The Minimum Speed of a Free Massive Particle

 

The Minimum Speed of a Free Massive Particle

Pavle I. Premović

Laboratory for Geochemistry, Cosmochemistry&Astrochemistry,

University of Niš, pavleipremovic@yahoo.com, Niš, Serbia

Abstract. The equations for the minimum speed and kinetic energy of a free massive particle are derived within the non-relativistic and special relativistic frameworks. These equations are based on de Broglie's relation between momentum and wavelength of this particle.

Keywords: Particle, de Broglie’s wavelength, speed, kinetic energy, relativistic.

The particle-wave duality of the matter is one of the biggest mysteries in science because a massive particle and a wave are opposite each other in every way. Indeed, this particle is a discrete entity enclosed to a relatively very small volume of space, while a wave propagates over a large region of space.

Introduction. In modern physics, it is now widely accepted that light (or in more general terms, electromagnetic radiation) has a dual nature. The wave-like nature of light explains most of its properties: reflection, refraction, diffraction, interference and Doppler effect. On the other hand, the photoelectric effect and Compton effect can be only explained based particle-photon nature of light. The wavelength of this photon λ can be expressed as follows

λ = h/p    … (1)

where h (= 6.62×10^-34 J sec^-1) is Planck’s constant and p is the momentum of photon.

Elementary physics shows that the momentum of a free massive particle p is

p = mυ

where m is the relativistic mass of this particle and υ is its speed relative to an observer at rest. De-Broglie postulated that a free massive particle shows the dualistic wave-particle character just like light and the eqn. (1) can be also applied to this particle

                                                                             λ = h/mυ    … (2).

aIn the first part of this paper, we will deal with a free[1] “non-relativistic” massive particle (hereinafter “non-relativistic” particle) and in the second part we will deal, in general with a free massive particle 

(hereinafter a free particle).

In general, the “non-relativistic” particles are those whose speed υ is far less than the speed of light or υ << c (= 2.99792×10⁵ km sec^-1). Physicists usually assume that the massive particles with υ/c ≤ 0.1 (or υ ≤ 0.1c) are “non-relativistic”. De Broglie’s particle-wave duality has been also verified experimentally for the “non-relativistic” (e. g. electron or neutron diffraction) and relativistic (e. g. electron diffraction) massive particles.

Derivation and Discussion. Suppose now that the “non-relativistic” particle is spherical and has a diameter and a mass at rest: D₀ and m₀. Obviously, de Broglie’s wave needs a medium through which to spread and that is of course - matter. From analogy with light, we hypothesize that the maximum de Broglie’s wavelength λ_max of a “non-relativistic particle” must be equal to its diameter.

Mathematically speaking

λ_max = D₀.

Applying the formula (2), we get

λ_max = h/m₀υ_min

where υ_min is the minimum speed of a “non-relativistic” particle. Substituting λ_max of this equation with

D₀ and after some rearrangement we get

υ_min = h/m₀D₀ = β    … (3).

According to this equation, in order for a massive “non-relativistic” particle to have a minimum velocity β equals zero its diameter must be infinitely large, which is, of course, impossible.

The minimum kinetic energy of a “non-relativistic” particle is

KE_min = 1/2(m₀β²)    … (4).

In sum, eqn. (3) appears also to set a lower limit for its speed, momentum and kinetic energy.

Examples:

(1) Alpha (α)-particle consists of two protons and two neutrons. The mass of the α-particle is about 6.65×10^-27 kg and its diameter is about 3.6 × 10^-15 m.  Employing eqn. (3) we calculate that the minimum speed of this particle is about 2.8 × 10⁷ m sec^-1 (or about 0.093c). In other words, this is roughly the lowest possible speed of the free α-particle.

The kinetic energy of the free α-particles emitted by the uranium isotopes (²²⁶U - ²³⁸U) ranges from about 4.2 MeV (= 6.7×10^-13 J) up to about 7.6 MeV (= 12.1×10^-13 J). Using the expression for the “non-relativistic” kinetic energy KE = 1/2(mυ_α²) we estimated that the speed of their α-particles υ_α range is about 1.4 × 10⁷ m sec^-1 up to about 1.9 ×10⁷ m sec^-1. These values agree in order of magnitude with the above rough estimate of the lowest possible speed of the free α-particle.

Alpha particles are relatively big and heavy and are not able to penetrate very far through a medium. As a result of scattering collisions with various nuclei of the medium through which the free α-particle passes, its kinetic energy is reduced and usually becomes a helium atom capturing two electrons from its surroundings.

