Heisenberg Uncertainty Principle – Definition, Equation, Significance

Heisenberg Uncertainty Principle is a basic theorem in quantum mechanics. It state that we can not measure position and momentum of a particle both at the same time with the same accuracy. It means that if we try to measure the accurate position of a particle, then at the same time we can’t accurately measure the momentum of the particle. Mathematically, the product of uncertainties in position and momentum is greater than h/4π, where h is Planck’s constant. The principle is named after Werner Heisenberg, who proposed this theory in 1927.

In this article, we will learn in detail about Heisenberg’s Uncertainty Principle, its origin, formula, derivation, and other equations related to it. We will also learn its importance, applications, and other related concepts.

What is Heisenberg’s Uncertainty Principle

The Heisenberg Uncertainty Principle, proposed by physicist Werner Heisenberg in 1927, is a fundamental concept in quantum mechanics. It states that there is a limit to how precisely certain pairs of physical properties of a particle, like position and momentum, can be known simultaneously.

In simpler words, the principle states that the more we know one of these characteristics (let’s say position of a particle), the less one knows the other (i.e., momentum), and vice versa. This limitation comes from the wave-particle duality of quantum mechanics, which deals with the wave-like and particle-like characteristics of particles.

Heisenberg’s Uncertainty Principle Formula

The Heisenberg Uncertainty Principle is mathematically expressed as:

Δx⋅Δp ≥ ℏ/2

where:

• Δx stands for the particle’s position uncertainty.
• Δp is used to signify the momentum uncertainty of a particle.
• ℏ here is the reduced Planck constant given by ℏ = h/2π which approximately equals to 1.054 × 10⁻³⁴Js.

Origin of the Heisenberg’s Uncertainty Principle

Heisenberg Uncertainty Principle was formulated by a German physicist, Werner Heisenberg, in the 1920s. It was a product of efforts by Heisenberg to comprehend how quantum theory explains subatomic particle behaviors. This principle was officially introduced in his 1927 paper. This principle was developed through his mathematical formulation of quantum mechanics that involved non-commutativity between some operators representing physical observables like position and momentum. Consequently, these sets of variables had limits on their measurement precision leading to the Uncertainty Principle.

Why is it Impossible to Measure Both Position and Momentum Simultaneously?

The Heisenberg’s uncertainty principle emerged from wave-particle duality, which is the central aspect of quantum mechanics theoretical framework. The Principle of Quantum Mechanics is fundamentally based on the wave-like behavior of matter. We usually make such attempts when we try to find out about a particle’s position using a probe (a photon, another particle, etc.) that is localized. The act of exchanging momentum between us and the particle will affect its movement. An opposite situation occurs when we deal with the particle’s momentum. We need to interact with it for a while so that this automatically interferes with its position.

From a mathematical point of view, the uncertainty principle is given by the inequality Δx·Δp ≥ ħ/2 (ħ is the reduced Planck’s constant). On the left-hand side, we have the product of uncertainties in position (Δx) and in momentum (Δp). This means that the product of uncertainties connected to position and momentum should be greater than or equal to a certain minimal value.

Heisenberg Uncertainty Principle Equation

Here are brief definitions for the equations associated with the Heisenberg Uncertainty Principle:

Position and Momentum

It states that the product of the uncertainties in position (Δx) and momentum (Δp ) must be greater than or equal to ℏ/2 or h/4π

Δx⋅Δp ≥ ℏ/2

Energy and Time

This ensures that ΔE × The uncertainty in time (∆t) must be greater than or equal to half of the reduced Planck constant.

ΔE⋅Δt ≥ ℏ/2

Position and Velocity

The quantity of Δx and Δv must be larger than or equal to the half value of its reduced Planck constant divided by m.

Δx⋅Δv ≥ ℏ/2m

Heisenberg’s γ-ray Microscope

Heisenberg’s gamma-ray microscope is a theoretical concept proposed by physicist Werner Heisenberg in 1927. The idea behind the microscope is to use gamma-ray photons to visualize objects with extremely high resolution, potentially even at the atomic scale.

