The Energy-Position and the Momentum-Time Uncertainty
Expressions
Pavle
I. Premović
Laboratory for
Geochemistry, Cosmochemistry&Astrochemistry,
University of Niš, pavleipremovic@yahoo.com, Niš, Serbia
Abstract. The energy-position uncertainty and the
momentum-time uncertainty expressions for a non-relativistic particle are derived from the
two mathematical expressions of the Heisenberg uncertainty principle. These
additional expressions are:
ΔEΔx < ћc (the energy-position uncertainty)
ΔpΔt > ћ/c (the position-time uncertainty)
where ћ is the reduced Planck constant and c ( ≈ 3×108 m sec-1) is the speed of light.
Keywords: Quantum
mechanics, uncertainty principle, particle,
energy-position, momentum-time.
Introduction. The Heisenberg uncertainty principle
is one of the fundamental postulates of quantum mechanics[2].
This principle is typically expressed in either of two mathematical forms
ΔpΔx ≥ ћ (the momentum-position
uncertainty) ...(1)
ΔEΔt ≥ ћ (the energy-time
uncertainty) ...(2).
In essence, the formal uncertainty
principle says that the momentum (Δp)
times the uncertainty in the position (Δx)
or the uncertainty in the energy (ΔE) times the uncertainty in the time (Δt) is greater or equal to ћ.[1] The h is the Planck constant (6.63 x 10-34 J sec) or ћ = h/2π = 1.05 × 10-34 J sec. The uncertainty principle states that some pairs of physical observables (these are the quantities of a state that we can determine in the lab) cannot be precisely measured simultaneously to
arbitrary accuracy {1}.
The purpose of this letter is to present a very
simple derivation of the energy-position uncertainty and the momentum-time uncertainty
expressions for non-relativistic particles via
the uncertaintyprinciple
inequalities (1) and (2). We assume that its content can be understood
by a wide audience of readers acquainted with a calculus-based introduction
to quantum mechanics.
Derivations and Discussion. Let us consider a non-relativistic particle of speed υ (say, approximately, < 0.5c), with the rest mass m0, a momentum p (= m0v)
which moves along the x-axis. The
total energy E of this particle is
the following sum
E = m0c2 (the rest energy) + p2/2m0
(the kinetic energy).
Taking the first derivative of the total E (i. e. kinetic energy) with respect p, one get
ΔE = (p/m0)Δp
or
ΔE = υΔp
If
we substitute Δp with ΔE/υ in the expression (1), ΔpΔx, and afterward
reshuffle υ to the right side, we
obtain
ΔEΔx ≥ ћv.
ΔEΔx = aћv ... (3).
Since Special relativity limits all particles to the speed of light c, we arrive at
ΔEΔx < ћc ... (4)
for υ < 0.5c or, in general, for a(υ/c) <
1.
This represents the energy-position
uncertainty expression for a non-relativistic particle. Of course, if
the speed of a non-relativistic particle υ << c then ΔEΔx
~ 0 since the rest energy of this particle m0c2 is
much greater than its kinetic energy p2/2m0.
The expression (4) informs us that the
maximum of a non-relativistic particle is on the order of about 10-26
J m or that ΔEΔx of a particle ranges
from ~ 0 J m - about 3.15 × 10-26
J m. It is worthy of note that the Planck length lP is roughly 1.6 × 10-35 m so the maximum
energy uncertainty ΔE is about 2 × 109 J or about the Planck
energy EP.
Multiplying both sides of the above expression ΔE = υΔp
with Δt and rearranging terms, we
obtain
ΔEΔt = υΔpΔt ≥ ћ
or
ΔpΔt ≥ ћ/υ.
As ћ/v > ћ/c then
ΔpΔt > ћ/c
... (5).
aThis represents the momentum-time uncertainty
expression for a non-relativistic particle. The minimum of ΔpΔt is about 3 ×10-43 kg m. As in the case of the energy-position
uncertainty expression (4), the expression (5) is only correct for
non-relativistic limit v < 0.5c.
It is worth of noting that the Planck time tP
= 5.39 × 10-44 sec so the minimum momentum uncertainty is roughly about
6 kg m sec-1.
