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.