Understanding the experimental and mathematical derivation of Heisenberg's Uncertainty Principle. Simple application for estimating single degree of freedom particle in a potential free environment is also discussed.

1. “It's okay to be honest about not knowing rather than spreading falsehood. While it
is often said that honesty is the best policy, silence is the second best policy.”
Anshul Goyal, The University of Texas at Austin
ABSTRACT
Quantum mechanics is widely regarded as a theory which gives fundamental and universal
description of the physical world. It differs from the classical mechanics and the differences
become significant when we talk at the atomic level and about particles moving with very high
velocities. Classical terms such as position and momentum, energy and time best described as the
“conjugate canonical variables” take inherent fuzziness where the accuracy can be best regarded
to “minimum uncertain states”. This is one of the striking differences between the Newtonian
Mechanics and the Quantum Mechanics. This report is an attempt to understand the Heisenberg’s
Uncertainty Principle and start with a brief literature background followed by the application of
the probabilistic concepts to further estimate this fuzziness.
INTRODUCTION
Heisenberg Uncertainty Principle is the consequence of the wave and the particle duality of the
matter. Particle nature signifies position measurable with infinite and arbitrary precision.
Mathematically, probability density function is a spike centered at the location where the particle
is present. However, wave nature signifies the momentum measurable which can be calculated
using the De-Broglie Equation. Quantum physics states that matter behaves both as particle and
wave and because of this behavior, irrespective of the precision of the measuring equipment,
position and momentum cannot be measured simultaneously. However, the broader definition
states this for any conjugate canonical variables.
Another interpretation of the wave and particle duality is the quantum object. A quantum
object/particle is the collection of several waves known as wave packet. As we add more and
more waves, uncertainty in position is reduced because of stronger constructive interference
among several waves to create a sharper probability density. However, the price paid is the
uncertainty in momentum since several waves means variety of wavelengths, which in turn is
associated to the momentum, thanks to the De-Broglie equation. Therefore, uncertainty principle
is a very logical outcome of this explanation.
The first form of uncertainty principle was presented by German physicist Werner Heisenberg in
the year 1927. The formal inequality relating the standard deviation of position x and the
standard deviation of momentum p was derived by Earle Hesse Kennard [1].
σ σ ≥
h
4π

2. Like every theory needs an experimental justification, Heisenberg did a thought experiment to
convince his claim known as “The Gamma Ray Microscope.” The experiment had some flaws
which were corrected by Bohr. Heisenberg pictured a microscope that obtains very high
resolution by using high-energy gamma rays for illumination. Though such a microscope does
not exist but can be constructed in principle. The microscope is used to obtain the position of the
electron using short wavelength (high energy) gamma rays. In the corrected version of the
thought experiment, a free electron sits directly beneath the center of the microscope's lens. The
circular lens forms a cone of angle 2A from the electron. The electron is then illuminated from
the left by gamma rays. According the principle of wave optics, the microscope can resolve
objects to a size of Dx, which is related to the wavelength, L of gamma rays as
Dx =
L
2 sinA
The high energy photons impart a kick to the electron, which causes a change in the linear
momentum of the electrons. For measuring the uncertainty in linear momentum of the electron
along the x direction, two extreme cases of the gamma ray recoil are considered and the principle
of conservation of linear momentum of the system is used to obtain the following relation.
Dpx ~
h
Dx
or Dpx Dx ~ h
Similar explanations are also obtained from the single slit and multiple slit electron diffraction
experiments. The more refined version of the uncertainty principle is stated as “The simultaneous
measurement of two conjugate variables (such as the momentum and position or the energy and
time for a moving particle) entails a limitation on the precision (standard deviation) of each
measurement. Namely: the more precise the measurement of position, the more imprecise the
measurement of momentum, and vice versa. In the most extreme case, absolute precision of one
variable would entail absolute imprecision regarding the other”.
MATHEMATICAL EXPLANATION
In this section, we look at the assumption made by Heisenberg to actually derive the uncertainty
relation which was eventually relaxed by Kennard to give inequality in the equation which is a
better known form of uncertainty principle today.
Heisenberg assumed the repeatability hypothesis [2] which was developed by Von Neumann and
Schrödinger. He considered a state just after measurement of the position observable Q, to
obtain an outcome q, with a mean error of (Q) and examine what relation would hold between
(Q) and (P), which is error in momentum observable, when p is the outcome. He supposed that
the state, is a Gaussian wave functionwith  (q) and  (p) being the conjugate Fourier
pairs. Kennard relaxed this assumption, the state  after the position measurement, can be
arbitrary wave function which satisfy the approximate repeatability hypothesis.
σ(Q) ≤ ε(Q)