(2) Macroscopic non-relativistic objects would have a minimum speed extremely low. For the golf ball with a mass of about 0.05 kg and a diameter of 0.05 m we estimate [using eqn. (3)] its minimum speed would be about 2.7×10^-31 m. The average speed of a golf ball is about 50 m sec^-1.

In the next part of this paper, we will consider a relativistic particle and the consequences of that consideration on the case of a “non-relativistic” particle.

Strictly speaking, all free massive particles are relativistic. Even if their speed is much less than the speed of light (or υ << c), they are still relativistic. So, there is only a relativistic free massive particle or better to say a free massive particle, in general. This is why the term non-relativistic is put under quotation marks.

Special theory of relativity sets the light speed c as an upper limit to the speed of a massive particle. According to this theory, the mass of a relativistic particle

m = m₀/√(1 – υ²/c²)   … (5).

However, Special relativity states that the above spherical “non-relativistic” particle traveling at a relativistic speed would contract in the direction of motion becoming the prolate spheroid-shaped relativistic particle, Fig 1.  Its diameter at rest D₀ would be shortened in the direction of its motion, by the factor (1– υ²/c²)^1/2.

 

In equation form,

L = D₀√(1 – υ²/c²)    … (6)

where L is the length of a relativistic particle[1] (hereinafter length) along its direction of motion Fig.

1.

Multiplying eqns. (5) and (6) we have

 

mL = m₀D₀.

Eqn. (6) shows that in contrast to the previous non-relativistic particle whose diameter D₀ is assumed to be constant; the length of a relativistic particle L depends on the speed of this particle υ. Obviously, its wavelength λ cannot be larger than the length L. In equation form,

a

Fig. 1.  The shape and dimensions of the free spherical particle at relativistic speed.

λ ≤ L = D₀√(1 – υ²/c²).

Combining eqn. (2) and the left end of this equation we have

υ ≥ h/mL.

Substituting into this equation m₀D₀ instead mL we obtain

υ ≥ h/m₀D₀

having a minimum value

υ_min  = h/m₀D₀ = β.

So, its minimum kinetic energy

KE_min [= (1/2mυ_min²)] = 1/2(m₀β²)

These equations are identical to eqn. (3) and eqn. (4) derived for υ_min and KE_min of a “non-relativistic” particle. In other words, our approach to the minimum speed and kinetic energy of  “non-relativistic” massive particle in the first part of this work sounds reasonable.

Conclusion. The equations for the minimum speed and kinetic energy of a free massive particle are derived. These equations are based on the non-relativistic and special relativistic formulations using de Broglie^’s relation between linear momentum and wavelength of this particle.

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[1] It is a free particle in the sense that it is experiencing no net force. 

[2] In fact, its equatorial axis.

 

 

The Reduced Compton Wavelength and the Energy-Position/Momentum-Time Uncertainties

 

The Reduced Compton Wavelength and

the Energy-Position/Momentum-Time Uncertainties

Pavle I. Premović

Laboratory for Geochemistry, Cosmochemistry&Astrochemistry,

University of Niš, pavleipremovic@yahoo.com, Niš, Serbia

The Compton scattering is an elastic collision of the x-ray or gamma photons with free electrons (or loosely bound valence shell electrons). This scattering is demonstrated in Fig. 1. In this communication, we are combining Heisenberg's uncertainty principle and the Compton scattering. Note, that some of the expressions (including their derivations) of this communication can be found in modern physics textbooks.

Figure 1. Schematic representation of Compton scattering, in which an incoming photon scatters from an electron at rest.

The electron of the rest mass m_e has the rest energy of E_r = m_ec². The Compton scattered electron (hereinafter the Compton electron) possesses relativistic kinetic energy, the relativistic speed υ_e and the linear momentum p_e = m_eυ_e.

Before scattering, let the photon move along the positive x-axis having an energy E₀ = hν₀, frequency ν₀, wavelength λ₀ and linear momentum p₀ = hν₀/c, where h ( = 6.63 × 10^-34 J s) is the Planck constant and c (= 2.99792×10⁸ m s^-1) is the speed of light. After scattering, the photon moves at an angle θ off the x-axis with an energy E = hν, frequency ν, wavelength λ and momentum p = hν/c.

The conservation of linear momentum of the electron after the collision yields,

p_e² = p₀² + p² - 2p₀pcosθ.

