Some Further Notes on the Energy-Position/Momentum-Time Uncertainty Expressions for a Non-Relativistic Particle

 

Some Further Notes on the Energy-Position/Momentum-Time Uncertainty Expressions for a Non-Relativistic Particle

Pavle I. Premović

Laboratory for Geochemistry, Cosmochemistry&Astrochemistry,

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

Abstract. A brief discussion related to the energy-position and the momentum-time uncertainty relations is presented.

Keywords: Energy-position uncertainty, momentum-time uncertainty, non-relativistic particle.

Derivation and Discussion. 

We recently introduced {1} the energy-position uncertainty and the momentum-time uncertainty expressions for a non-relativistic particle:

ΔEΔx < ћc (the energy-position uncertainty) … (1)

ΔpΔt > ћ/c (the position-time uncertainty) … (2),

where ћ (= h/2π = 1.05 × 10^-34 J s)[1] is the reduced Planck’s constant, h is the Planck’s constant (6.63 X 10^-34 J s), and c (= 2.99792×10⁸ m s^-1) is the speed of light. These expressions are derived from of two Heisenberg’s uncertainty relations:

aΔpΔx ≥ ћ (the momentum-position uncertainty)    ... (3)

ΔEΔt ≥ ћ (the energy-time uncertainty)                  ... (4).

Our derivation of the expression (1) is based on the two assumptions: the speed of a non-relativistic particle v < 1/2c and ΔpΔx ranges from ћ up to 2ћ. 

If the energy uncertainty is defined as ∆E[2] = v∆p, one can then easily derive the expression (1). According to Special relativity, v < c, so ∆x/v > ∆x/c. We can now write the following series of the expressions: ΔEΔx/c < ΔEΔx/v = vΔpΔx/v = ΔpΔx. Since  ΔpΔx ≥ ћ, we get

Energy-position uncertainty relation: ΔEΔx < ћc    … (5).

The strong nuclear force is one of the four basic forces in nature (the others being gravity, the electromagnetic force, and the weak nuclear force) and it is the strongest one. However, this force also has the shortest range, holding together nucleons: a proton and a neutron. The strong nuclear force is created between these nucleons by the exchange of particle called the pion. The pion has a rest mass energy ΔE = mc² = 140 MeV = 2.24 X 10^-11 J. When the pro­ton transfers its pion to the neutron, the neutron “borrows” the pion rest mass energy of 140 MeV. That energy pro­duces a force be­tween these two nucleons. According to the expression (5), the distance between them must then be < 1.5 X 10^-15 m., which is about inner nuclear distances. For comparison, the proton size is about 8.4 X 10^-16 m.

Dividing the inequality (5) with Δx, we get ΔE < ћc/Δx. Obviously, ћc/Δx cannot be equal to zero because then ΔE would be less than zero or negative Thus, 0 < ΔE < ћc/Δx. Similar reasoning indicates that 0 < Δx < ћc/ΔE.

The inequalities (3) and (5) can be written as ΔpΔx = aћ, where a ≥ 1 and vΔpΔx < ћc. Substituting ΔpΔx with aћ in vΔpΔx < ћc we get aћv < ћc or av < c.  For a non-relativistic particle with a speed v < 1/2c, v = bc/2, where b < 1. Now av = abc/2 < c or ab < 2.  This last inequality is only possible if and only if a ≤ 2. Thus, for a non-relativistic particle is the following inequality is right:

2 ≥ a ≥ 1    … (6).

aIn general, for a non-relativistic particle with a speed v < c/q, where q < 2, ΔEΔx < ћc is only possible for q ≥ a ≥ 1.

Depending of the values of various Planck’s constants, now we can write for ΔpΔx the following limits (ћ/2 ≤ ΔpΔx ≤ ћ; ћ ≤ ΔpΔx ≤ 2ћ; h/2 ≤ ΔpΔx ≤ h, and h ≤ ΔpΔx ≤ 2h) and inequalities (ΔpΔx ≥ h > h/2 > ћ > ћ/2).

Dividing the expressions (3) and (4), we get ΔpΔx/ΔEΔt = v/v =1 or ΔpΔx = ΔEΔt. So, for v < 1/2c, ΔEΔt also ranges ћ to 2ћ, as ΔpΔx. In this case, we can write the following limits (ћ/2 ≤ ΔEΔt  ≤ ћ; ћ ≤ ΔEΔt ≤ 2ћ; h/2 ≤ ΔEΔt ≤ h, and h ≤ ΔEΔt ≤ 2h) and inequalities (ΔEΔt  ≥  ћ > ћ/2 and ΔEΔt > h > h/2).

In our previous report {1}, we derive the expression (2) using a simple mathematical procedure. For the sake of readers, we will repeat this derivation. The expression (4) can be written as ΔEΔt = vΔpΔt ≥ ћ. Dividing vΔpΔt ≥ ћ with v we get ΔpΔt ≥ ћ/v. Because v < c, ћ/v > ћ/c, we arrive at

Momentum-time uncertainty relation: ΔpΔt > ћ/c    … (7).

As we stated above, we can write the inequality (3) as ΔpΔx = aћ, where 2 ≥ a ≥ 1. The inequality (7) can be written as ΔpΔt = dћ/c, where d > 1. Dividing these new equations, we get that v = (a/d)c. In the case of non-relativistic particle with a speed v < 1/2c, after a little algebra, we get d >4. So, for a non-relativistic particle:

ΔpΔt > 4ћ/c    … (8).

Thus, depending of the values of various Planck’s constants, now we can write for ΔpΔt the following limits (ΔpΔt > 2ћ/c; ΔpΔt > 4ћ/c; ΔpΔt > 2h, and ΔpΔt > 4h) and inequalities (ΔpΔt > 4ћ/c > 2ћ/c and ΔpΔt > 4h/c > 2h/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] The value ћ, h/2 and h are the most common values, but the absolute minimum value is ћ/2.

[2] The total energy of a non-relativistic particle E = mc² + E_k, where m and E_k (= 1/2mv²) are, respectively, its rest mass and kinetic energy. Hence, ΔE = ΔE_k < 1/2mv². Thus, for a non-relativistic particle with a speed v < 1/2c →  ΔE < 1/8 mc².