In Chapter 8, we devote a whole chapter to how quantum amplifiers work. We discuss phase-preserving and phase-sensitive amplification and how to realize these with heterodyne and homodyne measurements. Focusing on quantum superconducting circuits, we discuss three-wave and four-wave mixing, the different types of circuits to build in order to realize amplification, and how it can go wrong.

In Chapter 6, we discuss quantum jumps, or leaps, beginning with a historical discussion of quantum jumps, or leaps, and giving the motivating example of blinking atoms in trapped ions and quantum jumps in superconducting circuits. Despite the disruptive name, we discuss how jumps fit into the category of continuous measurement. In some cases, quantum jumps can be predicted, and even reversed. We build up the mathematical formalism by discussing the quantum Zeno effect, Lindblad-type master equations, leading to jump and no-jump dynamics - an inseparable combination of discrete and continuous dynamics. In fact, the discrete nature of the jump is illusory. We discuss the dynamics of a quantum jump and the transition from jumps to diffusion.

This first chapter gives an introduction to the book and guides the reader through much of the history of the field of quantum mechanics, focusing on what we view as the most important episodes in the field. We discuss the essential properties of quantum mechanics and how they have been reimagined in the decades that followed the era of the founders.

Chapter 9 explores many interesting phenomena that arise in the physics of continuous quantum measurement. We discuss measurement reversal (or “quantum uncollapse”), the most likely path of continuous quantum measurements, the joint simultaneous measurements of noncommuting observables, and entanglement of distant quantum systems by continuous measurement.

A groundbreaking event in the development of the quantum theory of atomic structure is the remarkable discovery by Johann Jacob Balmer in 1884. He figured out that Anders Jonas Angstrom's measurements of the wavelengths = 4101.2, 4340.1, 4860.74, and 6562.10 Å of the four prominent lines in the spectrum of the hydrogen atom fit the formula, where b = 3645.6 Å, and n = 3, 4, 5, 6. This is an outstanding illustration of the arithmetic genius that Balmer was, being an arithmetic teacher in a girls’ school. Balmer's formula was independently rediscovered by Johannes Robert Rydberg six years later, and rationalized in the planetary model of the old quantum theory proposed by Niels Bohr in 1913. Orbits, however, are intangible, position and momentum measurements not being compatible. Solution to the Schrödinger equation with appropriate boundary conditions satisfactorily account for the Balmer–Rydberg–Bohr formula, explaining the discrete eigenenergies of the hydrogen atom. In Section 5.1, we discuss the bound-state solutions of the Schrödinger equation for the hydrogen atom. However, the Schrödinger equation does not fully account for the degeneracy of the energy levels of (even the nonrelativistic) hydrogen atom. This bemusing condition is addressed in the Sections 5.2 and 5.3 of this chapter, and the continuum eigenfunctions are discussed in Section 5.4.

5.1 Eigenvalues and Eigenfunctions of the Hydrogen Atom

As the very first atom in the periodic table, the hydrogen atom is the simplest one. It is the prototype of one-electron central field atomic systems (including ions) for which exact analytical solution to the Schrödinger equation can be obtained. The electron–proton two-particles Schrödinger equation for the hydrogen atom is

where and are respectively the notional position vectors in a laboratory frame of reference of the electron and the proton, and me and mp their respective masses. The gradient operators and seek space derivatives with respect to the electron and the proton coordinates respectively, and the interaction potential energy operator depends on the relative distance between the two particles. The center of mass of the two particles is located at

and the reduced mass

which is located at the position vector with respect to the center of mass.

A.1 Continuous, Dynamical, and Discrete Symmetries

The role of symmetry in natural laws has been underscored by many, notably by Eugene P. Wigner [1]. Symmetry plays a huge role in our understanding of the laws of nature. Plato (427–327 BCE) recognized that the only regular solids, whose faces are polygons having identical faces meeting at identical solid angles, are the (i) tetrahedron, (ii) cube, (iii) octahedron, (iv) dodecahedron, and (v) icosahedron. Their symmetry properties are celebrated in Euler's theorem

where V is the number of vertices, E the number of edges, and F the number of faces. In 1905, Einstein recognized the symmetry in Maxwell's laws of electrodynamics and formulated the special theory of relativity. This symmetry resulted from the mind-boggling invariance of the speed of light in all inertial frames of reference. It required for its account a non-Euclidean four-dimensional (flat) space–time continuum [Chapter 13 of Reference 2]. Maxwell's theory of electrodynamics engendered an unimaginable constancy of the speed of light in all inertial frames of reference, regardless of their mutual motion. Einstein harbored length-contraction and time-dilation to account for the invariance of the speed of light, which is a fundamental symmetry principle. About ten years later, Einstein interpreted gravity in terms of an equivalence principle, inspired yet again by symmetry. A few years later, in 1918, Emmy Noether established that each conservation law is in essence an equivalent expression of an underlying symmetry. Rudimentary introduction to some of these ideas is now found even in undergraduate texts, such as Reference [2]. In this chapter, we discuss the role of symmetry in atomic structure and dynamics.

The hallmark of quantum theory is the recognition of employing operators to represent dynamical variables of classical mechanics. The simplest symmetry operation is translation through an infinitesimal displacement in homogeneous space.

The GMRT (Giant Metrewave Radio Telescope) is an array of thirty fully steerable parabolic radio telescopes having a diameter of 45 meters for radio astronomical research at meter wavelengths. It is an extremely advanced facility set up by the National Centre for Radio Astrophysics at Khodad (near Pune, India).

Theories of the formation of structure in the Big-Bang Universe predict the presence of proto galaxies or proto clusters of galaxies made up of clouds of neutral hydrogen gas before their gravitational condensation into galaxies. It should in principle be possible to detect these through the well-known radio line emitted by neutral hydrogen at a frequency of 1420 MHz. (see Fig. 7.4 in Chapter 7). The line is, however, expected to be very weak and red-shifted to meter wavelengths because of the expansion of the universe between emission, billions of years ago, and detection at the present epoch. Visit: http://www.ncra.tifr.res.in/ncra/gmrt/ about-gmrt/goals-of-gmrt (courtesy: NCRA , TIFR, India). (downloaded on 26 March 2023).

In the previous seven chapters, we have introduced the barebones of quantum mechanics. The methodology we reviewed equips us to study the physical universe we live in. Spectroscopy is a powerful experimental technique used to investigate matter. Basically, spectroscopy investigates a target system through its interaction with a probe, which can be electromagnetic radiation, or elementary or composite particles, or their combination. Our prototype of the target system is an atom. Atomic spectroscopy, or more generally atomic physics and quantum physics developed synchronously. These disciplines continue to interpenetrate each other to jointly push the frontiers of science, engineering, and technology.

8.1 Spectroscopic Oscillator Strengths

Spectroscopy literature is embellished by terminology from classical, semi-classical, and quantum-mechanical models. An all too pervasive term employed in the description of spectral transitions is the oscillator strength [1, 2]. It has its origins in the earliest explanatory models that used classical mechanics. The phenomenology pertaining to the spectroscopic oscillator strength is now comprehensively integrated into the fully quantum mechanical scheme. It is insightful to track the metamorphosis of the term oscillator strength from its earliest avatar in semi-classical physics to its current usage within the quantum models.

Let us consider a spectral transition in an atom which results from the absorption of electromagnetic energy. It is natural to inquire how much power is pumped by the electric field into the atomic system.

From our day-to-day experience, we develop our notion of reality. However, our perception of physical properties, such as position and momentum, needs refinement to describe natural phenomena correctly. In this chapter we introduce fundamental principles of the quantum theory that does so with enduring cogency. The laws of nature that govern the functioning of the physical universe cannot be accounted for using classical physics of Newton, Lagrange, and Hamilton. Foundational principles and mathematical structure of quantum theory are introduced in this chapter.

