Elementary Particles
with Feynman

by Bruno Autin

Historical introduction

Atomic conception of matter and light

Democritus from Abdera (5-th century BC)

Lucretius from Pompei (1-st century BC): De rerum natura

Descartes, Snell and Newton(Opticks, Treatise on the reflexions, refractions, inflexions and colours of light, 1704)

Wave description of light

Huyghens: analogy between sound and light, Traité de la lumière (1678)

Thomas Young (1802): diffraction of light by two slits.

Augustin Fresnel (beginning of XIX-th century) adopted Huyghens’ viewpoint to re-interpret Newton’s experiments and extend it to diffraction and polarisation.

At the same time, Fourier developed the dualism of time and frequency domains.

Following Coulomb for electrostatics and Ampère for magnetism, Maxwell developed electromagnetism and the famous equations (1861-1862) which were considered as the accomplishment of physics. However thermodynamics had to blur the landscape.

Black body radiation

Low energies (long wavelengths) are well described by Maxwell’s equation.

At high energies (short wavelengths), Maxwell spectrum diverges to infinity: ULTRAVIOLET CATASTROPHE.

Extra constraint by Planck: dependence of energy E on frequency ⋁: E = h ν

({{({{h = 6.626 (10^(-34))}}) Joule.second}})

Energy density as a function of temperature T and frequency ⋁:

({{({{((ν^3)/(Exp[((hν)/kT)]-1))}})
Boltzmann constant: k = 1.37 (10^(-23)) Joule/(^0)K}})

Plot[x^3/(Exp[b x] - 1), {x, 1, 10^15}, Axes -> True, AxesLabel -> {Style["ν", Large], Style["ρ", Large]}, PlotLabel -> Style["Planck distribution", Large]]

Quantum mechanics

Photo-electric effect (Hertz 1887), theory by Einstein (1905)

Evidence of a planetary atom with a dense kernel at the center (Rutherford, 1911)

From spectroscopy (Kirchoff, 1850) to Bohr’s hydrogen atom (1913)

Louis de Broglie (1923) postulates that any particle of matter has a wave and point structure.  Electron diffraction in crystals (Davisson and Germer, 1927).

Uncertainty relation between conjugate variables (Heisenberg, 1927).

Classical wave equation (Schrödinger,1926 )

Relativistic wave equation (Dirac,1928 )

Feynman’s approach

Three mental experiments of diffraction by two holes:

Bullets

Waves in water

Electrons interacting with light

Bullets

The bullets can traverse the screen only in the holes and one at a time. However their diameter is smaller than the diameter of the holes so that they bounce back from the edge of the hole and reach the backstop located behind the screen at any place. Against the backstop sits a counter which accumulates the bullets which strike it in a finite interval of time, always the same, at each position x distributed regularly against the backstop. For simplicity, only one dimension x is considered.

Three experiments are then performed: one with hole 1 open and hole 2 closed, one with hole 2 open and hole 1 closed and one with the two holes open. The number of bullets collected at a given x is the probability of finding a bullet at x denoted (Subscript[P, 1]), (Subscript[P, 2]) and (Subscript[P, 12]) respectively for each of the three experiments and

(Subscript[P, 12]) = (Subscript[P, 1]) + (Subscript[P, 2])

There is no interference between bullets crossing holes 1 and 2. The next figure is obtained using the function:

Plot[{Callout[Exp[-(x-0.7)^2],Style["(Subscript[P, 1])",Large],1],
         Callout[Exp[-(x+0.7)^2],Style["(Subscript[P, 2])",Large],-1],
         Callout[Exp[-(x-0.7)^2]+Exp[-(x+0.7)^2],Style["(Subscript[P, 12])",Large], 0]},
           {x,-2,2},
            Axes->{True,False}]

Waves in water

The second experiment consists of creating waves in a basin regularly with the same maximum amplitude over time. The waves propagate over the surface of the water until they touch a wall pierced with two slits. Downstream the holes, two trains of waves in phase mix together in an interference region until they reach a beach where they disappear.

Just upstream the beach, a detector measures the height of water at regular intervals along the x axis. The square of the height, the intensity I of the wave, is recorded according to the procedure of the first experiment. This time, the intensity when the two holes are open is modulated by the interference process and (Subscript[I, 12]) is different from the sum of (Subscript[I, 1]) and (Subscript[I, 2]). In the next diagram, the first two functions of the first diagram are maintained but the third one is modulated by a periodic function:

Plot[{Callout[Exp[-(x-0.7)^2],Style["(Subscript[I, 1])",Large],1],
         Callout[Exp[-(x+0.7)^2],Style["(Subscript[I, 2])",Large],-1],
         Callout[Cos[Pi x]^2( Exp[-(x-0.7)^2]+Exp[-(x+0.7)^2]),Style["(Subscript[I, 12])",Large],0]},
           {x,-2,2},
            Axes->{True,False}]

Electrons

The last experiment has the same set-up as the first one but electrons replace bullets and the pattern of the electron impacts against the backstop is that of the second experiment. That result demands some explanation.

First, when one says that "electrons replace bullets", it is meant that they can be distinguished individually and this is justified by their detection as clicks in a Geiger counter. The click is always the same in duration and intensity for any position at any time. Only the number of clicks, the rate, varies as a function of x. The click is thus associated with the passage of an electron and the electron can be localized exactly like a bullet.