The concept uses the uncertainty principle, which Heisenberg himself formulated. According to this principle, there is a limit to the precision with which certain pairs of physical properties, such as the position and momentum of a particle, can be simultaneously known. In the case of the gamma-ray microscope, the uncertainty principle imposes a limitation on the accuracy with which one can determine the position of the particle being observed.

Heisenberg proposed that by using gamma-ray photons with extremely short wavelengths, it would be possible to confine the location of the object being observed to within a very small volume. This confinement would be achieved by scattering the gamma-ray photons off the object, which would cause the position of the object to become uncertain due to the momentum transfer from the photons. By measuring the scattered gamma rays, one could gather information about the object’s position with high precision.

Heisenberg Uncertainty Principle Derivation

The uncertainty in the position changes, Δx is related to that of momentum through the de Broglie wavelength. The de Broglie wavelength (λ) is given by:

λ = h/p

where:

• h is the Planck constant,
• p is the particle’s momentum.

Currently, the inaccuracy of position can be approximately described as Δx ≈ λ and that for momentum is

Δp ≈ mΔv

Substitute these into the de Broglie wavelength equation:

Δx⋅Δp ≈ λ⋅mΔv

Now, using the fact that λ = h/p

Δx⋅Δp ≈ h/p⋅mΔv

Next, utilize the fact that p = mv to express h/p in terms of Δx and Δp:

Δx⋅Δp ≈ h/(mv).mΔv

Cancel the mass terms:

Δx⋅Δp ≈ (h/v)⋅Δv

Now, recall that velocity (v) is the derivative of position (x) with respect to time (t):

v = dx/dt

Δx⋅Δp ≈ {h/(dx/dt)}⋅Δ(dx/dt)

Simplify:

Δx⋅Δp ≈ h/(dx/dt)⋅ d(Δx)/dt

Now, integrate both sides with respect to time:

∫(Δx⋅Δp) dt ≈ ∫ h/(dx/dt).d(Δx)/dt dt

This leads to the Heisenberg Uncertainty Principle:

Δx⋅Δp ≥ ℏ/2

Is Heisenberg’s Uncertainty Principle Noticeable in All Matter Waves?

Heisenberg’s Uncertainty Principle says that the accuracy of the positions and momentums of a particle has a scaling limit. This principle applies to all particles, including matter waves.

According to the theory of quantum mechanics, Matter waves are attached to particles. They exhibit the interaction of waves and particles. This implies that objects like electrons, protons, and so on, even particles that are as large as atoms and molecules, have the ability to represent wave characteristics. The frequency of these matter waves is inversely proportional to the momentum of the particle, according to the de Broglie equation.

Significance of Heisenberg Uncertainty Principle

Heisenberg Uncertainty Principle is one of the foundational principle of quantum mechanics. The significance of Heisenberg Uncertainty Principle is mentioned below:

Foundational Principles of Quantum Mechanics: As the basic principle in quantum mechanics, it serves as a guide for understanding particles’ behavior.

Wave-Particle Duality: Moreover, it shows the dual nature of particles, emphasizing their wave-particle property.

Quantum Measurement Problem: This solves the issue of measurement disturbance, which impacts property estimation accuracy.

Technology and Nanoscale Systems: Informs the design of technologies based on quantum systems, such as high-resolution microscopy and quantum computers.

Influence on Quantum Field Theory: It plays a role in the evolution of quantum field theories and defines how fields or particles act within a vacuum.

Limits on Precision Measurements: The principle limits joint knowledge of specific pairs as well. This concept serves as a foundation for the appreciation of measurement limits in the quantum field.

Heisenberg Uncertainty Principle Application

The Heisenberg Uncertainty Principle is used in several fields of physics, technology and even philosophy nowadays. Here are some notable applications:

Quantum Mechanics: The Uncertainty Principle is a fundamental principle of quantum mechanics. It provides a idea for describing the probabilistic nature of quantum systems and the limitations of classical physics in predicting the behavior of particles.

Atomic and Molecular Physics: In atomic and molecular physics, the Uncertainty Principle is used to understand the behavior of electrons within atoms and molecules. It provides details electron orbitals, energy levels, and chemical bonding which helps to explain the stability and structure of atoms and molecules.

