In this first lesson, we present the so-called “orthodox” formulation of quantum mechanics, generally associated with the Copenhagen interpretation¹. This formulation includes, in particular, the wave-function collapse postulate (Postulate 3) and the fundamentally probabilistic character of measurement outcomes (Postulate 4). They apply to an isolated system. For open systems, both the system and its environment must be described quantum mechanically. A sixth and final postulate, concerning the composition of systems, will be stated briefly here and then developed fully in Theme 4: “Composite systems”.

1. The six postulates

For an overview, one may also consult the Wikipedia page Postulates of quantum mechanics^[3], or Griffiths^[4]. The numbering of the postulates varies between presentations.

1.1. Quantum states

The first postulate contains the kinematics of the theory. Here we restrict ourselves to so-called pure states. We shall later encounter mixed states, which generalise them: these are statistical mixtures of several pure states.

This postulate thus associates the physical state with an abstract mathematical object, very different from the vectors in $\R^3$ used in classical mechanics. The question of what this vector actually represents will be examined in Lesson 3. The choice of a linear structure is what makes it possible to form superpositions of states, as discussed in Theme 1.

Note that identifying two kets that differ only by a global phase is consistent with the Born rule in Postulate 4 below: this transformation does not change any measurement probability. Mathematically, it gives the state space a geometric structure that we shall specify in Section 2.1.

Let us now turn to the so-called measurement postulates. For the moment, we shall not attempt to define what a measurement actually is or how it is triggered. We shall of course return to this essential point in Theme 4, devoted to composite systems, where the system, the measuring apparatus and the environment will all be treated quantum mechanically. For a system assumed to be isolated, the following postulates should be seen as attempting, at best, to establish a connection between the mathematical objects of the theory and experimentally observable results, such as a value displayed by a measuring instrument or the accumulation of impacts on a target. They call for many comments, developed among other places in the next lesson.

1.2. Observables

Observables are objects that encode the possible outcomes of a physical measurement. The mathematical notions used here were introduced in Theme 2.

Several points require clarification.

• The usual observables include the Hamiltonian $\hat H$, associated with energy and introduced below, as well as position $\hat X$ and momentum $\hat P$, which we shall study in Lesson 4. As we have seen, orthogonal projectors are also self-adjoint: they are therefore observables in the mathematical sense. For a normalised ket $\ket{a_n}$, the projector $\hat\Pi_n=\ket{a_n}\bra{a_n}$ represents the question “is the state $\ket{a_n}$?”: its eigenvalues $1$ and $0$ encode the answers “yes” and “no”. If the state is a superposition of kets $\ket{a_n}$, the probabilities of these answers are given by the Born rule (Postulate 4). This idea generalises to projectors onto subspaces, for example to ask the question “is the particle in this region of space?”.
• The postulate associates a self-adjoint operator with every measurable physical quantity, but does not state the converse. It is nevertheless common to call every self-adjoint operator an observable, although it should be noted that this mathematical convention does not guarantee a priori the existence of a device capable of measuring it. The available interactions and the symmetries of the system may indeed restrict this possibility². This question remains an active topic of research today.
  Note 2: On the constraints imposed on measurements by conservation laws, see Y. Kuramochi and H. Tajima, Wigner—Araki—Yanase Theorem for Continuous and Unbounded Conserved Observables, Physical Review Letters 131, 210201 (2023), doi:10.1103/PhysRevLett.131.210201^[5].
• The spectrum of $\hat A$ is real by virtue of its self-adjointness. In finite dimension, the spectral theorem, presented in Theme 2, Lesson 6, guarantees the existence of an orthonormal basis of eigenvectors. If $\hat P_n$ denotes the orthogonal projector onto the eigenspace associated with the eigenvalue $a_n$, we have the spectral decomposition $$\hat A=\sum_n a_n\hat P_n, \qquad \hat P_n=\sum_{i=1}^{d_n}\ket{a_n^i}\bra{a_n^i},$$ where $(\ket{a_n^i})_{1\leq i\leq d_n}$ is an orthonormal basis of this eigenspace. In infinite dimension, the spectrum may contain a continuous part. The spectral decomposition then uses projectors associated with intervals of values and spectral integrals; see Section 1.6 below.
• Some books, in particular the treatise by C. Cohen-Tannoudji, B. Diu and F. Laloë, present this postulate as two separate postulates^[6]: the first associates a self-adjoint operator with every measurable quantity; the second identifies the possible outcomes with the spectrum of that operator.

