Quantum Fields and Waves: From Classical to Quantum Mechanics - Prof. David Peak, Study notes of Physics

Download Quantum Fields and Waves: From Classical to Quantum Mechanics - Prof. David Peak and more Study notes Physics in PDF only on Docsity! BK 2 1 Background, II In the beginning: a little “string theory” This course begins and ends with “strings.” The first kind is familiar: they’re what you see on a violin or what kids jump rope with. They’re made of twine or hemp or catgut or some such material, all of which in turn is made of simpler stuff like electrons, protons, and neutrons. The second kind nobody has ever seen (yet). They are at present still a speculation, a hope: they are supposed (hoped) to form a unified theory of everything. They are made of fundamental “string stuff” and so (according to the hope) is everything else. In this picture, electrons are vibrations of fundamental strings, while quarks are different vibrations, and so are photons, and so forth. The first kind of string seems a bit mundane, while the second seems like science fiction. But understanding details of how the first kind works will help us prepare for understanding one of the most actively investigated physical theories in human history. The theoretical apparatus used in the conventional contemporary understanding of the structure of matter is called “quantum field theory (QFT).” In general, a field is a smoothly varying function of space and time: ! = !( ! r , t) . In the macroscopic world of classical physics, smoothly continuous fields are useful for describing things like the temperature distribution in a bar of iron or the pressure variations in the air at different places in a concert hall during a piano recital. The classical field at each ( ! r , t) can be a scalar (a single number, such as temperature), a vector (a set of numbers arranged in a row or a column, such as the electric field), or some more complicated geometrical object (tensors, which are “fatter” arrays). In a quantum field, !( ! r , t) not only has geometric properties (scalar, vector, tensor), but it also is an operator whose action takes place in the space of particle states. In essence, what this obscure statement means is that the operator part of Ψ has the capability of creating or annihilating (i.e., destroying) particles (“quanta”) of a given kind with a given amount of momentum and energy at ( ! r , t). Here’s how we think quantum fields work. A quantum field is a tool in a kit of tools used to describe the physical universe called quantum mechanics. Quantum mechanics is not a theory of any physical thing, but rather a general framework for understanding what outcomes are possible when physical things interact with one another. Quantum field theory on the other hand, describes actual bits of matter. A fundamental tenet of quantum mechanics is that reality is what is measured. Prior to a measurement being made there is only possibility. For example, suppose we have an electron detector capable of measuring “electron-ness.” We turn it on and it clicks (i.e., “an electron is present”). Prior to the measurement, according to quantum mechanics, there was only the possibility the detector might click. The “electron” was neither “there” nor “not there” prior to the measurement. What was there, however, was a quantum field capable of creating electron-ness. Such a quantum field carries the possibility of making a click in the electron detector (i.e., for “creating an electron”). In this view, the universe is permeated with quantum fields and particles pop up from place-to-place and time-to-time as a result of the quantum field operator actions. This view of the universe is radically different from the classical view. In classical physics a particle is a particle; it has permanent intrinsic properties, such as its mass. But, as we will learn, according to special relativity mass is not necessarily permanent: it can materialize from energy and vice versa. Thus, QFT is not only a part of quantum mechanics it is also a consequence of relativity. But we are dangerously close to leaping ahead of ourselves. For now we need to slow down and be sure we understand the properties of simple, classical fields. BK 2 2 How a field Ψ changes from place-to-place and from time-to-time requires some dynamical rules. The dynamics of a classical particle (not a field) describes how its position in space varies in time. Position in space is the dependent variable for a classical particle and time is the single independent variable. The dynamical rule arises from Newton’s Second Law of Motion and takes the form of an ordinary differential equation: d 2 ! r dt 2 = ! F net /m . But, a field, Ψ, depends on several independent variables, so the associated dynamics require partial differential equations. A partial derivative measures how a function of several independent variables changes as one of its variables changes. For simplicity, suppose Ψ is a function of two variables: ! = !(x, t) . Then “the partial of Ψ with respect to x,” !" !x, is defined as !" !x = lim dx#0 "(x + dx, t)$"(x, t) dx ; in other words, !" !x is just the usual derivative of Ψ with respect to its x variable at one fixed value of its t variable; !" !t is defined similarly (i.e., hold x fixed, vary t). Example 1: Suppose !(x, t) = Asin(kx"#t), a sine wave of amplitude A and phase kx!"t . At a fixed x, as t increases the phase of the wave changes. On the other hand, the same phase can be found at increasing values of x provided that (change in x/change in t) = ω/k. Thus, the phase of the sine wave appears to propagate in the +x direction with a (phase or wave) speed v W =! k , and consequently Ψ is said to be a traveling sine wave. Treating t as fixed and varying x leads to !" !x = kAcos(kx#$t), while treating x as fixed and varying t leads to !" !t = #$Acos(kx#$t) . Note that both !" !x and !" !t are also functions of x and t. Ψ has four second partial derivatives, denoted ! 2 " !x 2 , ! 2 " !t 2 , ! 2 " !t!x #!(!"/!x) !t , and ! 2 " !x!t#!(!"/!t) !x . The latter two second-partials are called “mixed partials;” they are equal, meaning it doesn’t matter in which order you take the derivatives. For the given Ψ, ! 2 " !t!x =! 2 " !x!t = #$kAsin(kx#$t) . The other two second-partials are: ! 2 " !x 2 = #k 2 Asin(kx#$t) and ! 2 " !t 2 = #$ 2 Asin(kx#$t) . These only differ by a constant multiple: ! 2" !t2 = #2 k 2 ! 2" !x2 $ % & ' ( ) = vW 2 ! 2" !x2 . This equation, relating the second-partials of Ψ to one another, is called the “wave equation” (in one spatial dimension, to be precise). Classical waves on a string It is very instructive to work through a relatively simple example of a classical field theory, namely, the case of transverse waves on an elastic string. The definition of the physical system is as follows. A uniform string is stretched a bit so that everywhere inside it pieces pull on one another with a constant tension, τ. The length of the stretched string is L and its mass is M; every little bit of the string has the same mass density µ = M L . In equilibrium, the string is straight (i.e., there’s no gravity). We put a coordinate axis x through the string when it is in equilibrium so that x = 0 is one end and x = L is the other. The string is set into motion so that its pieces are displaced from equilibrium in one plane (e.g., the x-y plane) by various amounts. At every point x between 0 and L a small piece of the string has a local transverse displacement, q; that is, the actual motion of the string is up-and-down only. Provided that q(x, t) << L (small