Matter waves: Ψ · PDF file...

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  • Matter waves:

    Probability density = P(x,t) = |Ψ|2 = Ψ*Ψ



    xL -L

    L-L x Probability of electron being in interval dx = P(x)·dx

    Wave function = Ψ(x,t)


    More general: Probability of finding electron between x1 and x2 at time t: P(x,t)dx = |Ψ(x,t)|

    2dx x1




    à This requires ‘normalization’ of ψ(x)

  • Normalization Probability density = P(x,t) = |Ψ|2 = Ψ*Ψ


    L-L x

    à “Normalized wave function”

    The probability of finding the particle anywhere in space (i.e. between –∞ and +∞) must be 100%!

    P(x,t)dx = |Ψ(x,t)|2dx ≡ 1 -∞


    ∞ , ∀t

    (except if the particle decays!)

  • Example for a probability density

    Probability density of grades: What is the probability of getting a score between 38-40?

    0 20 50

    0.06 What is Probability of getting between 42.5 - 45?

    Probability = P(x)*dx = 0.07*2.5 = 0.16



    (~16/100 students)










    0 2.5 5 7.5 10 12. 5 15 17.

    .5 20 22. 5 25 27.

    5 30 32. 5 35 37.

    5 40 42. 5 45 47.

    5 50 Score

    Fre que


    Observed grade distribution

  • All sort of funky… ØParticles are described by wave functions (Ψ)

    (generally Ψ can have both real and imaginary components) Ø|Ψ|2 tells us probability density ØElectron waves ~ waves of probability.

    Electron double slit experiment. Display=Magnitude2 of wave function

    Large Magnitude |Ψ|2 means: probability of detecting electron here is high

    Small Magnitude |Ψ| 2 means: probability of detecting electron here is low

  • • The “blob” in Quantum Wave Interference is a 2D wave packet.

    • In this simulation, intensity (i.e. |Ψ|2 or |E|2) is represented by brightness.

    Electron Photon

    |Ψ|2 |Ε|2

  • Review: Ψ, a wave of probability

    What is the value of normalization constant ‘c’, such that P(x) integrated over all space =1?

    A. L B. sqrt(1/L) C. 1/L D. sqrt(2/L) E. 2/L

    Example: Ψ(x,t=0) =



    0 L



    Normalization of ψ: Integral of |ψ|2 over all space must be = 1 (100% likely will find the particle somewhere in space)!

    0, for x=0{

  • Ψ : a wave of probability Ψ(x,t=0) = 0 for x

  • -L


    0 L


    a bdx

    Say an electron were described by the following wave function:

    How do the probabilities of finding the electron near of a and b compare (within an interval dx)?:

    A. a > 2b B. a = 2b C. a < 2b


    Ψ(x,t=0)= αx/L from -L to L Ψ(x,t=0)=0 elsewhere

    P(x,t=0) = |ψ(x,t=0)|2

  • 0-L


    0 L

    Ψ(x,t=0)=αx/L from -L to L

    a bdx

    How do the probabilities of finding the electron near (within dx) of a and b A. a>2b B. a=2b C. a 2 * (P(x) at b) Probability density is what we detect!!

    Say an electron were described by the following wave function:

  • Vibrations on a string: Electromagnetic waves:

    Wave Equations




    2 1 t y

    vx y

    ∂ ∂=

    ∂ ∂

    v = speed of wave




    2 1 t E

    cx E

    ∂ ∂=

    ∂ ∂

    c = speed of light




    Magnitude is non-spatial: = Strength of Electric field

    Magnitude is spatial: = Vertical displacement of String

    Solutions: E(x,t) Solutions: y(x,t)

  • What are matter waves? EM Waves (light/photons): Amplitude E = electric field E2 tells us probability of

    finding photon. Maxwell’s Equations:

    Solutions are real sin/cosine waves:

    Matter Waves (electrons, etc.): Amplitude ψ = “wave function” |ψ|2 tells us probability of

    finding particle. Schrödinger Equation:

    Solutions are complex sine/cosine waves:




    2 1 t E

    cx E

    ∂ ∂=

    ∂ ∂

    t i

    xm ∂ ∂=

    ∂ ∂− ψψ !! 2 22


    )cos(),( )sin(),( tkxAtxE tkxAtxE

    ω ω −= −=

    ))sin()(cos( ),( )(

    tkxitkxA Aetx tkxi

    ωω ψ ω

    −+− == −

  • Review of sinusoidal waves:

    Wave in time: cos(2πt/T) = cos(ωt) = cos(2πf·t)

    Wave in space: cos(2πx/λ) = cos(kx)

    k is spatial analogue of angular frequency ω. (We use k because it’s easier to write sin(kx) than sin(2πx/λ).)

