About ten years ago I was resting between session breaks of a busy American Physical Society March meeting. A colleague, whom I had not seen in years, was with me and inquired about my work. I told him I was working on understanding transport in nanoscale systems. He replied, “Aren't the most important facts already understood?”

As unsettling as that question was, I realized he was simply echoing a sentiment in the community: the field of mesoscopic systems – larger “cousins” of nanoscale systems – had provided us with a wealth of experimental results, and a theoretical construct – known as the single-particle scattering approach to conduction – that had almost assumed the characteristics of a “dogma”. Many transport properties of mesoscopic systems could be understood in terms of this approach. Books on the subject had appeared which enumerated the successes of this theory. Nanoscale systems were nothing else than smaller versions of mesoscopic systems. All we needed to do was transfer the established experimental knowledge – and proven theoretical and computational techniques – to this new length scale. Or so it seemed.

The past decade has shown that the field of transport in nanoscale systems is not a simple extension of mesoscopic physics. Thanks to improved experimental capabilities and new theoretical approaches and viewpoints, it has become clear that novel transport properties emerge at the nanometer scale. In addition, many physical assumptions and approximations we reasonably make to describe mesoscopic systems may not hold for nanoscale structures.

In the preceding Chapter I have discussed an alternative picture of transport, which I called micro-canonical. From this picture we have learned that electrons flow from one electrode to the other across a nanojunction like a liquid would do in confined geometries. This was illustrated in Fig. 7.3.

The junction is an unavoidable obstacle for the electrons, which need to change their momenta while crossing it. This forces the system to reach local equilibrium fast, greatly helping the effect of other inelastic processes whose role is also to force the system towards local equilibrium.

To the above properties we need to add the known fact that, due to Coulomb interactions, the electron liquid is also viscous, namely electronelectron interactions create an internal friction for an electron to propagate through the electron liquid, much like the friction experienced by an object moving across a viscous liquid or gas.

To be more specific, the shear viscosity of the electron liquid at the density of a typical metal, such as gold, is about 10-7 Pa s (from Eq. F.8 with rs = 3a0). For comparison the viscosity of water at room temperature is about 10-3 Pa s.

The viscosity of the electron liquid is thus very small but not zero. In fact, the smaller the viscosity, the less stable the flow with respect to perturbations, especially those provided by obstacles or constrictions along the path.

Photons and electrons

One of the great technical achievements of the second half of the 20th century was the invention of the laser. With its help extremely intense light could be produced. Furthermore, it even became posssible to excite a single resonator mode so that both the frequency and the propagation direction of the emitted radiation field are extraordinarily sharp. The mere existence of such high-intensity single-mode fields – the real problem that was ingeniously solved was to generate them – is no matter of surprise. We know from classical electromagnetic theory that the amplitude of a monochromatic electromagnetic field can take on arbitrarily large values. This is a direct consequence of the linearity of Maxwell's equations. What does this mean for quantized radiation fields? Certainly, the mentioned property of classical fields must also be retained in the quantum mechanical description. Since the electromagnetic energy is quantized in ‘packets’ of magnitude hν (photons), the number of photons in a given mode must be unlimited, in principle.

All this is very well known. However, it becomes noteworthy when we compare the behaviour of photons with that of electrons. Amazingly, it turns out that an electronic quantum state can be occupied by one electron, at maximum. (Strictly speaking, in the description of the electronic state the spin variable must be included.) This drastic difference in the collective behaviour of photons and electrons could, in fact, not be expected from what we know from single photons and electrons. Both of them have particle-like as well as wave-like properties. However, when many of them are present, they exhibit quite different a demeanour.

Prelude: a new constant enters the stage

One year before his epoch-making work on blackbody radiation in which he postulated the existence of an elementary quantum of action, Max Planck (1899) proposed a system of natural units that was based on four universal constants. They were the velocity of light c, the gravitational constant f, and two new constants a, b that he took from Wien's radiation law (which he, at the time, believed to be strictly valid). He determined the numerical values of a and b from measurements of both the total energy emitted from a blackbody source and the constant in the exponent of Wien's law. In retrospect, since we know that Wien's law is a good approximation to the true radiation law (found by Planck in 1900), it becomes clear that b is nothing but the famous elementary quantum of action known as Planck's constant h, while a is identical to the ratio of h and Boltzmann's constant k.

Requiring all four constants to take the numerical value unity, Planck arrived at a system of physical units which he was quite enthusiastic about. He wrote (Planck, 1899): ‘It might not be without interest to remark that with the aid of the two … constants a and b the possibility is given to establish units for length, mass, time and temperature, which, independently of special bodies or substances, necessarily retain their importance for all times and for all, even extraterrestric and extrahuman cultures and which therefore can be termed ‘natural measurement units’. However, at the time little (if any) attention was given to his proposal.

Quantum mechanics casts its shadows before it

It might be felt amazing that quantum effects show up in our macroscopic world. We know miraculous phenomena such as superconductivity, for instance, which classical physics is basically unable to explain. This makes it impossible to separate nicely the microcosm from the macroscopic world through a clear-cut borderline, or a more or less well defined transition region at least. It should be noticed that classical physics, notably statistical thermodynamics, in certain cases already runs into trouble in what should be its domain of validity. This became obvious from observations, mostly at low temperatures, that manifestly differed from basic classical predictions.