Thus, there are four expressions of the uncertainty relation for a single, non-relativistic
particle which can be arranged to those related to energy
ΔEΔx < ћc (the
energy-position uncertainty)
ΔEΔt ≥ ћ (the energy-time
uncertainty)
and momentum
ΔpΔx ≥ ћ (the momentum-position
uncertainty)
ΔpΔt > ћ/c
(the
momentum-time uncertainty).
It
is interesting to note that the two “new” expressions (4) and (5) link together
three universal constants: the speed of light c, the Planck constant h
and Ludolph’s number π.
A derivation
of the energy-position and momentum-time uncertainty expressions for a relativistic
particle is a bit “fuzzy” (like space and time in quantum mechanics). The total
energy E of the relativistic particle is given by the following equation
E = mc2 = (p2c2
+ m02c4)1/2.
In this case for c > v ≥ 0.5c
ΔEΔx < 2ћc
ΔpΔt > ћ/c
only
if the Heisenberg inequality (1) is non-relativistic (see, also, refs. 5 and 6].
In the case of a photon
E = pc
After the first derivation
of E with respect for p
ΔE = cΔp.
Multiplying both sides with Δx
Δx = cΔpΔx
The position Δx of a photon is its wavelength λ, i. e. Δx = λ. The momentum of a photon p is expressed by the ratio h/λ. For a finite change of p, we can write Δp = h/λ. By substituting h/λ for Δp and λ for Δx in the
last equation and after some simple algebra, we arrive at
ΔEΔx = hc.
Moreover, since Δt of a photon is λ/c then
pΔt = Δp(λ/c) = h/c.
Applications.
There are possibly many applications of the above
expressions for the energy-position uncertainty (4) and the momentum-time
uncertainty (5). It appears that in many applications the energy-position uncertainty
expression (3) for a non-relativistic particle: ΔEΔx < ћc is more convenient
and straightforward than the expression (1) of the momentum-position
uncertainty: ΔpΔx ≥ ћ. One of the illustrative examples is
the case of the non-existence of the electron in the hydrogen (atom) proton.
The radius of the hydrogen proton is
approximately 10-15 m. If an electron exists inside this proton,
then the uncertainty in the position of the electron is given by Δx ≈ 10-15 m. According to the expression (3) ΔEΔx < ћc. The uncertainty in energy is ΔE < ћc/Δx where Δx ~ 10-15 m ΔE < 1.05×10-34×3×108/10-15
~ 3×10-11 J ~ 200 MeV. If this rough estimation is correct,
then the energy (E) of the electron
in the hydrogen proton should be in the same order of magnitude, 10-11
J or 100 MeV. The experimental results however indicate that the electron in the
atom has E < 4 MeV. Therefore, an
electron cannot exist in the hydrogen proton.
The
second example is the case of the “zero-point” energy of liquid He (helium) at
temperature T = 0 K. In this case, all other energy is removed from a system
except the “zero-point” one. Indeed, according to the expression (4) if a He
atom is confined within the smallest possible distance Δx at T = 0 K then it
still has ΔE < ћc/Δx > 0 at this temperature. In other words, since the
location of this atom is not completely indefinite then its energy E cannot be
zero. Hence the He atom must possess finite kinetic energy even at 0 K, so-called
“zero-point energy”. This is the kinetic energy of the He atoms at 0 K, coming
from their vibrational motion.
References
{1} The main literature sources of this letter
were the standard introductory textbooks of quantum mechanics.
{3} J. D. Walecka, Introduction to Modern Physics: Theoretical Foundations, World
Scientific, 2008, pp. 477.
{4} K. Gottfried, T.-W. Yan, Quantum Mechanics: Fundamentals, Springer Science, 2003.
{5} D.-H. Gwo, A presentation at American Physical Society April Meeting 2011 (http://vixra.org/pdf/1411.0042v7.pdf).
[1] Except of the Klein–Gordon and Dirac versions, quantum mechanics is non-relativistic {2-6}.
[2 ] In practice, the absolute minimum uncertainty of ћ or h/2 (or even h) is far more common than the value ћ/2 {7}. It is worth noting here that Budzik and Kizowski {8} reported that the single slit diffraction pattern of electrons is in accord with the following momentum-position uncertainty expression: ΔxΔp = ћ.
[3] It appears that the formal momentum-position uncertainty equation (1), ΔpΔx ≥ ћ, allows a very large change in Δp. In nature, any particle system tends to maintain a minimum of ΔpΔx ≈ ћ (or ΔEΔt ≈ ћ) [see ref. 9].
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