Uncertainty quantification
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The document discusses Heisenberg's uncertainty principle, which states that position and momentum of a particle cannot be measured simultaneously with arbitrary precision due to the wave-particle duality of matter. It provides mathematical derivations and conceptual explanations of the principle, emphasizing the relationship between uncertainties in measurements of conjugate variables. The implications of this principle are explored through examples and analyses of wave functions, showcasing how uncertainty evolves over time.
![“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](https://image.slidesharecdn.com/uncertaintyquantification-151210214948/85/Uncertainty-quantification-1-320.jpg)
![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 functionwith (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)](https://image.slidesharecdn.com/uncertaintyquantification-151210214948/85/Uncertainty-quantification-2-320.jpg)
![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,](https://image.slidesharecdn.com/uncertaintyquantification-151210214948/85/Uncertainty-quantification-3-320.jpg)
![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)](https://image.slidesharecdn.com/uncertaintyquantification-151210214948/85/Uncertainty-quantification-8-320.jpg)
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Uncertainty quantification
- 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 functionwith (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)