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Electron Waves Are Probability Waves in the Ocean of Uncertainty. Long Ago…… De Broglie Wavelength H Λ = Mv

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Electron Waves Are Probability Waves in the Ocean of Uncertainty. Long Ago…… De Broglie Wavelength H Λ = Mv Richard Feynman: Electron waves are probability waves in the ocean of uncertainty. Long Ago…… De Broglie Wavelength h λ = mv hxJs= 6.626 10−34 ⋅ −11 λe = 2.4x 10 m Electron Waves leads to Quantum Theory 2L Waves: λ ==, n 1,2,3..... n n 2 h 1 2 p De Broglie: λ = →=Emv = p 22m hn22 Combine: E = Energy is Quantized! n 8mL2 Quantum Theory Assume electrons are waves. A wave function is defined that contains all the information about the electron: position, momentum & energy. The wave density function is the square of the wave function. The probability of finding an electron in a particular state is given by the square of the wave function. All we can know are probabilities. Probability = Ψ 22(xt , ) = (possibility) Where do the Wave Functions come from??? Solutions to the time-independent Schrödinger equation: h22d ψ −+=Uψ Eψ 2mdx2 OR d 2ψ 2mU( − E) = ψ dx22h Double Slit is VERY IMPORTANT because it is evidence of waves. Only waves interfere like this. Intensity Superposition of Sinusoidal Waves • Assume two waves are traveling in the same direction, with the same frequency, wavelength and amplitude • The waves differ in phase • y1 = A sin (kx - ωt) • y2 = A sin (kx - ωt + φ) • y = y1+y2 = 2A cos (φ/2) sin (kx - ωt + φ/2) Resultant Amplitude Depends on phase: Spatial Interference Term 1-D Wave Interference y = y1+y2 = 2A cos (φ/2) sin (kx - ωt + φ/2) ⎛⎞Δφ Resultant Amplitude: 2A cos⎜⎟ ⎝⎠2 Constructive Interference (Even πφ): Δ=2nn π , =0,1,2,3... Destructive Interference ()Odd πφ: Δ=(2nn + 1) π , =0,1,2,3... Constructive Destructive Interference φ = 0 φ = π φ Double Slit Interference 2π Phase Difference: Δφ =Δr Constructive: Δrm= λ λ Bright fringe: Δ=rdsinθ = mλ , m =0,1,2,3,... λL ymm==±±(012),, K bright d 1 Destructive: Δ=rm( + )λ 2 1 Dark fringe : Δ=rdsinθλ = ( m + ) m =0,1,2,3,... 2 λL ⎛⎞1 ymmdark =+⎜⎟(012) =±±,, K small angle approx: sinθ ≈ tanθ = yL / d ⎝⎠2 Light Waves: Intensity E = Emax cos (kx – ωt) B = Bmax cos (kx – ωt) E max = c Bmax EB E22 cB I ==S max max = max = max 2 av 222μ μ c μ ooo I ∝ Emax Intensity Distribution Resultant Field • The magnitude of the resultant electric field comes from the superposition principle – EP = E1+ E2 = Eo[sin ωt + sin (ωt + φ)] • This can also be expressed as ⎛⎞φφ ⎛ ⎞ EEPo=+2cossin⎜⎟ ⎜ωt ⎟ ⎝⎠22 ⎝ ⎠ – EP has the same frequency as the light at the slits – The amplitude at P is given by 2Eo cos (φ / 2) • Intensity is proportional to the square of the amplitude: 2 2 I ∝ A II=Δmax cos (φ / 2) Light Intensity Bright fringe: Δ=rdsinθ = mλ , m =0,1,2,3,... 2π Phase Difference: Δφ =Δr λ 2 II=Δmax cos (φ / 2) 22⎛⎞πd sin θπ ⎛⎞d II=≈max cos ⎜⎟Imax cos ⎜⎟y ⎝⎠λλ ⎝⎠L Double Slit: Light The intensity at a point on the screen is proportional to the square of the resultant electric field magnitude at that point. I ∝ E 2 Wave Function : E Amplitude Intensity: Amplitude squared Intensity is the REALITY for light waves. What about for light particles? PROBABILITY ~ Intensity ProbabilityProbability ConceptsConcepts The probability that a dart lands in A, B or C is 100%: Ptot=++ PPP A B C Suppose you roll a die 30 times. What is the expected numbers of 1’s and 6’s? A. 12 B. 10 C. 8 D. 6 E. 4 Suppose you roll a die 30 times. What is the expected numbers of 1’s and 6’s? A. 12 B. 10 C. 8 D. 6 E. 4 Do Workbook 40.2 #1 ConnectingConnecting thethe WaveWave andand PhotonPhoton ViewsViews The intensity of the light wave is correlated with the probability of detecting photons. That is, photons are more likely to be detected at those points where the wave intensity is high and less likely to be detected at those points where the wave intensity is low. The probability of detecting a photon at a particular point is directly proportional to the square of the light-wave amplitude function at that point: The figure shows the detection of photons in an optical experiment. Rank in order, from largest to smallest, the square of the amplitude function of the electromagnetic wave at positions A, B, C, and D. (The probability of detecting a photon !) A. D > C > B > A B. A > B > C > D C. A > B = D > C D. C > B = D > A The figure shows the detection of photons in an optical experiment. Rank in order, from largest