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Energy of Single Particle

Contents

Energy
Particle
Newtonian Mechanics
Relativistic Theory
Quantum Mechanics
Quantum Field Theory

Energy

Energy as some kind of potent force first appeared in 4th century BCE (see "History of Energy"). The scientific definition emerged in the 1850s with the "First Law of Thermodynamics" dU = dQ - dW (see Figure 01). It is a macroscopic description of the power holding inside substances and the work (causing organized motion) that can be derived from them. It is now realized that for mono-atomic gas the heat energy Q consists only the random translational motion. As the substance condenses from gas to liquid or solid, the inter-atomic/molecular forces become the important part of the internal energy U (Figure 01). This can be verified readily by coal burning which requires a lot of oxygen to break up the chemical bonds.
Energy of a System The change of heat energy dQ can be further specified for isothermal process (where the temperature T = constant, i.e., in equilibrium) as :
dQ = TdS + dN,
where dS is the change of entropy (randomness), is the chemical potential associated with the change of state such as phase change etc, and dN the change of particle number in the system.

Figure 01 Energy of a System [view large image]

 While dW = pdV signifies the change of volume V in response to pressure p.


Subsequently, the Maxwell-Boltzmann statistics shows that the temperature T is related to vp, the most probable velocity of the particles in the system, by the formula : kT = (m/2)(vp)2 (where m is the mass of the particles, and k = 1.38x10-16 erg/Ko the Boltzmann constant via the definition of entropy), providing a link between macroscopic and microscopic variables. At the limit of only one particle in the system, the kinetic energy of that particle would be just Ek = (m/2)v2. The statistical description becomes meaningless at such limit.

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Particle

A particle is a point mass with no spatial extent and thus no structure. There is a more practical approach on the definition of particle in physics such that a galaxy, a planet, electron, quark can be considered as a point as long as the scale of the system under deliberation is much larger than the size of the object. Figure 02 shows some examples with the ratio of object-size/system-extent = r. Even though the structure of the "point" is not important, it is often specified by some parameters such as mass, charge, spin, ... It is actually a reductionist's approach by ignoring finer details in the lower levels (of description) in order to learn something about the higher levels (see "Effective Theories"). The examples in Figure 02 are described briefly below :
    Particle, Definition
  1. Galaxy - The galaxies in the universe become points in the pressure-less fluid. Cosmology is a macroscopic domain in which those points are lumped into the density = Nm/V, where m is the averaged mass of the galaxies, N the total number, and V the volume of the universe. The size to system scale ratio R ~ 10-6. The dynamics is governed by the "Friedmann Equation".
  2. Planet - Its behavior is summarized in the Kepler's Law derived from Newtonian mechanics, and characterized by its mass m with the ratio R ~ 10-7.
  3. Electron - The electrons are considered as points in the hydrogen atom and other atomic nuclei. It is parameterized by its mass, charge, and spin. It is governed by the Schrodinger Equation, and can be solved analytically in hydrogen atom. The ratio R ~ 10-7 similar to the planetary motion since the interaction is long range for both cases with 1/r2 dependence.
  4. Quark - The point-like quarks are mediated by gluons to form nucleon. It is supposed to be

    5. Figure 02 Particle, Definition
    [view large image]

    		 characterized by its electric charge, mass, spin, and color charge. The ratio R ~ 10-3, which is much larger because the strong interaction in this case is a short range force. Actually, not much is known about this system.

The various facets of energy in different theory of a particle are described below for understanding its implication and to examine the problem with negative energy. Figure 03 and its caption shows briefly the formulas for the energy E in 4 different kinds of theory, where Ek and Ep are the kinetic and potential energy respectively, p the linear momentum, m the rest mass, and c the velocity of light.
    Particle, Definition
  1. Classical Mechanics : E = Ep + Ek for bound or free state.
  2. Relativity Theory : E = (m2c4 + p2c2)1/2 for free particle.
  3. Non-relativistic Quantum Mechanics : E = Ep + Ek for bound or free state.
  4. Quantum Field Theory : E = (m2c4 + p2c2)1/2 for free particle.

    5. Figure 03 The 4 Domains of Theory [view large image]

    		 Obviously, the expressions are different between non-relativistic and relativistic theories.


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Newtonian Mechanics

In classical mechanic, the interaction between masses is prescribed by Newton's gravitational force F = - GMm/r2, where r is the radial coordinate, G = 6.67x10-8 cm3/gm-sec2 is the gravitational constant for setting the interaction strength between the two mass points M, and m. Note that it depends only on r, and M >> m is assumed in the following, so that the mass point M is stationary acting only as a source of gravitation.