The conservation of energy gives,

                                                            (p_ec)² = ( hν₀ – hν)² + 2m_ec²(hν₀ – hν).

Combining these two expressions leads to the standard Compton formula

λ – λ₀ = (h/m_ec)(1 − cosθ).

Here h/m_ec is called the standard Compton wavelength of the electron, λ_c, and it has a value of 2.43 10^-12 m. This formula supposes that the scattering occurs in the rest frame of the electron. The energy hν₀ of the incident photon with this wavelength is equal to the rest mass energy m_ec² of the electron. In the following discussion, we will take the reduced Compton wavelength λ_rc = ћ/m_ec (= 3.87 × 10^-13 m), where ћ (= h/2π =1.05 × 10^-34 J s) is the reduced Planck constant.

If the Heisenberg uncertainty principle holds for the Compton electron[1] then

Δp_eΔx_e ≥ ħ.

This inequality can be expressed as

aΔp_eΔx_e = aħ    ... (1)

where a ≥ 1. In this case, quantum mechanics predicts that the uncertainty in position of the Compton electron Δx_e must be greater than the reduced Compton wavelength ħ/m_ec or Δx_e > λ_rc = ħ/m_ec.[2] Accordingly, we can write

Δx_e = αћ/m_ec    ... (2)

where α > 1.

If the uncertainty in energy Δp_e of the Compton electron is higher than m_ec then the uncertainty in its energy ΔE_e (> m_ec²) would be high enough to create an electron-positron pair. Suppose that

Δp_e = bm_ec    ... (3),

where b ≤ 1. Thus for the Compton electron the momentum-position expression we can be written as

Δp_eΔx_e = bαħ     ... (4)

with bα = a, see equation (1).

By differentiating the relativistic equation for the total energy E_e of the Compton electron  E_e² = (p_ec)² + (m_ec²)², we obtain

ΔE_e = υ_eΔp_e    ... (5).

If we multiply both sides of equations (5) with Δx_e and using equation (4) we get

ΔE_eΔx_e = υ_eΔp_eΔx_e = bαħυ_e     ... (6).

According to Special relativity, the speed of the Compton electron υ_e cannot be greater than c.  It is reasonable to assume that 1 ≤ a ≤ 2[3] and that υ_e is relativistic when it is equal or larger than 0.5c. In this case bαυ_e < 2c. So equation (6) can be rewritten

ΔE_eΔx_e < 2ћc ... (7).

aRecently, employing the uncertainty relation for momentum-position, we have derived the same energy-position uncertainty expression for a relativistic particle with the speed υ ≥ 0.5c {1}. This derivation was based on the same two assumptions as above for the Compton electron, and these are (A): the speed of light c is independent of the motion of the observer, as postulated by Special relativity and (B): 1 ≤ a ≤ 2.

We also derived the expression for the momentum-time position for a relativistic particle with the speed υ ≥ 0.5 c{1}:

Δp_pΔt > ħ/c    ... (8).

The energy time-uncertainty relation for the Compton electron is

ΔE_eΔt ≥ ħ    ... (9).

As we derived above ΔE_e = υ_eΔp_e. So, relation (9) can be rewritten

υ_eΔp_eΔt > ħ    ... (10).

Since the speed of the Compton electron υ_e after interaction with the photon cannot be greater than c. In this case (10) we arrive at

Δp_eΔt > ħ/c.

Thus, inequality (8) holds for the Compton electron.

As we pointed out in footnote 1 the value of ħ/2 is seldom attained; the values ħ and h are more common. Thus, the energy-position expression (7) can be written as ΔE_eΔx_e < ћc or ΔE_eΔx_e < 2hc. Similarly, the momentum-time expression (8) can be reformulated as Δp_eΔt > ħ/2c or Δp_eΔt > h/c.

Reference

{1} P. I. Premović, The Energy-position and the momentum-time uncertainty expressions. The General Science Journal, December 2021. 

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[1] This is a crucial assumption. In general, it is worth mentioning that the lower limit of ћ/2 for ΔpΔx is rarely attained; more usually ћ even h.

[2] Since the standard Compton wavelength λ_c (= h/m_ec) and the de Broglie wavelength (λ_DB = h/m_eυ) are both greater than λ_rc (= ħ/m_ec). So, if Δx_e > λ_DB or  > λ_c then Δx_e > λ_rc.

[3] It is reasonable to expect that 1 < α ≤ 2.