Chapter5_9 - root-mean-square deviation from the mean i.e...

Info icon This preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
PHY3063 R. D. Field Department of Physics Chapter5_9.doc University of Florida Opposite to Classical Mechanics! Heisenberg Uncertainty Principle (1927) Due to the nature of wave superposition and wave packets we see that π 2 k x and π ω 2 t . The energy and momentum of a particle are given by x x k p E h h = = ω and hence we see that h p x x and h E t where p x is the uncertainty in the x-component of the momentum and E is an uncertainty in the energy. Thus, there is a fundamental limit on the ultimate precision with which we can know both the position (x- coordinate) of a particle and its momentum (x-component) . In addition, a measurement of a particles energy performed during a time interval t must be uncertain by an amount E . Exact Uncertainty Relations: We made a crude approximation in deriving the above relations. Later we will derive the precise form of the uncertainty relations: 2 2 2 2 h h h h E t p z p y p x z y x , where corresponds to the
Image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: root-mean-square deviation from the mean ( i.e. standard deviation ). Note that this corresponds to a lower limit. It is always possible to know things less well. Uncertainty as a Function of Time: If at t = 0 we have localized the particle to within ∆ x , then at t = 0 ∆ p x ≈ h/ ∆ x and ∆ v x ≈ h/(m ∆ x ) . At a later time t, ∆ x = ∆ v x t ≈ ht/(m ∆ x ) and hence, 1. The better we know the particle’s position at t = 0 ( i.e. smaller ∆ x ), the worse we know it’s position at a later time t ( i.e. larger ∆ x ). 2. The uncertainty in the particle’s position ∆ x increases with time. In classical physics you can know precisely the position and momentum of a particle ( i.e. no limit on the precision). Furthermore, if you know the position and momentum of an isolated particle at t = 0, then the exact position of the particle can be predicted for all future times!...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern