Loss and Dissipation: The RLC Circuit

The effects of energy loss or dissipation is well-known and understood in classical systems. It is the source of heat in LCR circuits and in the application of brakes in a vehicle or why a struck bell does not ring indefinitely. Understanding quantum behavior begins with understanding the Hamiltonian for the problem. Classically, loss arises from a coupling of the Hamiltonian for an isolated quantum system to a continuum of states. We look at such a Hamiltonian and develop the equations of motion following the rules of quantum mechanics and find that even in a quantum system, this coupling leads to loss and non-conservation of probability in the otherwise isolated quantum system. This is the Weisskopf–Wigner formalism that is then used to understand the quantum LCR circuit. The same formalism is used in Chapter 15 for the decay of isolated quantum systems by coupling to the quantum vacuum and the resulting emission of a photon.

Scattering, Quantum Current, and Resonant Tunneling

Having examined free particles and particles that are confined in space by a potential energy term, we now consider the impact of a disturbance in the flat energy landscape for a free particle. By disturbance we means some kind of fixed “obstacle” which is either a positive (repelling) or negative (attractive) potential. We are interested in determining the impact on the free particle. Continuing to work mostly in one dimension, the particle described by a plane wave corresponding to momentum moving in the positive direction (a positive k−vector in the x−direction), we study elastic scattering. In one dimension, this means that we determine the probability that the particle is transmitted (continuing in the forward direction) or reflected (now moving in the backward direction.) We will also determine the nature of the solution inside the potential and in the case that the potential energy maximum is greater than the kinetic energy of the particle, we will show that the particle tunnels through the barrier. Interestingly, when we have two barriers, we can find conditions where the probability that the particle is transmitted is unity. This is the result of resonance, a feature of the wave-like nature of the particle’s wave function.

Bosons and Fermions: Indistinguishable Particles with Intrinsic Spin

If we imagine a Hamiltonian, H^(r1,r2), describing two identical particles at positions r1 and r2 and then we interchange the particles, the Hamiltonian will be unaffected, i.e. H^(r1,r2)=H^(r2,r1). If we introduce an exchange operator P^r1,r2 such that P^r1,r2H^(r1,r2)=H^(r2,r1)P^r1,r2=H^(r1,r2)P^r1,r2, we see that they commute, or [P^r1,r2,H^(r1,r2)]=0. We know then that P^r1,r2andH^(r1,r2) have common eigenfunctions. We can then easily show that the eigenfunctions of the exchange operator must be either even or odd. Experiments show that odd exchange symmetry corresponds to half-integer spin particles called fermions, while even exchange symmetry corresponds to integer spin particles called bosons. The notes then discuss the implications of the new postulate and then presents the Heitler–London theory and the Heisenberg exchange Hamiltonian which has been so successful in predicting molecular structure.

Heisenberg Operator Approach: Nano-Mechanical Oscillator (Part 2) and the Quantum LC Circuit

With the knowledge of the new design rules in Chapter 7, we use this new insight to find the eigenvectors for the nano-vibrator problem, and then we use the same approach to examine the quantum LC circuit. While the usual approach is to use Kirchhoff’s laws to analyze a simple circuit classically, we first see that Hamilton’s equations can in fact be used, giving the same classical result. But then, using the new design rules and the knowledge of the total energy in the circuit, we identify a canonical coordinate and a conjugate momentum that have nothing to do with real space and motion of a particle of mass m. At the same time, consistent with the Schrödinger picture, we continue to see that the time evolution of an observable such as position, x(t), or current, i(t), is not part of the solution. Given that Hamilton’s equations give the same result as Kirchhoff’s law but the quantum solution does not, reinforces the idea that the quantum description is showing features that cannot be imagined with a viewpoint based on classical (i.e. non-quantum) analysis.

Quantum Electromagnetics

Chapter 15 derived the fundamental theory and eigenstates for the quantized radiation field and then showed how the quantum vacuum gives rise to spontaneous emission. This chapter now goes more deeply into the meaning and implications of the quantized field. The polarization of a photon can be used as a qubit and the photonic qubit is called a flying qubit. It enables transmission of information from one node to another. Spontaneous emission is shown to enable creation of an entangled state between a photonic qubit and the spin of an electron. Spontaneous emission can also degrade the performance of some device designs and in other devices it can enhance performance such as for a single photon emitter. In this we show how to engineer the vacuum to control spontaneous emission.

