) (also assumed normalised), this amounts to \(p_n = \vert \braket{\phi_n|\psi}\vert^2\).
There is a second part to the measurement postulate, which states that, if the measurement is immediately repeated (without intermediate dynamics, as described by the next postulate), then the same measurement outcome is obtained. Because the measurement outcome with respect to the initial state \(\ket{\psi}\) is probabilistic and can yield different results, this requires that after the first measurement, the state changes is changed. This is the well-known collapse of the wave function. More specifically, if a measurement of observable \(\hat{A}\) is performed in a system with state \(\ket{\psi}\) and the measurement value \(\lambda_n\) is obtained, then the state of the system changes to
Note that the denominator cannot vanish, as otherwise the probability of having obtained measurement outcome \(\lambda_n\) would have been zero in the first place.
3.4. Postulate 4: Dynamics#
During time intervals without measurements, the state of an isolated quantum system evolves unitarily according to the (first order linear) differential equation
known as the Schr”{o}dinger equation, where \(\hat{H}(t)\) is the Hamiltonian of the system, which may itself be time-dependent. In the case of a time-independent Hamiltonian, we can define the evolution operator
which relates states at different times via \(\ket{\psi(t)} = \hat{U}(t, t') \ket{\psi(t')}\) and is clearly a unitary operator. Clearly, we need to know the Hamiltonian of a system in order to even start thinking about modelling its quantum properties. We will always assume that the Hamiltonian is given. In practice, however, the situation can be much more complicated. Typically, we want to build only an effective quantum description of the system (e.g. only the electrons, only certain electrons, \(\ldots\)) and not start all the way down at the level of fundamental particles and the standard model (which is also only an effective model valid up to some energy scale).