ch39 young freedman - Waves As we have seen, light has a duality of being a wave and a particle. By a symmetry argument, de Broglie in 1924 proposed that all form of matter should

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  • Heisenbergs Uncertainty Principle

    Probability in finding where they land requires a wave description

    Photons going through the slit as particles

    THEN the spreading-out of the photons implies that the photons reaching the screen must have a small uncertainty in the vertical ( ) direction of its momentum as it exits the slit.y

    yp

  • We will try to estimate the uncertainty for these photons by

    Heisenbergs Uncertainty Principle

    yp

    yp

    The central max dominates the diffraction pattern and a majority of the photons will land in the central max.

  • Considering the diffraction pattern, the width of the central max is given by the location of its 1st order minimum ,

    Heisenbergs Uncertainty Principle

    1sin a

    1

    Assuming that so that is small so that we have 1a 1 1sin a

    yp

  • yp

    - And, from the diagram,

    Heisenbergs Uncertainty Principle

    - For the photons reaching , they must have a range of y-dir momentum:

    ,y yp p

    1y xp p

    So, we can estimate the uncertainty needs to be as large as 1xp yp

    1

    1

    1 1tany

    x

    pp

    for small

  • Heisenbergs Uncertainty Principle

    With Then, putting in , we have 1y x xp p p a 1 a

    The incoming photons moving along the x-dir with momentum in the x-dir

    will have a wavelength given by . Substituting this into the

    inequality, we have

    xp

    y xp p x

    hp

    haa

    xh p

    1y xp p

    yp

  • Heisenbergs Uncertainty Principle

    For photons going through the slit with width a, one can also say that the y-

    position of the photons has an uncertain of

    yy p h

    y a

    Putting everything together, we arrive at a constraint relating the conjugate pair

    of dynamic variables such that, , yy p (where h is the Plancks constant)

    yp

  • Heisenbergs Uncertainty Principle

    Our discussion is not specific to the direction. In general, we will have

    similar uncertainty relations for all three spatial directions, i.e.,

    2yy p

    We have use heuristic argument in getting to previous inequality. A more

    precise inequality can be derivate from full QM calculations and it should read

    where / 2h

    , ,2 2 2x y z

    x p y p z p

    y

    This is called the Heisenbergs Uncertainty Principle.

    But, there are no restrictions for unconjugated variables: , .yx p or x y etc

  • Uncertainty Relation for E & t

    Intuitively, the Heisenbergs Uncertainty Principle enforces an inverseproportional relation on the two conjugate pairs of dynamics variables:

    ,xp x ,E t and

    By decreasing the uncertainty in one of the variables (x or t), its corresponding conjugate variable must increases accordingly ! orxp E

    2xp x

    2E t

    And, additionally, there is an uncertainty relation for and :E t

    similar to

  • Heisenberg Uncertainty Principle againThe Heisenberg Uncertainty Principle is not a statement on the accuracy of the experimental instruments. It is a statement on the fundamental limitations in making measurements !

    To understand this, there are two important factors:

    1. The wave-particle duality2. The unavoidable interactions between the observer and the object

    being observed

  • Uncertainty Principle: Conceptual ExampleSuppose we want to measure the position of a particle (originally at rest) by a laser light.

    The measurement is accomplished by the scattering of photons off this particle (shining a light) :

    photon

    particle

    scattered phone

  • Uncertainty Principle: Conceptual ExampleBut, photons carry momentum and the particle after scattering will recoil !

    particlescattered phone

    recoil

    While the scattered photon gives us a precise position of the particle, the photon will also inevitably impart momentum onto the particle.

  • Uncertainty Principle: Conceptual ExampleIf the measurement is made by a photon with wavelength , then at best,

    we expect our position measurement to be accurate up to an uncertainty,

    x

    And, the photon will transfer parts of its momentum to the particle along the measurement direction x. The amount transfer is given by,

    xhp

    Combining these two expressions gives: xx p h

    Note: By improving x using a smaller does not help because px ~ h/gets larger in inverse proportion to !.