Escape Velocity

Figure 04 Escape Velocity
[view large image]
The electro-static force F = Qq/r2 is similar to the Newtonian gravity with the interaction constant absorbed into the definition of charge.
Electro-static Force Electro-static Charges The mainly difference is the existence of two different kinds of charge, i.e., the positive (+) and negative (-) varieties. As shown in Figure 05, like charges are repulsive while opposite charges are attractive to each other. Thus, the letter case behaves the same as gravity with GMm replaced by Qq. In case Q = - q = e (the charge of the electron), Ep = - e2/r. For the case of like charge Ep = + e2/r, there is no bound state (Figure 06).

Figure 05 Electro-static Forces [view large image]

Figure 06 Electro-static PE

The force for harmonic oscillator F = - kx (where k is the spring constant restraining the particle from equilibrium) is attractive with potential energy given by :
Potential Curve .
Then E = kx2/2 + mv2/2 which would never become negative. Instead, the degree of localization of the particle is determined by the boundary a = (2E/k)1/2, which is obtained by setting v = 0 in E (Figure 07(b)). It also shows the values of Ep and Ek inter-changed between x = 0 and a. The same kind of energies swapping also occurs in planetary motion Between perihelion and aphelion (Figure 07(a)).

Figure 07 Potential Curves [view large image]

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Relativistic Theory

In non-relativistic theory the energy and momentum are invariant separately under the Galilean transformation. While in relativistic theory the energy and momentum together forms a 4-vector (4 components vector) which is invariant under the Lorentz transformation.
4-Momentum

Figure 08 4-Momemtum [view large image]

 In addition, the velocity of the particle v > c for p2 > |p0|2 with m2 < 0 (Figure 08). It implies that the particle becomes a "tachyon", which does not exist.

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Quantum Mechanics

The energy equation E = p2/2m + V, where the linear momentum p = mv in Ek and V represents the potential energy, serves as the starting step in the formulation of non-relativistic QM (Quantum Mechanics).
Non-relativistic QM

Figure 09 Non-relativistic Quantum Mechanics [view large image]

 Analytic solutions have been obtained for many potential forms including the hydrogen atom and harmonic oscillator. The novel features include discrete energy level and there is a certain probability for penetrating the boundary (Figure 09).


In particle physics, the energy of a particle is often associated with a length scale via the de Broglie relation :
Energy/Length Scale p = h/,
which can be rewritten as L(cm) = hc/E ~ 4x10-4/E(ev) (Figure 10).
This formula is adopted from telescope resolution, which reveals finer detail with shorter wavelength. The similarity is related to probing the particle structure in scattering experiments, i.e., higher energy of the incident particle can reveal finer detail within.

Figure 10 Energy/Length Scale
[view large image]

 Note that the energy scale is referred to binding energy for some composite systems such as molecules, atoms, and nuclei.

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Quantum Field Theory

Time Machine Quantum Field Theory (QFT) is devised for studying high energy particle collision. As such, the energy has to be relativistic. The field equation is the starting point for its formulation. The Maxwell's Equations from the era of 1800's are the most ancient one. It is now assigned to cover the photon. The Klein-Gordon Equation for the spin-0 particle (such as the Higg's) was derived in 1926 by quantizing the relativistic energy equation E2 = m2c4 + p2c2 according to the rules mentioned earlier (also see "Klein-Gordon Equation of the Scalar Field"). This formulation avoids the issue of negative energy by using the squared version. In 1928 P.A.M. Dirac tried to resolve the problem with negative energy by splitting the energy equation into two linear forms (see "Derivation of the Dirac Equation and the Weyl Spinor"). The end products are two coupled linear equations, one of which turns out to associate with negative energy in the wave function (see "Interpretation of the Dirac Equation"). He then invented a sea of un-seen particles filling out all the negative energy levels in an attempt to solve the problem. The accepted interpretation now is to identify the negative energy particle with positive energy "anti-particle" moving backward in time (see the colorful illustration below). It is something like treating the -|E| traveling wave exp[(px+|E|t)/] = exp{[(px-(-|E|)t]/]} = exp{[px-|E|(-t)]/}. This is known as "Feynman–Stueckelberg interpretation". A lot of questions arise since then about "moving backward in time" like going back to the Jurassic Era (Figure 11) via such mechanism (aka Time Machine).

Figure 11 Time Machine
[view large image]

 Feynman diagram for electron-positron annihilation.

Here's an attempt to visualize the rather obscure term of "negative time":
A particle with charge q in a constant magnetic field B moves in circular path according to the formula for the angular velocity d/dt, i.e.,
Anti-Particle

Figure 12 Negative Energy=Anti-Particle, and Backward Time [view large image]

 In this example, the circular orbit is equivalent to the path of the traveling wave, the anti-fermion with charge "-q" and "-t" behaves similarly to a normal fermion. The negative time has no practical significance and can be banished as just a mathematical trick (Figure 12).
In a more abstract level, the CPT Theorem asserts that time reversal of fermion has to accompany with anti-fermion in order to maintain invariance under those transformations.


Strictly speaking, there is no such thing as "A" particle in QFT. Virtual particles pop up briefly all the time over all space. It is inherent in the theory (see "Quantum Vacuum").