    λ = Wavelength = length of one cycle

    k = 2π/λ = wave number = number of radians per meter


    T = Period = time of one cycle

    ω = 2π/T = angular frequency = number of radians per second


  • Plane Waves • Most general kinds of waves are plane waves

    (sines, cosines, complex exponentials) – extend forever in space

    • ψ1(x,t) = exp(i(k1x-ω1t))

    • ψ2(x,t) = exp(i(k2x-ω2t))

    • ψ3(x,t) = exp(i(k3x-ω3t))

    • ψ4(x,t) = exp(i(k4x-ω4t))

    • etc… Different k’s correspond to different energies, since E = ½mv2 = p2/2m = h2/2mλ2 = !2k2/2m

  • Superposition principle • If ψ1(x,t) and ψ2(x,t) are both solutions to

    wave equation, so is ψ1(x,t) + ψ2 (x,t). → Superposition principle

    • E.g. homework (HW8, Q7b) – superposition of waves one traveling to the left and to the right create a standing wave:

    • We can make a “wave packet” by combining plane waves of different energies:

    )sin()sin(2)sin()sin(),( tkxAtkxAtkxAtxE ωωω −=+−−= ???

    ψ (x,t) = ΣnAnexp(i(knx-ωnt))


  • 35


  • Plane Waves vs. Wave Packets Plane Wave: ψ(x,t) = Aexp(i(kx-ωt))

    Wave Packet: ψ(x,t) = ΣnAnexp(i(knx-ωnt))

    Which one looks more like a particle? • In real life, matter waves are more like wave packets.

    Mathematically, much easier to talk about plane waves, and we can always just add up solutions to get wave packet.

    • Method of adding up sine waves to get another function (like wave packet) is called “Fourier Analysis.” You will explore it with simulation in the homework.

  • 37

  • Plane Waves vs. Wave Packets

    A. p most well-defined for plane wave, x most well-defined for wave packet.

    B. x most well-defined for plane wave, p most well-defined for wave packet.

    C. p most well-defined for plane wave, x equally well-defined for both.

    D. x most well-defined for wave packet, p most well-defined for both.

    E. p and x equally well- defined for both.

    Plane Wave: Ψ(x,t) = Aei(kx-ωt) :

    Wave Packet: Ψ(x,t) = ΣnAnei(knx-ωnt) :

    For which type of wave are position x and momentum p most well-defined?

  • Plane Waves vs. Wave Packets Plane Wave: Ψ(x,t) = Aei(kx-ωt)

    – Wavelength, momentum, energy: well-defined. – Position: not defined. Amplitude is equal everywhere,

    so particle could be anywhere!

    Wave Packet: Ψ(x,t) = ΣnAnei(knx-ωnt)

    – λ, p, E not well-defined: made up of a bunch of different waves, each with a different λ,p,E

    – x much better defined: amplitude only non-zero in small region of space, so particle can only be found there.

  • 3


  • Three deBroglie waves are shown for particles of equal mass.

    I II III x

    The highest speed and lowest speed are: a. II highest, I & III same and lowest b. I and II same and highest, III is lowest c. all three have same speed d. cannot tell from figures above

    ans b. shorter wavelength means larger momentum = larger speed. III is largest wavelength, I and II are same. amplitude of wave is not related to speed.

    A, 2f 2A, 2f A, f


    Review question

    A: Amplitude. f: frequency

  • E=hc/λ…

    A. …is true for both photons and electrons. B. …is true for photons but not electrons. C. …is true for electrons but not photons. D. …is not true for photons or electrons.

    E = hf is always true but f = c/λ only applies to light, so E = hf ≠ hc/λ for electrons.

    c = speed of light!

  • Today:

    • Heisenberg’s uncertainty relation

    • Intro to Schrödinger's equation

  • Heisenberg Uncertainty Principle • In math: Δx·Δp ≥ !/2 (or better: Δx·Δpx ≥ !/2) • In words: Position and momentum cannot both

    be determined precisely. The more precisely one is determined, the less precisely the other is determined.

    • Should really be called “Heisenberg Indeterminacy Principle.”

    • This is weird if you think about particles. But it’s very clear if you think about waves.

  • Heisenberg Uncertainty Principle

    Δx small Δp – only one wavelength

    Δx large Δp – wave packet made of lots of waves

    Δx medium Δp – wave packet made of several waves

  • A slightly different scenario:



    Plane-wave propagating in