Short introduction to Bohmian mechanics

Sheldon Goldstein in a Foundations of Physics paper [1] indicates (p. 341): “Since macroscopic objects are normally regarded as built out of microscopic constituents …there can be no problem of macroscopic reality per se in Bohmian mechanics.” Goldstein in the same paper remarks (p. 342) that this |ψ|2 has “a status very much the same as that of a thermodynamic equilibrium.”

Basil Hiley the closest collaborator of David Bohm, says the following (Hiley [2] (p. 2)): “What Bohm (1952a; 1952b) did was to show how to retain a description of all the usual properties of a classical world and yet remain completely within the quantum formalism.” The Bohm (1952) [3] [4] references in that quote refer to the original work of Bohm in which he sets out the basics of the Bohmian mechanics.

As we will see later in the next section of this chapter, the appearance of an “additional term” in the Hamilton-Jacobi equation, is the hallmark of Bohmian mechanics and it is very often termed the “quantum potential.” Hiley [2] (p. 2) remarks that since the Bohmian momentum is a well-defined function of position and time, an ensemble of trajectories can be found when the quantum potential is non-zero. A unique, classical path is found when the quantum potential is zero. From the outset, we hope the reader can savor the beauty of this simple but very powerful result. Bohmian mechanics indeed shows this very gentle transition from quantum mechanics to classical mechanics via the value of the quantum potential.

In this chapter we describe a technique to deal with identical particles that is called second quantization. Despite being a technique, second quantization helps a lot in understanding physics. One can learn and endlessly repeat newspaper-style statements particles are fields, fields are particles without grasping their meaning. Second quantization brings these concepts to an operational level.

Since the procedure of second quantization is slightly different for fermions and bosons, we have to treat the two cases separately. We start with considering a system of identical bosons, and introduce creation and annihilation operators (CAPs) that change the number of particles in a given many-particle state, thereby connecting different states in Fock space. We show how to express occupation numbers in terms of CAPs, and then get to the heart of the second quantization: the commutation relations for the bosonic CAPs. As a next step, we construct field operators out of the CAPs and spend quite some time (and energy) presenting the physics of the particles in terms of these field operators. Only afterward, do we explain why all this activity has been called second quantization. In Section 3.4 we then present the same procedure for fermions, and we mainly focus on what is different as compared to bosons. The formalism of second quantization is summarized in Table 3.1.

In the same series of experiments which led to the discovery of superconductivity, Kamerlingh Onnes was the first who succeeded in cooling helium below its boiling point of 4.2 K. In 1908, he was cooling this liquefied helium further in an attempt to produce solid helium. But since helium only solidifies at very low temperatures (below 1 K) and high pressures (above 25 bar), Kamerlingh Onnes did not succeed, and he turned his attention to other experiments.

It was, however, soon recognized that at 2.2 K (a temperature reached by Kamerlingh Onnes), the liquid helium underwent a transition into a new phase, which was called helium II. After this observation, it took almost 30 years to discover that this new phase is actually characterized by a complete absence of viscosity (a discovery made by Kapitza and Allen in 1937). The phase was therefore called superfluid. In a superfluid, there can be persistent frictionless flows – superflows – very much like supercurrents in superconductors. This helium II is, up to now, the only superfluid which is available for experiments.

The most abundant isotope of helium, 4He, is a spinless boson. Given this fact, the origin of superfluidity was almost immediately attributed to Bose condensation, which was a known phenomenon by that time.

Preface

Courses on advanced quantum mechanics have a long tradition. The tradition is in fact so long that the word “advanced” in this context does not usually mean “new” or “up-todate.” The basic concepts of quantum mechanics were developed in the twenties of the last century, initially to explain experiments in atomic physics. This was then followed by a fast and great advance in the thirties and forties, when a quantum theory for large numbers of identical particles was developed. This advance ultimately led to the modern concepts of elementary particles and quantum fields that concern the underlying structure of our Universe. At a less fundamental and more practical level, it has also laid the basis for our present understanding of solid state and condensed matter physics and, at a later stage, for artificially made quantum systems. The basics of this leap forward of quantum theory are what is usually covered by a course on advanced quantum mechanics.

Most courses and textbooks are designed for a fundamentally oriented education: building on basic quantum theory, they provide an introduction for students who wish to learn the advanced quantum theory of elementary particles and quantum fields. In order to do this in a “right” way, there is usually a strong emphasis on technicalities related to relativity and on the underlying mathematics of the theory. Less frequently, a course serves as a brief introduction to advanced topics in advanced solid state or condensed matter.

Most introductory courses on physics mention the basics of atomic physics: we learn about the ground state and excited states, and that the excited states have a finite life-time decaying eventually to the ground state while emitting radiation. You might have hoped for a better understanding of these decay processes when you started studying quantum mechanics. If this was the case, you must have been disappointed. Courses on basic quantum mechanics in fact assume that all quantum states have an infinitely long life-time. You learn how to employ the Schrödinger equation to determine the wave function of a particle in various setups, for instance in an infinitely deep well, in a parabolic potential, or in the spherically symmetric potential set up by the nucleus of a hydrogen atom. These solutions to the Schrödinger equation are explained as being stationary, the only time-dependence of the wave function being a periodic phase factor due to the energy of the state. Transitions between different quantum states are traditionally not discussed and left for courses on advanced quantum mechanics. If you have been disappointed by this, then this chapter will hopefully provide some new insight: we learn how matter and radiation interact, and, in particular, how excited atomic states decay to the ground state.

Our main tool to calculate emission and absorption rates for radiative processes is Fermi’s golden rule (see Section 1.6.1). We thus start by refreshing the golden rule and specifying its proper use. Before we turn to explicit calculations of transition rates, we use the first part of the chapter for some general considerations concerning emission and absorption, those being independent of the details of the interaction between matter and electromagnetic radiation. In this context we briefly look at master equations that govern the probabilities to be in certain states and we show how to use those to understand properties of the equilibrium state of interacting matter and radiation. We show that it is even possible to understand so-called black-body radiation without specifying the exact interaction Hamiltonian of matter and radiation.

In this chapter (summarized in Table 4.1) we present a simple model to describe the phenomenon of magnetism in metals. Magnetism is explained by spin ordering: the spins and thus magnetic moments of particles in the material tend to line up. This means that one of the spin projections is favored over another, which would be natural if an external magnetic field is applied. Without the field, both spin projections have the same Zeeman energy, and we would expect the same numbers of particles with opposite spin, and thus a zero total magnetic moment. Magnetism must thus have a slightly more complicated origin than just an energy difference between spin states.

As we have already seen in Chapter 1, interaction between the spins of particles can lead to an energy splitting between parallel and anti-parallel spin states. A reasonable guess is thus that the interactions between spin-carrying particles are responsible for magnetism. The conduction electrons in a metal can move relatively freely and thus collide and interact with each other frequently. Let us thus suppose that magnetism originates from the interaction between these conduction electrons.

We first introduce a model describing the conduction electrons as non-interacting free particles. Within this framework we then propose a simple wave function which can describe a magnetic state in the metal. We then include the Coulomb interaction between the electrons in the model, and investigate what this does with the energy of our trial wave function.