<< Chapter < Page Chapter >> Page >
By the end of this section, you will be able to:
  • Describe the physical meaning of the position-momentum uncertainty relation
  • Explain the origins of the uncertainty principle in quantum theory
  • Describe the physical meaning of the energy-time uncertainty relation

Heisenberg’s uncertainty principle    is a key principle in quantum mechanics. Very roughly, it states that if we know everything about where a particle is located (the uncertainty of position is small), we know nothing about its momentum (the uncertainty of momentum is large), and vice versa. Versions of the uncertainty principle also exist for other quantities as well, such as energy and time. We discuss the momentum-position and energy-time uncertainty principles separately.

Momentum and position

To illustrate the momentum-position uncertainty principle, consider a free particle that moves along the x -direction. The particle moves with a constant velocity u and momentum p = m u . According to de Broglie’s relations, p = k and E = ω . As discussed in the previous section, the wave function for this particle is given by

ψ k ( x , t ) = A [ cos ( ω t k x ) i sin ( ω t k x ) ] = A e i ( ω t k x ) = A e i ω t e i k x

and the probability density | ψ k ( x , t ) | 2 = A 2 is uniform and independent of time. The particle is equally likely to be found anywhere along the x -axis but has definite values of wavelength and wave number, and therefore momentum. The uncertainty of position is infinite (we are completely uncertain about position) and the uncertainty of the momentum is zero (we are completely certain about momentum). This account of a free particle is consistent with Heisenberg’s uncertainty principle.

Similar statements can be made of localized particles. In quantum theory, a localized particle is modeled by a linear superposition of free-particle (or plane-wave) states called a wave packet    . An example of a wave packet is shown in [link] . A wave packet contains many wavelengths and therefore by de Broglie’s relations many momenta—possible in quantum mechanics! This particle also has many values of position, although the particle is confined mostly to the interval Δ x . The particle can be better localized ( Δ x can be decreased) if more plane-wave states of different wavelengths or momenta are added together in the right way ( Δ p is increased). According to Heisenberg, these uncertainties obey the following relation.

The heisenberg uncertainty principle

The product of the uncertainty in position of a particle and the uncertainty in its momentum can never be less than one-half of the reduced Planck constant:

Δ x Δ p / 2 .

This relation expresses Heisenberg’s uncertainty principle. It places limits on what we can know about a particle from simultaneous measurements of position and momentum. If Δ x is large, Δ p is small, and vice versa. [link] can be derived in a more advanced course in modern physics. Reflecting on this relation in his work The Physical Principles of the Quantum Theory , Heisenberg wrote “Any use of the words ‘position’ and ‘velocity’ with accuracy exceeding that given by [the relation] is just as meaningless as the use of words whose sense is not defined.”

Questions & Answers

what is force
Afework Reply
The different examples for collision
Afework
What is polarization and there are type
Muhammed Reply
Polarization is the process of transforming unpolarized light into polarized light. types of polarization 1. linear polarization. 2. circular polarization. 3. elliptical polarization.
Eze
Describe what you would see when looking at a body whose temperature is increased from 1000 K to 1,000,000 K
Aishwarya Reply
how is tan ninety minus an angle equals to cot an angle?
Niicommey Reply
please I don't understand all about this things going on here
Jeremiah Reply
What is torque?
Matthew Reply
In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment, moment of force, rotational force or turning effect, depending on the field of study.
Teka
Torque refers to the rotational force. i.e Torque = Force × radius.
Arun
Torque is the rotational equivalent of force . Specifically, it is a force exerted at a distance from an object's axis of rotation. In the same way that a force applied to an object will cause it to move linearly, a torque applied to an object will cause it to rotate around a pivot point.
Teka
Torque is the rotational equivalence of force . So, a net torque will cause an object to rotate with an angular acceleration. Because all rotational motions have an axis of rotation, a torque must be defined about a rotational axis. A torque is a force applied to a point on an object about the axis
Teka
When a missle is shot from one spaceship towards another, it leaves the first at 0.950c and approaches the other at 0.750c. what is the relative velocity of the two shipd
Marifel Reply
how to convert:m^3/s^2 all divided by kg to cm^3/s^2
Thibaza Reply
Is there any proof of existence of luminiferious aether ?
Zero Reply
mass conversion of 58.73kg =mg
Proactive Reply
is Space time fabric real
Godawari Reply
What's the relationship between the work function and the cut off frequency in the diagram above?
frankline Reply
due to the upthrust weight of the object varise with force in which the body fall into the water pendincular with the reflection of light with it
Gift
n=I/r
Gift
can someone explain what is going on here
falanga
so some pretty easy physics questions bring em
falanga
what is meant by fluctuated
Olasukanmi Reply
If n=cv then how v=cn? and if n=c/v then how v=cn?
Natanim
convert feet to metre
Mbah Reply
what is electrolysis
Mbah
Electrolysis is the chemical decomposition of electrolyte either in molten state or solution to conduct electricity
Ayomide
class ninekasindhtextbookurdusave
Ayesha Reply
can someone help explain why v2/c2 is =1/2 Using The Lorentz Transformation For Time Spacecraft S′ is on its way to Alpha Centauri when Spacecraft S passes it at relative speed c /2. The captain of S′ sends a radio signal that lasts 1.2 s according to that ship’s clock. Use the Lorentz transformati
Jennifer
Practice Key Terms 3

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, University physics volume 3. OpenStax CNX. Nov 04, 2016 Download for free at http://cnx.org/content/col12067/1.4
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'University physics volume 3' conversation and receive update notifications?

Ask