1.1 Quantum versus Classical Physics Theories

In classical physics, the mechanical state of a system is represented by a point in the position– velocity phase space, or equivalently in the position–momentum phase space. The entire theoretical formalism of Newtonian–Lagrangian–Hamiltonian mechanics is based on this Galilean conjecture. The classical hypothesis seems appropriate in a large number of physical situations that concern our day-to-day experiences. With this ansatz, the temporal evolution of a system is described by the trajectory of the point in the phase space. The trajectory is obtained from the equation of motion that governs the time dependence of position and velocity, represented respectively by q and (or alternatively the time dependence of position and momentum, represented respectively by q and p). The alternative equations of motion – Newton’s, Lagrange’s, and Hamilton's – are equivalent. Their applicability must be reconciled with the upper limit on accuracy with which conjugate physical properties of an object, viz. position and momentum, are simultaneously measured. There is, however, an inverse relation between the accuracy of simultaneous measurement of these two properties. The more accurately you measure either, the less accurately can you measure the other. The act of measuring either of the conjugate properties requires an observation. Observations resulting in accurate measurements of conjugate variables are, however, not compatible with each other. Heisenberg's principle of uncertainty is the quantitative expression of this law of nature. It is expressed as a rigorous mathematical inequality that is neatly written in a compact form. Nonetheless, we refrain from advancing its mathematical expression too soon. It has no classical analogue. It cannot be written in any terms of what we are familiar with from classical mechanics.

A graduate course on quantum mechanics is a daunting task – for both students and teachers. Students come for such a course with a fair amount of background in classical physics, classical in the sense that it is time-tested. They are familiar with the works of Newton, Lagrange, Hamilton, Euler, etc. In this scheme, an object's physical state is described by its position q and momentum p, and temporal evolution by Hamilton's equations of motion for the time rates and. Their experience with classical physics entrenches their faith in it and builds their intuition, but they must now be taught that a physical theory that requires simultaneous knowledge of position and momentum is fundamentally untenable. Students must now settle with the fact that classical mechanics ‘works’ only when it is a very good approximation to a more appropriate theory of Nature, which is quantum mechanics. The foundational principles of quantum mechanics conflict with those of classical physics, causing confusion and doubt. Overcoming the resulting befuddlement involves learning what seems like an abstract formalism, which nonetheless turned out to be an unassailable theory of practical value. It changed our lives in the last century with quantum devices, and is now all set to take another leap into the second quantum revolution. It ushers in mind-blowing technology driven by entanglement and quantum computing.

The route to diligent applications of quantum mechanics begins with a shock. Students must grapple with formidable challenges on the path to comprehending consequential principles in a mystified territory. They have to develop proficiency in new methodologies involving abstract mathematics before they can see for themselves that quantum theory simply works; nothing succeeds like success. They can then use the theory to propel the frontiers of sciences, engineering, and technology. Amid this bewilderment, a graduate course in quantum mechanics is as romantic as it is challenging. One must learn to see beyond the corners of your vision, acquire rigorous capabilities in mathematics, enjoy luminous discourses between brilliant minds, cultivate an inventiveness to develop new technology that impacts human life, and understand the cosmos. Quantum mechanics: formalism, methodologies, and applications is a vast subject, very young compared to classical physics, but a very rich field to which some of the most outstanding intellectuals have made dazzling contributions during the past hundred odd years.

Statistics is intrinsic to quantum theory. Description of nature requires it. In classical physics, we take recourse to statistical analysis either when we are not interested in a detailed solution, or when we are unable to cope up with a large amount of data. Statistical methods enable us to extract important and useful information about the system to understand its physical properties. For example, rather than keeping track of kinetic energies of individual molecules, we determine their average, and benefit from the notion of temperature. The great debate between Albert Einstein and Niels Bohr in the late 1920s and in the 1930s eventually established (especially after the work of John Bell in 1964, discussed in Chapter 11) that the role of statistics in describing nature is far deeper than was suspected earlier. Statistics must be invoked to describe physical properties of nature even when we are dealing with a single particle, or for that matter even vacuum. In this chapter, we study another important manifestation of the role of statistics in quantum theory; this comes from the identity of particles in nature.

9.1 Symmetry Properties of Wavefunctions for Bosons and Fermions

If we have ten electrons, classical physics would regard the particles as distinguishable; the ten electrons could be numbered from one through ten, and one could, at least in principle, follow the dynamics of a particular, say the seventh, electron. Electrons are fundamental particles in nature. They are not amenable to their description by classical statistics. They must be described by quantum theory. In an N-electron system, it is impossible to tag any of these electrons by a number, such as the seventh. This impacts how the electrons occupy energy levels available to them.