1.3. Measurement

The third postulate is the very famous “wave-function collapse”, at the heart of the conceptual problems posed by this theory. It specifies the effect of such a measurement on the physical state of the measured system.

This statement is written here for a measurement outcome belonging to the discrete spectrum. Its adaptation to the case of a continuous spectrum is presented in Section 1.6 below. In the discrete case, there is a very common simplification: if the eigenvalue $a_n$ is non-degenerate, the associated eigenspace has dimension $1$, and the corresponding projector is simply $\hat{P}_n = \ket{a_n}\bra{a_n}$. By Hermitian symmetry of the inner product, we have:

Postulate 3 therefore becomes in this case:

where we have removed a global phase $e^{i\phi}$ with no observable physical consequence, as stated above. Thus, in the non-degenerate case, we may retain the following essential rule: when the outcome $a_n$ has been observed, immediately after the measurement the state of the system may be taken to be $\ket{a_n}$.

It remains only to give a rule for the probabilities of observing one eigenvalue $a_n$ or another. This is known as the Born rule (1926):

When $\hat A$ is measured, one of the possible outcomes is necessarily obtained: the sum of their probabilities is therefore $1$. In the orthonormal basis of eigenstates, the Born rule gives

The equality $\sum_n|\braket{a_n}{\psi}|^2=\braket{\psi}{\psi}$ follows from Parseval's identity applied to the kets $\ket{a_n}$, which form an orthonormal Hilbert basis. The normalisation condition on the state vector thus ensures that the sum of the probabilities is $1$.

We now extend this formula to the degenerate discrete case. Here we introduce the $\ket{a_n^i}$, with $i$ ranging from 1 to $p$, the dimension of the eigenspace. Using $\hat P_n=\sum_{i=1}^{p}\ket{a_n^i}\bra{a_n^i}$, we obtain

Indeed, $\sum_{i=1}^{p}|\braket{a_n^i}{\psi}|^2=\sum_{i=1}^{p}\braket{\psi}{a_n^i}\braket{a_n^i}{\psi}=\bra\psi\left(\sum_{i=1}^{p}\ket{a_n^i}\bra{a_n^i}\right)\ket\psi=\bra\psi\hat P_n\ket\psi$.

The state reduction after obtaining the outcome $a_n$ may therefore also be written, when $P_\psi(a_n)\neq0$,

The continuous-spectrum case is studied in Section 1.6. For another presentation of measurement probabilities and projectors in the presence of degeneracy, see the notes for MIT course 8.321, Lecture 3, Section 3.2^[7].

1.4. The dynamical postulate

Quantum mechanics postulates the existence of a privileged observable representing the total energy of the system: the Hamiltonian, denoted by $\hat H$, which is the quantum version of the classical Hamiltonian in analytical mechanics, with which the reader is assumed to be familiar. This operator has a dual role. Like any observable, its spectrum gives the possible outcomes of an energy measurement, but through the dynamical postulate it is also the generator of the time evolution of the state between two measurements:

Note that here we restrict ourselves to isolated systems. Thus, no external action perturbs the system, and the Hamiltonian is therefore taken not to depend explicitly on time. The case of a time-dependent Hamiltonian will be covered later. Note that the exponential $e^{-i\hat H(t-t_0)/\hbar}$ is defined on all of $\H$ through the spectral calculus, which also guarantees its unitarity. For the moment, we shall accept that, when the initial state belongs to $D(\hat H)$, the integrated form (10) and the differential equation (9) are equivalent. By contrast, for an arbitrary initial state in $\H$, the integrated form remains well defined, whereas the derivative in the Schrödinger equation may not exist. We shall clarify these statements and their converse, given by Stone's theorem, below in the section “Further study: existence and uniqueness”. These results explain why the statement of the postulate presents both the Schrödinger equation and unitary evolution.

1.5. Composition of systems

The final postulate tells us how to bring several systems together within a single quantum description.

This brief statement is sufficient to describe a system, an apparatus and an observer jointly. The construction of the tensor product, composite observables and entanglement will be studied systematically in Theme 4, devoted to composite systems.

1.6. The case of a continuous spectrum

The preceding measurement postulates were written with sums over discrete eigenvalues. In infinite dimension, the spectrum of an observable may also contain a continuous part. This is the case, for example, of the position operator associated with measuring the position of a point particle; see also Lesson 4 and the following lessons.