Those discrepancies were found, above all, in thermodynamics. In particular, the specific heat capacity of materials with decreasing temperature did not obey the rule that followed from classical (i.e., Boltzmann) statistics. This rule states that the energy can be calculated by assigning to any constituent the average energy kT/2 (k Boltzmann constant) per degree of motional freedom. (In the case of vibrations the potential energy gives rise to an extra contribution kT/2.) For instance, in an ideal gas formed from atoms, any atom has three translational degrees of freedom. Hence each atom contributes the energy 3kT/2, and we simply get the total energy of the sample on multiplication of this figure by the number of atoms. When the constituents of a gas are molecules, additional rotational and vibrational degrees of freedom come into play, which further enhance the average energy of a molecule.

Schrödinger's cat states

A prerequisite to getting quantitative information on the microscopic world is the existence of measuring apparatuses that indicate definite measuring results.The idea that a pointer of a measuring device is itself in an indeterminate state waiting for someone to make the wavefunction for the combined system (object + measuring apparatus) collapse by simply having a look on the apparatus, thus reading it out, appears absurd. I think, there is no doubt that measuring devices will behave like other objects in the macroscopic world in which there is no place for ‘intrinsic’ uncertainty.

Actually, this fact becomes questionable, at least it needs further theoretical investigation, when it is claimed that the quantum mechanical description applies also to macroscopic bodies, in particular measuring devices. (There are physicists who do not shrink back from attributing a single wavefunction even to the whole universe!) The dilemma originates from the quantum mechanical rule following from the linearity of the Schrödinger equation, that a pure quantum state remains a pure quantum state during its evolution. So, assuming that both the microscopic system to be observed and the measuring apparatus are initially in a pure state each, the combined system (object + measuring apparatus) will still be in a pure state after the interaction has taken place. Hence one will need ‘someone’ to bring about the collapse of the wavefunction. (In a realistic approach to this problem, the agent is recognized to be the environment with its virtually infinite number of degrees of freedom.) Schrödinger, being fully aware of this dilemma, illustrated it ingeniously through a fictitious ‘diabolic device’.

This is an unconventional introduction to quantum mechanics. Emphasis is laid on the physical aspects rather than the formal apparatus. (As a side-effect, the book will be easier to read than usual textbooks in which mathematics is predominant.) To elucidate the novel features displayed by the quantum world, comparison is made with classical physics whenever possible, thus emphasizing the anti-intuitive (in German: ‘unanschaulich’) character of the new physics.

It is my goal to discuss thoroughly the basic quantum mechanical concepts, such as quantum states and their preparation, quantum mechanical uncertainty, quantum correlations and quantum measurement. In addition, selected experiments are reported that show up the potential of quantum theory and, by the way, tell us of the ingenuity of the researchers. Many of those experiments are taken from quantum optics which, in fact, is a wonderful playing ground for quantum physicists. In particular, exciting theoretical concepts, such as the famous gedanken experiment of Einstein, Podolsky and Rosen, and quantum teleportation, could be realized experimentally. Those achievements, of course, find their due place in this book.

Attention is also given to the mysterious interrelationship between spin and statistics, which makes the behaviour of Fermi and Bose gases so different. A special paragraph is devoted to the experimental verification of Bose–Einstein condensation, one of the highlights of experimental research in the last two decades.

Furthermore, fundamental interaction processes, notably scattering, are treated, as well as macroscopic quantum effects, such as superconductivity and the Josephson and quantum Hall effects. Finally, the intriguing concept of quantum computers is briefly discussed.

Why quantum computing?

As we all know very well, conventional (electronic) computers have enormous power which, moreover, is constantly increasing. With their help gigantic quantities of data can be handled and evaluated. This is of crucial importance, for instance, in modern high energy physics experiments. On the other hand, computers can be used to model very complex systems with the aim of predicting, for example, future developments in our environment, notably climatic changes. Thanks to their incredible speed computers are, of course, destined to treat mathematical problems for which specific algorithms exist very efficiently. To mention a simple example, the digits of π are readily evaluated up to a very high order. Last, but not least, their potential to solve differential equations numerically, even when they are nonlinear, makes computers indispensable in modern research.

So, one might ask, is there any reason to look for fundamentally new ways of computing as they are seemingly offered by quantum processes? What advantages can be expected when the laws of quantum mechanics, rather than those of classical physics, are exploited for computation? Two answers were given: first, the simulation of complex quantum systems on classical computers meets serious difficulties that might be overcome with the help of quantum computers. This idea was suggested in 1982 by R. Feynman, and in the 1990s it was indeed shown by several groups of researchers that quantum mechanical systems for which no efficient simulation on a classical computer is known can be efficiently simulated on a quantum computer.

As in classical physics, also in quantum mechanics: physical processes are bound to obey conservation laws. However, there is an important difference. While in classical physics we can suppose that all variables have definite values (irrespective of whether we know them or not), quantum mechanics allows them to be intrinsically uncertain. In those cases conservation laws apply only to ensemble averages (expectation values), and hence are less informative. They show their full strength in measurement processes. Then the measured values satisfy the respective conservation laws in any individual experiment. Let me remind you of spin measurements on a system of two spin-½ particles, being in a state with zero total spin, which we used in Section 5.5 to illustrate the EPR paradox. The spin components, with respect to an arbitrarily chosen direction, of the two particles are indefinite; however, measuring them yields definite values (+½ or −½) that add up to zero, thus confirming the angular conservation law.

In the following, we will consider some realistic experiments for which conservation laws provide a simple explanation.

Energy and momentum conservation

For an understanding of the interaction between matter and radiation, Einstein's photon concept (actually, he spoke of ‘light quanta’) proved to be very helpful. He assumed photons to be particle-like constituents of electromagnetic radiation, in particular light. They have the energy hν, where ν is the frequency of the radiation field. (Since the electromagnetic spectrum extends over an extremely wide range, there is a huge number of different sorts of photons.) The crucial point is that photons are generated or annihilated only as a whole. This property gives rise to typical quantum effects.