The smallest many-particles system is a pair of identical particles. If we subject the pair of particles to an interchange, effected by the operator, and then interchange them yet again, we must get the original state. We expect the double interchange to be the identity operator. We therefore write

where the two-particle state is represented by, with representing the coordinates of the two particles. These coordinates would include not only the three space coordinates but also the spin coordinate.

Approximations are made in mathematics, physics, engineering, and life sciences for a variety of reasons such as (a) obtaining an exact solution is not possible and (b) an exact solution is not necessary, subject to a given context. Both of these factors are important in the development of approximations. We have already seen in Chapter 2 that out of the infinite alternative paths accessible to a system due to their entanglement/superposition, the classical path that makes action stationary results from the condition that action is much larger than the PEB constant. When this condition is satisfied, laws of classical physics provide an excellent approximation to quantum mechanics. In Chapter 2, we observed that the interpretation of the wavefunction as probability amplitude became the basis of the path integral formulation of quantum mechanics, in which the propagator's phase is the action. Considering this relationship, it is appropriate to wonder if the Schrödinger equation could have been predicted from classical mechanics, if only the wavefunction (Eq. 6.22, below) was introduced, even just as a definition. In Section 6.1, we demonstrate, in hindsight, how this could have been possible. Of course, historical development of quantum mechanics took place on a different course. We also demonstrate the inverse of this process, viz., how the classical equation of motion is obtainable as an approximation from quantum mechanics in the limit h→0. An approximation scheme often implemented to solve the Schrödinger equation, called the Wentzel–Kramers–Brilloiun–Jeffrey (WKBJ) method, is based on the connections between classical and quantum mechanics. It is introduced in Section 6.1. In Section 6.2, we discuss perturbation methods to obtain approximate solutions to unsolvable problems. Time-independent perturbation methods are discussed in Section 6.2, and time-dependent perturbation methods in Section 6.3. Major non-perturbative approaches to attack complex problems include the adiabatic approximation, the sudden approximation, and the method of variation. These are introduced in Section 6.4.

6.1 Classical Limit of the Schrödinger Equation and the WKBJ Approximation

Readers who are well conversant with the Hamilton–Jacobi (HJ) theory in classical mechanics [1] will be able to skip a significant part of the discussion in this section. It is, however, included here, even if only briefly, for the sake of completeness.

In this chapter we show that a reformulation of quantum mechanics, namely Feynman's path integral approach, is equivalent to the earlier ones, due to Schrödinger and Heisenberg. We shall introduce and apply it in this chapter to interpret the Aharonov–Bohm effect, which is considered to be a “quantum wonder.” It underscores the importance of electromagnetic potentials. We shall find that in addition to the dynamical phase introduced in Chapter 1, another angle, namely the geometric phase (Pancharatnam–Berry phase), is an important physical property of the wavefunction whose consequences are measurable.

2.1 Propagator: Propagation of a Quantum of Knowledge

Notwithstanding the equivalence of wave mechanics and matrix mechanics, Schrödinger and Heisenberg did not appreciate each other's methodologies [1]. In 1926, Schrödinger said: “I knew of [Heisenberg’s] theory, of course, but I felt discouraged, not to say repelled, by the methods of transcendental algebra, which appeared difficult to me, and by the lack of visualizability.” Heisenberg's dislike for Schrödinger's wave mechanics was equally intense. In the same year, in a letter to Pauli, he wrote “The more I think about the physical portion of Schrödinger's theory, the more repulsive I find it…. What Schrödinger writes about the visualizability of his theory ‘is probably not quite right,’ in other words it's crap.” In the mean time, Paul A. M. Dirac, who had made path-breaking contributions [2] in that eventful year (1926) to the formulation of quantum mechanics, had drawn a penetrating analogy [3] between classical and quantum mechanics. Dirac's remark drew the attention of Richard P. Feynman, who used it to build the path integral approach to quantum mechanics based on the variational principle.

Feynman was introduced to the principle of extremum action [4] rather early, while in high school. He reveals in his book [5] that “my Physics teacher … Mr. Bader … told me something … absolutely fascinating…. Every time the subject comes up, I work on it … the principle of least (rather, ‘extremum’) action.”