Postulates 3 and 4: position measurement.

We shall see that the state of a particle on a line belongs to an infinite-dimensional Hilbert space whose elements may be represented by functions of the position $x\in\R$. Position is associated with a self-adjoint observable $\hat X$ whose spectrum is continuous. For each $x\in\R$, we then introduce a position state $\ket{x}$ formally satisfying an eigenvalue equation:

We shall specify the mathematical meaning of this notation later. For now, this equation means that $x$ is the spectral value associated with the position state $\ket{x}$ and therefore, according to Postulate 2, a possible outcome of a position measurement. At time $t$, the formal projection of the state $\ket{\psi(t)}$ onto $\ket{x}$ defines the amplitude called the position-space wave function:

The Born formula (3) must then be adapted to the continuous nature of the spectrum. The quantity $|\psi(x,t)|^2$ is not the probability of obtaining exactly the value $x$ at time $t$, but a probability density. The elementary probability of finding the particle between $x$ and $x+dx$ at that time is $|\psi(x,t)|^2dx$. Thus, for an interval $I=[a,b]$,

Since the particle must be somewhere on the line, the total probability is one:

At every time $t$, the squared modulus of the wave function must therefore be integrable: the function $x\mapsto\psi(x,t)$ belongs to the space $L^2(\R)$; see Theme 2.

Now suppose that a measurement made at time $t$ localises the particle in the interval $I=[a,b]$. Wave-function collapse projects onto the eigenspace and then renormalises. We must therefore retain the part of the wave function located in $I$, set its exterior part to zero, and then renormalise the state. This, incidentally, explains the name “wave-function collapse”. When $P_{\psi(t)}(x\in I)\neq0$, the wave function immediately after the measurement therefore becomes

The denominator is indeed the square root of $P_{\psi(t)}(x\in I)$, as in the discrete case. This ensures that the new post-measurement wave function is still normalised (check it!). In the idealisation of a measurement yielding the exact position $x$, the reduction is formally written $\ket{\psi(t)}\longrightarrow\ket{x}$, but note that a real apparatus always has finite resolution.

2. First consequences of the postulates

With the postulates established, we can derive several general consequences that will be used throughout the following lessons: the identification of physical states, the calculation and evolution of expectation values, and then the exact form of the dynamics in an eigenbasis of the Hamiltonian.

2.1. The projective state space

Two normalised vectors that differ only by a global phase represent the same physical state. Indeed, all probabilities given by the Born rule are invariant under the transformation $\ket\psi\mapsto e^{i\chi}\ket\psi$. The physical state is said to be the ray spanned by $\ket\psi$, and the set of these rays forms the projective space associated with $\H$. In Lesson 10, we shall see that, for a two-level system, these rays are represented by the points on the Bloch sphere. In due course, we shall describe the abstract mathematics of projective spaces in detail.

This identification should not obscure the fact that relative phases are physically relevant. The states $(\ket{0}+\ket{1})/\sqrt2$ and $(\ket{0}-\ket{1})/\sqrt2$, for example, are not proportional and represent two distinct quantum states, as shown in Theme 1.

2.2. Expectation value and standard deviation

The preceding calculation uses a discrete spectral decomposition. In the presence of a continuous spectrum, sums are replaced by spectral integrals. For an unbounded observable, the existence of the moments must also be checked. The expression $\bra\psi\hat A\ket\psi$ remains defined provided that $\psi\in D(\hat A)$. For the position operator $\hat X$, this condition is written explicitly as

It is equivalent to requiring that the second moment of position be finite:

The expectation value and variance are then well defined, and are equal to

2.3. Evolution of expectation values

The dynamical postulate directly determines the time evolution of expectation values. This is the content of the following theorem:

An important corollary is that an observable with no explicit time dependence that commutes with the Hamiltonian has a constant expectation value.

