Towards a demystification of Quantum Mechanics
By: Giovanni Organtini
Tags: Physics, Science, Quantum Mechanics, State, Teaching

The first step is to re-examine classical mechanics in view of an introduction to quantum mechanics. This episode is devoted to clarifying the concept of “state”. In the following episodes we will deal with other, often neglected, concepts in classical physics.

Figure taken from Wikimedia made by Geek3

The concept of state

In classical physics we sometimes use the concept of “state”. In particular, in most textbooks you can find that:

1. the state of a pointlike particle is often said to be given when its position and velocity are known;

2. the state of a gas is given by the ideal gas law (also called the gas’ equation of state) stating that pV﹦nRT.

What is intended for “state”? Looking for its definition in a physics textbook is a waste of time: as a matter of fact, I could not find a single textbook in which it is defined. Let’s then try to find its definition on a vocabulary. The Merriam-Webster dictionary defines a state as a “mode or condition of being” (interestingly enough, one of the items refer to the state of an atomic system: see below). Indeed, this is the actual meaning to be given to this word in physics, too.

When we want to describe a physical system (being it a pointlike particle, a gas, a circuit or a solid), we need to characterise it providing a set of (mutually independent) quantities obtained measuring them. The “state” of a system is then given when a complete list of measurable physical quantities are given for it. In other words it comprises all and only those quantities needed to fully characterise the system at time t such that we can predict its state at another time t’.

For an ideal gas, for example, four physical quantities determine its state: its pressure, its volume, its temperature and its quantity. They are related to each other by means of the “equation of state”, such that only three are needed to specify its state. In principle, there could be other measurable quantities of interest: its color, for example, could be one of the state variable. However, as long as we are not dealing with the changes in the color of a gas, its inclusion in the state is useless, similarly to what happens for the mass in mechanics, where the latter is nearly always considered constant. The state, then, only comprises “interesting” quantities, i.e. quantities that are going to be measured and do not depend on each other. In summary, borrowing quantum mechanical notation, we can specify the state of an ideal gas as |p, V, T〉as well as |p, V, n〉or any other combination of p, V, n and T.

In kinematics, the state of a particle is often said to be given when position and velocity are known. In fact, in this case, the state depends on the choice of the reference frame in which coordinates are expressed (note that when talking about gases we implicitly use a reference frame at rest with respect to the gas container). Physics cannot depend on our choices, then there must be a way to express the state of a system independently of it. One can choose a “privileged” reference frame consisting in the frame in which the particle is at rest when t=0. In this case the state can be fully characterised giving the velocity v. If the speed of the particle is constant, in the given reference frame it remains in the |x=0, v=0〉state, otherwise its state changes into a different one.

Of course we could make similar arguments for various systems. For example, the state of a capacitor can be characterised when the charge Q and its voltage ΔV are known. In this case, the definition of capacitance C=Q/ΔV plays the role of the equation of state in the physics of gases. You can do the exercise of identifying the state on every physics system you know.

So, the state depends on the system we are interested in, not only because it comprises only quantities relevant to the specific problem: depending on the system, the list of potentially interesting variables changes. Manifestly, we cannot characterise the state of a gas using position and velocities: these quantities simply do not make sense for a gas (they make sense for its constituents, not for the gas as a system — note, also, that kinetic theory of gases is a relatively recent discovery). A gas can be contained in a volume: we can identify the position (of a point) of the container, not of the gas. It has a certain temperature: it does not move, then it has no velocity (its constituents have it, but they are not a gas; they are pointlike particles). It must be noted that here we are talking about an ideal gas in equilibrium. When we do the physics of a gas flowing in a tube, its velocity makes sense (but it has a somewhat different definition) and in fact is part of the state of the fluid, being possible to predict its value using the Bernoulli’s equation and knowing its state at t=0.

It turns out that the state of an electron in an atom cannot be represented by the same variables used for a pointlike particle. Simply because in this case the position and the velocity of an electron are meaningless: they cannot be measured, due to the Heisenberg principle, so they cannot be part of the state.

Of an electron in an atom (a quantum mechanical system) we can measure its energy (e.g. using photoelectric effect) and its angular momentum (from the analysis of absorption and emission spectra). So, energy and angular momentum are meaningful quantities for such a system and can be the quantities to be included in its state.

Figure taken from Wikimedia made by Ranjithsiji

For a free subatomic particle like a muon (again a quantum mechanical system), we can measure its position and its velocity in certain conditions, so it makes sense to include them in the state. A better choice (the system is often relativistic) is to specify the kinematical state of the muon providing its energy and momentum. Muons are unstable and they decay into an electron and two neutrinos. The state of the system is then characterised by the number and the flavour of the particles in it (that can initially be given by the mass of the muon). After a certain time the state evolved such that, at sufficiently long times, the state comprises three particles: an electron and two neutrinos, each of which has its own energy and momentum.

In summary, the concept of state traditionally arises in quantum mechanics as a central concept that most people have difficulties to grasp. Indeed, it is a rather simple concept, existing in classical physics, too, and the difficulties arise only because its role is not enough emphasised when we are exposed to classical physics. If we were used to write (classical) physics laws in terms of the evolution of a state, we were not be surprised when we switch to a complete different, quantum mechanical, description of an electron in an atom. Simply stated, the Bohr’s model of the atom is wrong. Continuing teaching it is like to continue teaching the “caloric theory”. It is instructive to mention it, but none of us believe it is useful to discuss it in details. Classical physics is to quantum mechanics as kinetic theory is to thermodynamics. It is just a change in the way we describe the state, in turn due to fact that physics quantities used in classical physics loose their meaning in certain conditions (and, note, this is not peculiar of quantum mechanics: even in the physics of gases, position and velocity loose their meaning).