3. σ(P) ≤ ε(P)
Therefore the Heisenberg’s inequality follows immediately from the Kennard’s inequality and
approximate repeatability hypothesis. Equations are presented below to reiterate the fact clearly.
σ(Q)σ(P) ≥ Kennard Inequality
ε(Q)ε(P) ≥ Heisenberg inequality
To consider a simple case (for the purpose of analytical solutions, though quantum mechanics
has very few such cases; most famous being particle in a box [3]), free particle is chosen whose
potential does not change and is zero, it moves with a velocity v in the x direction in some
classical sense. The wave function, (x, t) can be derived using the Schrödinger Equation.
ψ(x, t) = A exp(i(kx − ωt))
If we observe the system at time t = 0, the wave function x, t) reduces to
ψ(x, t) = ψ(x) = A exp(ikx)
The states of the particle represented by the above wave function are completely delocalized.
These can be made more localized by adding more and more waves whose wave numbers, k are
well spread. This ensures localization in space. These are typically the Fourier Transform pairs
represented by the following Equation.
ψ(x) =
1
√2π
ϕ(k)e dk
where
ϕ(k) =
1
√2π
ψ(x)e dx
The localized  (x), also known as the wave-packet can be obtained as
ψ(x) =
1
2πα
/
e e /
By definition, the probability density function is obtained by f(x) = |ψ| such that
ψ∗
ψ = 1
PDF thus obtained in this case is a Gaussian distribution,

4. f (x) = ψ(x)ψ∗
(x)
1
2πα
e /
, μ = 0; σ = √α
ϕ (k) is Fourier Transform pair and once it is calculated, similar to the above process the PDF
for wave number, k is obtained.
ϕ(k) =
2α
π
e ( )
f (k) = ϕ∗(k)ϕ(k) =
2α
π
e ( )
, μ = k ; σ =
1
2√α
From the above equations the product of σ σ = ∆X ∆K = 1/2. The wave number can be
expressed in terms of momentum using ∆P = ∆K.
This proves the equality relation which is a special case for Gaussian Wave-packet and assumes
the form of uncertainty relation attributed to minimum uncertain states. This result is very similar
to what was actually derived by Heisenberg.
∆X ∆P =
h
4π
Fig 1 and Fig 2 below illustrates the idea of the delocalized and the localized wave-packet
respectively. The governing parameters and ko control the shape of the envelope (standard
deviation, ) and localization of the particle respectively. In Fig 1, the value is more and hence
the spread is also more. The wave-packet has been localized by reducing the value of alpha and
increasing the wave number, k which creates sharp peaks in local region. This completes the
fundamental mathematical explanation of the uncertainty principle. Depending upon the physics
of the problem, the solution of the Schrödinger equation may yield complicated ψ(x, t).

5. Fig 1: Delocalized Wave-packet
Fig 2: localized Wave-packet

6. DECISION ANALYSIS: RESEARCH
The decision may be to estimate the position of the quantum particle keeping in mind the
uncertainty which exist at the quantum level.
Consider the same problem described above where we are now interested to predict the
uncertainty in position and velocity at any general time t. Let Xo be the random variable
describing the position of the particle at time t = 0; Xt describes the position at any time t and Vo
be the velocity measured simultaneously with Xo at time t = 0. The position of particle at any
time t can be written as
X = X + V t
This is a linear function where the mean and standard deviations can be obtained as
μ = μ + t μ
The uncertainties in Xo and Vo can be obtained from the model described above.
σ = √α ; σ =
h
4πm√α
Assuming that the position and velocity of the particle are statistically independent, the
uncertainty in position at any time t can be given by
σ = α + t
h
16π m α
Following the Heisenberg’s Uncertainty principle, the uncertainty in velocity at any time t will
be
σ =
h
4πmσ
=
h
4πm
α + t
h
16π m α
The equations derived above helps the researcher to design his experiment based on the
localization in position of the particle he wish to achieve. For example consider a particle with
mass, m =10 kg , uncertainty, σ = 10 cm, average velocity, v = 10 m/sec . The
calculation show that after the time interval of 3.3 × 10 sec, σ becomes 100% of σ . Fig 3
and Fig 4 shows the pdf of position as time increases. A physical explanation can be that as time
increases the localized wave packet becomes more diffusive and it becomes difficult to track the
particle. However, the momentum/wavelength becomes more localized. The conclusion is that its
not very long that a quantum particle can be localized. The mean position of the particle is
governed by classical principles but the uncertainty spreads out in time and can be attributed to
the quantum effect as shown in Fig 5.

7. Fig 3 Localized Position at t = 0
Fig 4 Non-Localized Position at = .

8. Fig 5: Variation in with time
REFERENCES
[1] "One Thing Is Certain: Heisenberg's Uncertainty Principle Is Not Dead", Scientific American
[2] Ozawa, M, “Heisenberg’s original derivation of the uncertainty principle and its universally
valid reformulations”
[3] Linden, N, Malabarba, A and Short, A. "Particle in a Box." University of Bristol
X
(m)