2.4. Eigenstates of the Hamiltonian

Let us begin by solving the time evolution for an energy eigenstate. If at the initial time $t_0$ the system is in an eigenstate of the Hamiltonian, $\hat{H} \ket{\psi_n} = E_n \ket{\psi_n}$, then its subsequent evolution is given by the Schrödinger equation, which in this case becomes trivial,

and has the solution

Note that this solution may also be recovered directly from the evolution operator. Indeed, the definition of the spectral functional calculus implies that, for any function $f$ and any eigenvector $\ket{\psi_n}$ of $\hat H$ associated with $E_n$,

Taking $f(\lambda)=e^{-i\lambda(t-t_0)/\hbar}$ gives

Since the global phase of a ket is not physical, we may say that the ray remains constant, represented by $\ket{\psi_n}$ at every later time. This is why such states are also called stationary states. Care is nevertheless required with relative phases: if instead the initial state is a superposition of energy eigenstates,

then its evolution becomes

If energy is measured on this state, these relative phases play no role. According to the Born rule, assuming here that the energy eigenvalues are non-degenerate, the probability of obtaining $E_n$ is

Thus, the entire probability distribution of the energies is constant, which also implies, by its definition, that the expectation value of the energy is constant. On the other hand, the fact that the different eigenstates accumulate different phases over time is directly observable experimentally and leads to quantum interference. We shall demonstrate this in Lesson 11.

Note that conservation of the expectation value of energy follows more generally from Ehrenfest's theorem. Taking $\hat A=\hat H$ and using $[\hat H,\hat H]=0$, we obtain

When the Hamiltonian has no explicit time dependence, this derivative vanishes and the expectation value of the energy is conserved: this is the quantum analogue of the theorem of conservation of mechanical energy.

For more on dynamics and energy eigenstates, see, for example, J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics^[8]. The MIT notes, Lecture 6^[9], also present evolution and energy eigenstates.

3. Further study: existence and uniqueness

The Schrödinger equation is a differential equation in a Hilbert space. When projected onto elements of a basis, it becomes an ordinary differential equation. The following questions then arise naturally: does a solution exist for every initial condition, is that solution unique, and what is the regularity of these solutions in the variable of interest, here time? For a time-independent Hamiltonian (also called autonomous), the answer is given by the following theorem, part of which we shall accept without proof.

Several remarks:

• To say that $t\mapsto\ket{\psi(t)}$ is of class $C^1$ in norm means that, for every $t$, there exists a vector $\ket{\dot\psi(t)}\in\H$ such that $$\left\| \frac{\ket{\psi(t+h)}-\ket{\psi(t)}}{h} -\ket{\dot\psi(t)} \right\|\longrightarrow0 \qquad\text{as }h\longrightarrow0,$$ and that this derivative is itself norm-continuous: $$\left\|\ket{\dot\psi(t)}-\ket{\dot\psi(s)}\right\| \longrightarrow0 \qquad\text{as }t\longrightarrow s.$$ The notion of norm continuity was introduced in Theme 2, in Lessons 1 and 2.
• A strongly continuous unitary group is a family of unitary operators satisfying $U(0)=I$ and $U(t+s)=U(t)U(s)$ such that, for every $\ket\psi\in\H$ and every time $s$, $$\left\|U(t)\ket\psi-U(s)\ket\psi\right\|\longrightarrow0 \qquad\text{as }t\longrightarrow s.$$ The group law means that an evolution for a duration $s$, followed by an evolution for a duration $t$, is equivalent to a single evolution for the total duration $t+s$. It expresses the fact that, for a time-independent Hamiltonian, the evolution depends on the elapsed duration rather than on the instant chosen as the origin. Moreover, $U(-t)=U(t)^{-1}$ shows that the evolution is reversible. Strong continuity means that a sufficiently small variation in time cannot abruptly alter the state of the system, nor therefore the resulting measurement probabilities. In other words, the path traced by $\ket{\psi(t)}$ in $\mathcal{H}$ is norm-continuous between two measurements: this unitary evolution contains no “quantum jumps”. By contrast, the state reduction postulated during an ideal measurement may produce a discontinuous change, as described by Postulate 3 in Section 1.3 above. We shall discuss this dichotomy in Lesson 3.

Without giving the complete proof of this non-trivial result, we can explicitly verify the group law: since $\hat H$ is self-adjoint, the spectral calculus gives

Since the exponentials are functions of the same operator $\hat H$, they commute and their exponents add:

In particular, $U(0)=I$ and $U(t)^\dagger U(t)=U(-t)U(t)=U(0)=I$: each $U(t)$ is therefore unitary. The remainder of the proof is taken as given here.

This theorem explains why self-adjointness is the appropriate condition for generating unitary evolution. For proofs and details concerning domains, see G. Teschl, Mathematical Methods in Quantum Mechanics^[10], Section 5.1, “The time evolution and Stone's theorem”, in particular Theorems 5.1 and 5.2 (text available on the author's website).

4. References