Lev I. Deych

Fine Structure of the Hydrogen Spectra and Zeeman Effect

You might still have a vague recollection of me mentioning the spin–orbit coupling in Sect. 9.5.1, where I introduced the tensor product of spin and orbital spaces as a means to construct vectors representing both orbital and spin components of a quantum state (if you do not remember that, you would do yourself a favor by going back and rereading that part of the book). More specifically, the issue of spin–orbit coupling came up in the discussion of generic vectors in the tensor product space, which could be presented as a superposition of basis vectors, in which different spin states are paired with different orbital components. Such states can be called spin–orbit coupled because the orbital properties of a system in such a state can be changed by affecting its spin and vice versa. However, practically, such states can only be realized in systems with actual spin–orbit interaction contributing a special term containing a combination of spin and orbital operators to their energy and, correspondingly, quantum Hamiltonian. This interaction is quite common. It appears in many ordinary systems, such as atoms or semiconductors, and is responsible for a number of important phenomena. In atoms it gives rise to the spectral features known in the early days of quantum mechanics, while in semiconductors it brings about the relatively recently discovered effects allowing, for instance, to use the spin to control electron spatial flow. Combining spin and orbital phenomena in such nontrivial situations is never a simple task, even if merely because it doubles the number of equations that must be solved. At the same time, the phenomena resulting from the spin–orbit interaction are way too important to be simply ignored and shall be discussed even if you only start getting comfortable with intricacies of the quantum description of the world. Therefore, in this section, I am giving you a chance to learn about some aspects of the spin–orbit interaction in a relatively non-threatening environment by considering, again, a simple model of a single electron in a hydrogen-like atom.

Representations of Vectors and Operators

We have managed to get through four chapters of this text without specifying any concrete form of the state vectors, and treating them as some abstractions defined only by the rules of the games that we could play with them. This approach is very convenient and rewarding from a theoretical point of view as it emphasizes the generality of quantum approach to the world and allows to derive a number of important general results with relative ease. However, when it comes to responding to experimentalists’ requests to explain/predict their quantitative experimental results, we do need to have something a bit more concrete and tangible than the idea of an abstract vector. The similar situation actually arises also in the case of our regular three-dimensional geometric vectors. It is often convenient to think of them as purely geometrical objects (arrows, for instance) and derive results independent of any choice of coordinate system. However, at some point, eventually, you will need to get to some “down-to-earth” computations, and to carry them out, you will have to choose a coordinate system and replace the “arrows” with a set of numbers—the vector components.

Observables and Operators

The version of classical mechanics based on forces and Newton’s laws resists any meaningful reformation into a quantum theory because it depends critically on such concepts (trajectory, acceleration, etc.) that do not correspond to any observable reality in the quantum world. More productive for finding links between classical and quantum realms is an alternative formulation, where energy rather than force takes the central role. There are two essential elements in this formulation of classical mechanics. One is the idea of canonical coordinates in the so-called phase space (as opposed to regular three-dimensional configuration space), and the other is the concept of Hamiltonian.

Spin 1/2

The model of a pure spin 1/2, detached from all other degrees of freedom of a particle, is one of the simplest in quantum mechanics. Yet, it defies our intuition and resists developing that pleasant sensation of being able to relate a new concept to something that we think we already know (or at least are used to thinking about). We call this feeling “intuitive understanding,” and it does play an important albeit mysterious role in our ability to use new concepts. The reason for this difficulty, of course, lies in the fact that spin is a purely quantum phenomenon with no reasonable way to model it on something that we know from classical physics. While the only known to me bulletproof remedy for this predicament is practice, I will try to somehow ease your pain by taking the time to develop the concept of spin and by providing empirical and theoretical arguments for its inevitability.

Free Electrons in Uniform Magnetic Field: Landau Levels and Quantum Hall Effect

The Zeeman effect, which we discussed in some detail in Sect. 14.2, originates from the interaction between the magnetic moment of an electron bound to an atom and a uniform magnetic field. Experimentally this effect is often observed in atomic gases but can also manifest itself with bound electrons in semiconductors and dielectrics. What is important is that the quantum states of the electrons in all these cases belong to discrete energy eigenvalues. In metals and in the conduction band of semiconductors, on the other hand, the energy levels of electrons belong to the continuum spectrum, and in some instances, electrons can even be treated as free particles. The interaction between such unbound, almost free electrons and the uniform magnetic field results in some fascinating effects which had played and are still playing an important role in physics.

Non-interacting Many-Particle Systems

Quantum mechanical properties of a single particle are an important starting point for studying quantum mechanics, but in real experimental and practical situations, you will rarely deal with just a single particle. Most frequently you encounter systems consisting of many (from two to infinity) interacting particles. The main difficulty in dealing with many-particle systems comes from a significantly increased dimensionality of space, where all possible states of such systems reside. In Sect. 9.4 you saw that the states of the system of two spins belong to a four-dimensional spinor space. It is not too difficult to see that the states of a system consisting of N spins would need a 2 N -dimensional space to fit them all. Indeed, adding each new spin 1/2 particle with two new spin states, you double the number of basis vectors in the respective tensor product, and even the system of as few as ten particles inhabits a space requiring 1024 basis vectors. More generally, imagine that you have a particle which can be in one of M mutually exclusive states, represented obviously by M mutually orthogonal vectors (I will call them single-particle states), which can be used as a basis in this single-particle M-dimensional space. You can generate a tensor product of single-particle spaces by stacking together M basis vectors from each single-particle space. Naively you might think that the dimension of the resulting space will be M N , but it is not always so. The reality is more interesting, and to get the dimensionality of many-particle states correctly, you need to dig deeper into the concept of identity of quantum particles.

One-Dimensional Models

One-dimensional models might appear in quantum mechanics in two, in a way, diametrically opposite situations. In one case, you can pretend that the potential energy of a particle changes only in one direction, such as a potential energy of a uniform electric field. Classically, this would mean a motion characterized by acceleration in one direction and constant velocity in perpendicular directions. By choosing an appropriate inertial coordinate system, you can always eliminate the constant velocity component and consider this motion as straightlinear. Quantum mechanically, this situation has to be described in the coordinate representation, and the respective coordinate wave function can be presented in the form

Harmonic Oscillator Models

It is as difficult to overestimate the role of harmonic oscillator models in physics in general and in quantum mechanics in particular as the influence of Beatles and Led Zeppelin on modern popular music. Harmonic oscillators are ubiquitous and appear every time when one is dealing with a system that has a state of equilibrium in the vicinity of which it can oscillate, i.e., in a vast majority of physical systems—atoms, molecules, solids, electromagnetic field, etc. It also does not hurt their popularity that the harmonic oscillator is one of the very few models which can be solved exactly.

Two-Level System in a Periodic External Field

I have already mentioned somewhere in the beginning of this book that while vectors representing states of realistic physical systems generally belong to an infinite-dimensional vector space, we can always (well, almost, always) justify limiting our consideration to a subspace of states with a reasonably small dimension. The smallest nontrivial subspace containing states that can be assumed to be isolated from the rest of the space is two-dimensional. One relatively clean example of such a subspace is formed by two-dimensional spinors in the situations when one can neglect interactions between spins of different particles as well as by the spin–orbital interaction. An approximately isolated two-dimensional subspace can also be found in systems described by Hamiltonians with discrete spectrum, if this spectrum is strongly non-equidistant, i.e., the energy intervals between adjacent energy levels △ i = Ei+1 − E i are different for different pairs of levels. Two-level models are very popular in various areas of physics because, on one hand, they are remarkably simple, while on the other hand, they capture essential properties of many real physical systems ranging from atoms to semiconductors.