2. Formal quantum mechanics and the path integral

When we consider classical mechanics, we start from Newton's laws and derive the behaviour of moving bodies subject to forces from these laws. This is a nice approach as we always like to see a structured presentation of the world surrounding us. We should however not forget that people have thought for thousands of years about motion and forces before Newton's compact formulation of the underlying principles was found. It is not justified to pretend that physics only consists of understanding and predicting phenomena from a limited set of laws. The 'dirty' process of walking in the dark and trying to find a comprehensive formulation for the phenomena under consideration is an essential part of physics.

This also holds for quantum mechanics, although it was developed in a substantially shorter amount of time than classical mechanics. In fact, quantum mechanics started at the beginning of the twentieth century, and its formulation was more or less complete around 1930.

The previous chapter contained a brief review of quantum mechanics at a level where you should feel comfortable. Now we make a step partly into new material by considering quantum mechanics from a formal viewpoint. Part of this material can be found in Griffiths' book (chapter 3), in particular for section 2.1, where we introduce quantum mechanics by formulating the postulates on which the quantum theory is based. In sections 2.2, 2.3 and 2.4, we establish the link between the classical mechanics and quantum mechanics via Poisson brackets and via the path integral.

2.1. The postulates of quantum mechanics

Quantum theory can be formulated in terms of a set of postulates which however do not have a canonised form similar to Newton's laws: most books have their own version of these postulates and even their number varies.

We now present a particular formulation of these postulates.

Postulates of quantum mechanics

1. The state of a physical system at any time $t$ is given by the wave function of the system at that time. This wave function is an element of the Hilbert space of the system. The evolution of the system in time is determined by the Schrödinger equation: $${\rm i}\hslash \frac{\partial}{\partial t}\left| \psi(t) \right\rangle= \hat{H} \left| \psi(t) \right\rangle.$$ Here $\hat{H}$ is an Hermitian operator, called the Hamiltonian.

2. Any physical quantity $Q$ is represented by an Hermitian operator $\hat{Q}$.

   When we perform a measurement of the quantity $Q$, we will always find one of the eigenvalues of the operator $\hat{Q}$. For a system in the state $\left| \psi(t)\right\rangle$, the probability of finding a particular eigenvalue $\lambda_i$, with an associated eigenvector $\left| \phi_i\right\rangle$ of $\hat{Q}$ is given by $$P_i = \frac{\left| \left\langle \phi_i | \psi(t) \right\rangle \right|^2}{ \left\langle \psi(t) | \psi(t) \right\rangle \left\langle \phi_i | \phi_i \right\rangle}.$$ Immediately after the measurement, the system will find itself in the state $\left| \phi_i\right\rangle$ corresponding to the value $\lambda_i$ which was found in the measurement of $\lambda_i$.

Several remarks can be made.

1. The wave function contains the maximum amount of information we can have about the system. In practice, we often do not know the wave function of the system.

2. Note that $\hat{Q}$ being Hermitian implies that the eigenstates $\left| \phi_i\right\rangle$ always form a basis of the Hilbert space of the system under consideration. Thus the state $\left| \psi(t)\right\rangle$ of the system before the measurement can always be written in the form $$\left| \psi(t) \right\rangle= \sum_i c_i(t) \left| \phi_i \right\rangle.$$ The probability to find in a measurement the values $\lambda_i$ is therefore given by $$P_i = \frac{\left| c_i \right|^2}{\sum_j \left| c_j\right|^2}$$ where we have omitted the time-dependence of the $c_i$. For a normalised state $\left| \psi(t)\right\rangle$ it holds that, if the eigenvectors $\left| \phi_i\right\rangle$ are orthonormal: $$\sum_i |c_i|^2 = 1.$$ In that case $$P_i = \left| c_i \right|^2.$$

3. So far we have suggested in our notation that the eigenvalues and eigenvectors form a discrete set. In reality, not only discrete, but also continuous spectra are possible. In those cases, the sums are replaced by integrals.

4. In understanding quantum mechanics, it helps to make a clear distinction between the formalism which describes the evolution of the wave function (the Schrödinger equation, postulate 1) versus the interpretation scheme. We see that the wave function contains the information we need to predict the outcome of measurements, using the measurement postulate (number 2).

It now seems that we have arrived at a formulation of quantum mechanics which is similar to that of classical mechanics: a limited set of laws (prescriptions) from which everything can be derived, provided we know the form of the Hamiltonian (this is analogous to the situation in classical mechanics, where Newton's laws do not tell us what the form of the forces is).

However there is an important difference: the classical laws of motion can be understood by using our everyday experience so that we have some intuition for their meaning and content. In quantum mechanics, however, our laws are formulated as mathematical statements concerning objects (vectors and operators) for which we do not have a natural intuition. This is the reason why quantum mechanics is so difficult in the beginning (although its mathematical structure as such is rather simple). You should not despair when quantum mechanics seems difficult: many people find it difficult, and the workings of the measurement process are still the object of intensive debate. Sometimes you must switch your intuition off and use the rules of linear algebra to solve problems.

Above, we have mentioned that quantum mechanics does not prescribe the form of the Hamiltonian. In fact, although the Schrödinger equation, quite unlike the classical equation of motion, is a linear equation, which allows us to make ample use of linear algebra, the structure of quantum mechanics is richer than that of classical mechanics because in principle any type of Hilbert space can occur in Nature. In classical mechanics, the space of all possible states of an $N$-body system is a $6N$ dimensional space (we have $3N$ space and $3N$ momentum coordinates). We may extend this with 2 angles per particle if the particles carry a magnetic or electric dipole moment. In quantum mechanics, wave functions can be part of infinite-dimensional spaces (like the wave functions of a particle moving along a one-dimensional axis) but they can also lie in a finite-dimensional space (for example in the case of spin, which has no classical analogue).