The pattern of the impacts on the backstop reveals a wave behaviour that seems incompatible with the localization of the electron since the electron should cross the two holes at the same time to create an interference.

Tracking the electrons

Compton effect . A source of light  is located behind the screen midway between the holes and when an electron passes through a hole, a flash of light is emitted from its position and if it crosses the two holes simultaneously there should be two flashes of light.

The experiment reveals that each click is associated with one flash, never two. The electron is thus localized. When the statistics of the impacts of electrons is measured on the backstop, there is no more interference pattern and this is not mysterious. The scattering is random and plays the role of the edges of the slit  in the case of the bullets.

However, the position of the electron is obtained at the cost of a perturbation of the trajectory. This is unavoidable but two variants can be tested to limit the perturbation: one by reducing the number of collisions between electrons and photons, the other by reducing the amplitude of the kick given by the photon to the electron

Influence of intensity

With a dimmer and dimmer light, the scattering is weaker and weaker. In fact the number of flashes diminishes but the intensity of each flash is constant.

A flash is associated with a photon in the same way a click is associated with an electron. With a faint light, some electrons may not meet a photon and reach the counter without emitting a flash.

One can thus set a table with three columns. Each row corresponds to a value of x. The first column contains the number of electrons passing through slit 1, the second column the electrons coming from slit 2 and the third column the electrons for which no flash has been emitted. The patterns for the first and second column are of type (Subscript[I, 1]) and (Subscript[I, 2]). There is interference ((Subscript[I, 12])) for the third column.

Influence of wavelength

The wavelength λ is related to the momentum p of the photon

λ = (h/p)

In the figure, the holes are separated by several millimeters and one should plan to vary the light wavelength down to microwaves.

From visible light to far infrared, no special effect is observed with respect to the previous experiment.

Suddenly, as soon as the wavelength approaches the distance Δx between the slits, a broad flash is emitted from the space located between the holes. One can no longer say which hole the electron has traversed : its position is blurred. An interference pattern appears and becomes more and more contrasted with longer wavelengths. It is thus impossible to correlate electron position and momentum under a certain value of the momentum.

That indetermination was already noticed for the microscope: two points near to one another cannot be distinguished when the light wavelength is too large. Here appears the interest of particle accelerators: the higher the particle momentum is, the smaller an object can be resolved:

Δx ≥ ℏ / (2p)

Statistics

Position

In a position measurement, the probability of finding a particle in a region dr surrounding position r is P(r)dr and the probability of finding a particle in a volume V is:

P(V) = (∫_V)P(r) dr = 1

P(r) is the squared norm of the wave function Ψ(r) which measures the probability of presence of a particle at r:

P(r) = (SuperStar[Ψ])(r)Ψ(r) = ((|Ψ(r)|)^2)

and, from the definition of P(V),  the norm of  Ψ satisfies the condition:

N = (∫_V)((|Ψ(r)|)^2)dr= 1

The function  Ψ(r) is given by Schrödinger’s equation for a non relativistic motion:

i ℏ (∂/(∂r))Ψ = H Ψ

where H is the Hamiltonian operator of the field which guides the particle. The wave packet ψ(x) of wave vector k-(Subscript[k, 9]) occupies a region of extension ξ about (Subscript[x, 0]) and can be represented by a gaussian function

ψ(x)=(e^(i (Subscript[k, 0]) x -((((x-(Subscript[x, 0])))^2)/(2(ξ^2)))))

Wave vector

A particle at position r is associated with a plane wave (e^(i k.r))where k is the wave vector. For a wave packet of different k vectors, the probability of finding a particle of momentum p is Π(p) dp and the probability of finding the particle in a domain D of the momentum space is

Π(D) = (∫_D)Π(p) dp = 1

The wave packet in the k-space is given by the Fourier transform ϕ(k)  of ψ(x):

ϕ(k) = (1/(Sqrt[2π]))(∫_(-∞)^(+∞))ψ(x)(e^(i k x))dx

The definition of the Fourier transform  g(ω) of a function f(t) in Mathematica is:

g(ω) = (1/(Sqrt[2π]))(∫_(-∞)^(+∞))f(t)(e^(i ω t))dt

and is coded as

FourierTransform[f(t,ω),t,ω]

The symbolic code to find ϕ(k) is

Clear[ψ,ϕ,ξ,t0, k0]
ψ[x_]=Exp[I k0 x-(x-x0)^2/(2ξ^2)];
ϕ[k_]=Simplify@FourierTransform[ψ[x],x,k ,Assumptions->ξ>0]

((E^(-(1/2) (k+k0) (-2 I x0+(k+k0) (ξ^2))))/(Sqrt[(1/(ξ^2))]))

By regrouping the terms in k-(Subscript[k, 0]), the previous expression becomes

ϕ(k)=ξ (e^(i (k-k0)x0-(1/2)(((k-k0))^2)(ξ^2)))

Heisenberg’s uncertainty relation

The σ-values of the real parts of the Ψ and ϕ  functions are respectively ξ and 1/ξ so that

(Subscript[σ, x]) (Subscript[σ, k]) =1

That expression is better known in the form of Heisenberg’s uncertainty relation

(Subscript[σ, x]) (Subscript[σ, p]) =ℏ

That uncertainty applies also to time and energy variables and more generally to any set of conjugate variables.