  • Waves and UncertaintyConsider a plane wave with polarization in the y-dir and propagating in the x-dir:

    , sinyE x t A kx t

    at a fixed time: 0 0t t ,0 siny kxE x A

    at a fixed location: 0 0x x 0, sinyE t A t

    x

    yE

    t

    yE

    wave with a definite wavelength :2 k

    wave with a definite frequency :2f

    f

  • Waves and UncertaintyRecall that is the wave number and is the angular frequency of the wave and we can express the x-momentum and energy of a photon in terms of

    22x

    hp kh

    k

    andk

    22

    hf fE h

    and

    Substituting these relations into our equation for , ,yE x t

    , sinyE x t A kx t

    This wave function for a (plane wave) photon expressed in this way will have definite values of x-momentum and energy , i.e.,xp E

    0xp 0E and

    There is no uncertainty in the values for and for a plane wave.xp E

    , siny xp EE x t A x t we have,

  • x

    yE

    Waves and UncertaintyOne might then ask if , can be still be satisfied?

    YES! In fact, the Heisenbergs Uncertainty Principle implies that:

    so that the uncertainty in the position for a plane wave must be infinite !

    0xp

    x

    Looking back at our plane wave with a definite wavelength (x-momentum ), it indeed extends over the entire range of x and one cannot localize where the photon might be.

    2xp x

    xp

  • t

    yE

    Waves and UncertaintyCorrespondingly, one can argue that a similar uncertainty relation applies to the uncertainties in energy and time, i.e.,

    And, in particular, for a plane wave, we have,

    For a plane wave with a definite frequency (energy E), it will extend over all time t and one cannot localize the photon in time.

    2E t

    0E t and

    (Heisenbergs Uncertainty Principle for energy and time)

  • Waves and UncertaintyObviously, a plane wave with infinite extents in space & time is an idealization.

    In practice, wave function of a photon will have limited extents in space & time (i.e., we know roughly where it is located in space and time),

    x t and

    Then, correspondingly, according to the Heisenbergs Uncertainty Principle, the uncertainties for the x-momentum and energy for a realistic photon cannot be zero,

    0xp 0E and

    such that and are satisfied !2x

    p x 2

    E t

  • Waves and Uncertainty

    Intuitively, the Heisenbergs Uncertainty Principle enforces an inverseproportional relation on the two conjugate pairs of dynamics variables:

    ,xp x ,E tand

    By decreasing the uncertainty in one of the variables (x or t), its corresponding conjugate variable must increases accordingly ! orxp E

    2xp x

    2E t

  • Waves and UncertaintyTo visualize a localized wave function, one can consider a superposition of multiple plane waves

    1 1 1 2 2 2, sin siny x xE x t A p x E t A p x E t

    For simplicity, let consider the sum of two of the same plane waves that we have been considering BUT with slightly different x-momentum and E.

    Looking at the wave at a fixed time and0 0t t

    2 1A A we have

    (black) (red)(green)

  • Waves and Uncertainty

    The resultant electric field are localized (beat pattern). With superposition of waves with many wavelengths/frequencies (x-momentum/energy), the localization can be stronger. The result is called a wave packet.

    NOTE: This wave packet is localized in space BUT it contains many and as required by the HUP.

    x xp 0xp

  • PHYS 262George Mason University

    Prof. Paul So

  • Chapter 39: Particles Behaving as Waves Matter Waves Atomic Line-Spectra

    and Energy Levels Bohrs Model of H-

    atom The Laser Continuous Spectra &

    Blackbody Radiation

  • Matter WavesAs we have seen, light has a duality of being a wave and a particle.

    By a symmetry argument, de Broglie in 1924 proposed that all form of matter should process this duality also.

    Recall for photons, we have:

    For a particle with momentum p = mv (or mv) and total energy E, de Broglie proposed:

    h h hp mv mv

    Efh

    (de Broglie wavelength)

    hp

  • Electron DiffractionClinton Davisson and Lester Germer (1927)

    sin 50od

    Nickel crystal acts as a diffraction grading.

    Constructive interference is expected at:

    sin , 1, 2,d m m

    2 / 2

    / / 2a

    a

    eV p m

    h p h meV

    54aV V peak

    Va

  • Diffraction of e off Aluminum FoilIn 1928, G.P. Thomson (son of J.J. Thomson) performed another demonstrative experiment in showing electrons can act as a wave and diffract from a polycrystalline aluminum foil.

    Debye and Sherrer: X-Ray diffraction experiment done a few years earlier using a similar set