Quantum Computing: Quantum computers use the principles of quantum mechanics, including uncertainty, to perform computations using quantum bits (qubits).

Electron Microscopy: In electron microscopy, the Uncertainty Principle sets limits on the spatial resolution of images obtained using electron beams. It helps to determine the minimum size of features that can be resolved in a sample

Particle Physics: In particle physics, the Uncertainty Principle influences the study of subatomic particles and their interactions. It provides details about phenomena such as particle decay, scattering processes, and the uncertainty in the measurement of particle properties.

Solved Numerical Problems on Heisenberg’s Uncertainty Principle

Example: A particle’s position is known with an uncertainty of Δx=10⁻¹⁰ meters. Calculate the minimum uncertainty in its momentum according to the Heisenberg Uncertainty Principle.

Solution:

According to the Heisenberg Uncertainty Principle:

Given:

Δx = 10^-10 meters

Using the uncertainty principle equation.

Δp ≥ ℏ/2Δx

Δp ≥ (1.05×10^-34m²kg/s) / (2×10^-10m)

Δp ≥ (1.05×10^-34m) / (2×10^-10) kg m/s

Δp ≥ 5.25×10^-25 kg m/s

So, the minimum uncertainty in momentum is Δp ≥ 5.25×10^-25 kg m/s.

JEE Questions on Heisenberg’s Uncertainty Principle (with solution)

Question: The position of an electron is known to an accuracy of 10^-10 m. What is the minimum uncertainty in its velocity?

Given h = 6.63×10^-34 JS and m_{e =} 9.1×10^-31 kg.

Solution:

We know Heisenberg’s Uncertainty Principle

Δx⋅Δp ≥ ℏ/2

Where:

• Δx = Uncertainty in position
• Δp = Uncertainty in momentum
• ℏ = Reduced Planck constant

Given Δx=10^-10 m, ℏ=6.63×10^-34, and find Δp=?.

Δp ≥ ℏ/2Δx

Δp ≥ 6.63×10^-34/2×10^-10 kg m/s

Δp≥3.315×10^-24 kg m/s

This is the minimum uncertainty in momentum. Now, using the relation p=mv, we can find the minimum uncertainty in velocity. Since

9.1×10^-31 kg for an electron.

3.315×10^-24 = (9.1×10^-31)v

v = 3.315×10^-24/9.1×10^-10 m/s

v ≈3.64×10⁶ m/s

So, the minimum uncertainty in velocity of the electron is 3.64×10⁶ m/s.

Frequently‌ ‌Asked‌ ‌Questions‌(FAQ)

State Heisenberg Uncertainty Principle.

According to the Heisenberg Uncertainty Principle, there is a fundamental limit in accuracy by which property pairs such as position (Δx) and momentum (Δp ) can be known simultaneously. It is given mathematically as Δx⋅Δp≥ ℏ/2, where h is the reduced Planck constant.

How to express Heisenberg Uncertainty Principle mathematically?

We can mathematically express Heisenberg Uncertainty Principle as Δx⋅Δp ≥ ℏ/2, where h is the reduced Planck constant.

What are the components of Heisenberg Uncertainty Principle?

The components of the Heisenberg Uncertainty Principle are:

• Δx: Uncertainty in position.
• Δp: Uncertainty in momentum.
• ℏ: Reduced Planck constant (≈1.054×10^-34 J⋅s

What is Heisenberg uncertainty principle and Schrodinger equation?

Heisenberg Uncertainty Principle: It states the limits on precision of position and momentum measured at a time, which is shown as Δx⋅Δp≥ℏ/2.

Schrödinger Equation: Gives quantum state evolution over time [Tex]\hat H\Psi = E\Psi[/Tex], with as the wave function [Tex]\Psi[/Tex] and [Tex]\hat H[/Tex]– Hamiltonian typical operator .

How do we define the Heisenberg Uncertainty Principle?

We can define Heisenberg Uncertainty Principle as a fundamental principle which states that we can not precisely measure position and momentum of the particle at the same time.