Abstract This chapter begins its exploration of field quantization with an illustration of how a harmonic oscillator may be quantized, which motivates and preludes the subsequent discussion on canonical quantization, also known as the second quantization. Then it uses the formalism to quantize the Klein-Gordon and Schrodinger equations. Afterwards, goes on to present the quantization of the Maxwell equations, which establishes the celebrated photoelectric effect through a derivation of the Planck-Einstein relation and Compton-Debye formula. Finally, the chapter concludes with a comparison study of the thermodynamic properties of a harmonic oscillator with respect to its classical and quantum-mechanical descriptions.
The study of quantum properties of the harmonic oscillator is important in physics as many real-world systems oscillate harmonically. Motion of systems in a confined space is often modeled as being a quantized harmonic motion or, in first instance, is approximated by a harmonic motion. This chapter helps the reader to solve the equation employing an algebraic operator technique, which is based on the Dirac notation. This approach has several definite advantages and exploits the commutation relations among the operators involved and their properties. The chapter transforms the stationary Schrodinger equation into a second-order differential equation and finds the solution to the equation with the aid of a more advanced mathematical technique that involves special functions. The energy of a harmonic oscillator is quantized, with the sequence of values. The difference in energy between adjacent energy levels is equal to the energy of a single photon, ħω.
The time-dependent Schrödinger equation is a cornerstone of quantum physics and governs all phenomena of the microscopic world. However, despite its importance, its origin is still not widely appreciated and properly understood. The quantization of the forced harmonic oscillator is studied with the quantum variable (
x ,
v^) , with the commutation relation [
x ,
v^] =
iℏ/
m , and using a Schrödinger’s like equation on these variable, and associating a linear operator to a constant of motion
K(
x,v,t) of the classical system, The comparison with the quantization in the space (
x,p) is done with the usual Schrödinger’s equation for the Hamiltonian
H(
x,p,t), and with the commutation relation [
x ,
p^] =
iℏ. It is found that for the non-resonant case, both forms of quantization bring about the same result. The difference means that the H-approach case brings about more complex behavior in the quantum dynamics than the K-approach case, and that the H-approach case losses more information than the K-approach case.However, for the resonant case, both forms of quantization are different, and the probability for the system to be in the exited state for the (
x ,
v^) quantization has fewer oscillations than the (
x ,
p^) quantization, the average energy of the system is higher in (
x ,
p^) quantization than on the (
x ,
v^) quantization, and the Boltzmann- Shannon entropy on the (
x ,
p^) quantization is higher than on the (
x ,
v^) quantization. The parameter does not appear to be suitable for experimental measurement because the variation in the average value of the energy is quite modest. Entropy is a nice parameter to investigate experimentally because it differs significantly.
The quantization of the forced harmonic oscillator is studied with the quantum variable (x, v∧), with the commutation relation , and using a Schrödinger’s like equation on these variable, and associating a linear operator to a constant of motion K (x, v, t) of the classical system, The comparison with the quantization in the space (x, p) is done with the usual Schrödinger’s equation for the Hamiltonian H(x, p, t), and with the commutation relation . It is found that for the non-resonant case, both forms of quantization bring about the same result. However, for the resonant case, both forms of quantization are different, and the probability for the system to be in the exited state for the (x, v∧) quantization has fewer oscillations than the (x, p∧) quantization, the average energy of the system is higher in (x, p∧) quantization than on the (x, v∧) quantization, and the Boltzmann-Shannon entropy on the (x, p∧) quantization is higher than on the (x, v∧) quantization.
Abstract The purpose of this article is to construct an explicit relation between the field operators in Quantum Field Theory and the relevant operators in Quantum Mechanics for a system of N identical particles, which are the symmetrised functions of the canonical operators of position and momentum, thus providing a clear relation between Quantum Field Theory and Quantum Mechanics. This is achieved in the context of the non-interacting Klein–Gordon field. Though this procedure may not be extendible to interacting field theories, since it relies crucially on particle number conservation, we find it nevertheless important that such an explicit relation can be found at least for free fields. It also comes out that whatever statistics the field operators obey (either commuting or anticommuting), the position and momentum operators obey commutation relations. The construction of position operators raises the issue of localizability of particles in Relativistic Quantum Mechanics, as the position operator for a single particle turns out to be the Newton–Wigner position operator. We make some clarifications on the interpretation of Newton–Wigner localized states and we consider the transformation properties of position operators under Lorentz transformations, showing that they do not transform as tensors, rather in a manner that preserves the canonical commutation relations. From a complex Klein–Gordon field, position and momentum operators can be constructed for both particles and antiparticles.
This chapter begins by developing the relation between the quantum theory and the linear algebra. It discusses the most fundamental postulates of the theory and provides a phenomenological development of the Schrödinger wave equation (SWE) although the results might be gleaned from the classical Lagrangian and Hamiltonian with the Poisson bracket relations. Afterward some simple examples for the infinitely and finitely deep wells help clarify the basic postulates. The simple harmonic oscillator has perhaps the most sweeping implications and applications for the quantum theory. For this reason, we present the most applicable formalism for the Harmonic oscillator that uses the operator approach rather than the classical method of partial differential equations. The angular momentum and spin are presented for the study of the atom and for applications for nanodevices, and quantum computing. Next the representation formalism clarifies the distinction between the dynamical operators and the wave functions and their interrelations. The chapter discusses both timeindependent and time-dependent perturbation theories. The density operator combines the quantum theory with the classical theory for finding the behavior of a system. The density operator represents one of the basic concepts in the quantum theory. The remainder of the chapter introduces the more advanced material meant to introduce the quantum field theory starting with the second quantization and the propagator.