to smallest, the square of the amplitude function of the electromagnetic wave at positions A, B, C, and D. (The probability of detecting a photon !) A. D > C > B > A B. A > B > C > D C. A > B = D > C D. C > B = D > A Spontaneous Emission Transition Probabilities Fermi’s Golden Rule Transition probabilities correspond to the intensity of light emission. ProbabilityProbability DensityDensity We can define the probability density P(x) such that the photon probability density is directly proportional to the square of the light-wave amplitude: In one dimension, probability density has SI units of m–1. Thus the probability density multiplied by a length yields a dimensionless probability. NOTE: P(x) itself is not a probability. You must multiply the probability density by a length to find an actual probability. Do Workbook 40.2 #2 In Sum: Light Wave model: Interference pattern is in terms of wave intensity Photon model: Interference in terms of probability The probability of detecting a photon within a narrow region of width δx at position x is directly proportional to the square of the light wave amplitude function at that point. Prob(in δ x at x) ∝ A(x) 2 δ x Probability Density Function: Px() ∝ A(x)2 The probability density function is independent of the width, δx , and depends only on x. SI units are m-1. Double Slit for Electrons shows Wave Interference Interference pattern builds one electron at a time. Electrons act like waves going through the slits but arrive at the detector like a particle. Double Slit: Electrons A light analogy….. There is no electron wave so we assume an analogy to the electric wave and call it the wave function, psi, : Ψ()x The intensity at a point on the screen is proportional to the square of the wave function at that point. P()xx=Ψ ()2 The Probability Density Function is the “Reality”! WARNING! There is no ‘thing’ waving! The wave funtion psi is “a wave-like function” that oscillates between positive and negative values that can be used to make probalisitic predictions about atomic particles. Probability: Electrons The probability of detecting an electron within a narrow region of width δx at position x is directly proportional to the square of the wave function at that point: Prob(in δ x at x) =Ψ (x) 2 δ x Probability Density Function: Px() =Ψ (x)2 The probability density function is independent of the width, δx , and depends only on x. SI units are m-1. Note: The above is an equality, not a proportionality as with photons. This is because we are defining psi this way. Also note, P(x) is unique but psi in not since –psi is also a solution. Where is the electron most likely to be found? Least likely? At what value of x is the electron probability density a maximum? This is the wave function of a neutron. At what value of x is the neutron most likely to be found? A. x = 0 B. x = xA C. x = xB D. x = xC This is the wave function of a neutron. At what value of x is the neutron most likely to be found? A. x = 0 B. x = xA C. x = xB D. x = xC In Sum….. Where do the Wave Functions come from??? Solutions to the time-independent Schrödinger equation: h22d ψ −+=Uψ Eψ 2mdx2 OR d 2ψ 2mU( − E) = ψ dx22h NormalizationNormalization • A photon or electron has to land somewhere on the detector after passing through an experimental apparatus. • Consequently, the probability that it will be detected at some position is 100%. • The statement that the photon or electron has to land somewhere on the x-axis is expressed mathematically as • Any wave function must satisfy this normalization condition. The Probability that an electron lands somewhere between xL and xR is the sum of all the probabilities that an electron lands in a narrow strip i at position xi: NN 2 ∑∑P()xxiiδ = ψδ() x x ii==11 The Probability that an electron lands somewhere between xL and xR is: xxRR Prob(xxx≤≤ ) = Pxdx() = ψ () xdx2 LR∫∫ xxLL Normalization ∞∞ ∫∫P()xdx== ψ () x2 dx 1 −∞ −∞ Total Area must equal 1. The value of the constant a is A. a = 0.5 mm–1/2. B. a = 1.0 mm–1/2. C. a = 2.0 mm–1/2. D. a = 1.0 mm–1. E. a = 2.0 mm–1. The value of the constant a is A. a = 0.5 mm–1/2. B. a = 1.0 mm–1/2. C. a = 2.0 mm–1/2. D. a = 1.0 mm–1. E. a = 2.0 mm–1. De Broglie Wavelength h λ = mv hxJs= 6.626 10−34 ⋅ −11 λe = 2.4x 10 m Wave Packet: Making Particles out of Waves h p = cf= λ λ p = hf/ c Superposition of waves to make a defined wave packet. The more waves used of different frequencies, the more localized.
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