Quantum Radiation Field: Spontaneous Emission and Entangled Photons

One of the greatest successes in quantum theory, and certainly one of the more important parts for application to devices and applications is the prediction of the emission of light through the quantization of an electromagnetic field. Broadly, this is the field of quantum electrodynamics. In this chapter, we develop the Hamiltonian for the classical electromagnetic field. It is seen that the Hamiltonian for each mode (identified by the k-vector and polarization of the field) of the plane wave electromagnetic field is identical to that of the harmonic oscillator. One unit of energy, ℏω, in a mode is a called a photon. The eigenkets for the system are number states (Fock states). We then consider a two-level system described by a Hamiltonian which couples the two-level quantum system to the quantized electromagnetic field. Using the Weisskopf–Wigner formalism developed in Chapter 14, we solve the equations of motion for the time dependent Schrödinger equation assuming the system starts in the excited state with no radiation present in the vacuum field. The results show the creation of one unit of energy in an electromagnetic mode corresponding to the emission of a photon. The excited state probability decays exponentially with the emission of this photon. We consider the important and special case of such a two-level system but in a cavity restricting the radiation field to a single mode. The Jaynes–Cummings Hamiltonian shows that this system, if started in the excited state, Rabi oscillates with no radiation incident on the system.

Quantum Measurement and Entanglement: Wave Function Collapse

The postulates presented at this point are generally agreed upon as being the primary set. But in the course of these postulates, there is no mention of the consequences of measurement. This chapter discusses this problem and the solution as provided by the Von-Neumann postulate. The concept of the projection operator is introduced, and this leads naturally to the study of the quantum entangled state. The results show in part the origin of the struggle that Einstein and others had with quantum, and the Einstein, Podolsky, and Rosen (EPR) paradox. Quantum entanglement is the key to advanced ideas in quantum encryption, teleportation, and quantum computing.

Periodic Hamiltonians and the Emergence of Band Structure: The Bloch Theorem and the Dirac Kronig–Penney Model

Crystals are defined by their atomic makeup and their crystal symmetry. The spatial symmetry leads to a periodic potential that localizes the electron or carrier at the lattice sites corresponding to the location of the atoms. But because of the periodicity of potential the particles wave function extends throughout the entire crystal, not localized at a specific atom. This delocalization of the wave function gives rise to the band structure in crystals such as those that are semiconductors. This chapter explores the physics that emerges in the wave function when the time independent Schrödinger equation is solved for a periodic potential.

Two-State Systems: The Atomic Operators

In the digital world, the concepts of on and off or high and low or 0 and 1 are common classical two-state systems. Quantum systems can be similarly configured, as we saw in Chapter 9 with the demonstration of Rabi oscillations. Two-state or few-state systems are so important that a powerful algebra has been developed to study and explore these systems. A similar algebra emerged from the algebra developed for spin ½ particles. While Chapter 10 discussed the spinors and spin matrices and the corresponding Pauli matrices, in this chapter the corresponding commutators are determined for the various atomic operators first introduced in Chapter 15. We then move to the Heisenberg picture including the operators for the vacuum field. The Heisenberg equations of motion are derived following the rules in Chapter 8 when a classical electromagnetic field is present and then in the presence of the quantum vacuum to include the effects of decay. This provides the first means of handling the return of an excited population back to the ground state which is very challenging to deal with in the amplitude picture. This chapter is enormously important because it sets the stage for much more advanced studies in advanced texts that determine the impact of fluctuations of the field and correlations measured from single photon emitters.

Free Particle, Wave Packet and Dynamics, Quantum Dots, Defects and Traps

This chapter continues with a study of the time independent Schrödinger equation and seeks to contrast the quantum behavior of a free particle with that of a particle localized in a potential quantum well. A free particle can exist over all space or can be localized in a wave package. The wave packet is a coherent superposition of the plane waves that make up the wave function that localizes the particle because of constructive and destructive interference. The wave packet spreads out in time because the waves leading to constructive interference get out of phase. In Chapter 2, the particle was localized by a quadratic potential energy. Here, the potentials are described as piecewise constant. The approach is based on assuming a one-dimensional space, x, which is relevant to many problems in the laboratory. The solution is easily generalized to higher dimensions (x-y or x-y-z), but the physics remains the same. The objective is to understand the shape of the eigenfunctions in space and to be able to relate this to the probability density of locating the particle, as well as understanding the relevance of these systems to today’s technology.