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Physics 334 Modern Physics Credits: Material for this PowerPoint was adopted from Rick Trebino’s lectures from Georgia Tech which were based on the textbook “Modern Physics” by Thornton and Rex. Many of the images have been used also from “Modern Physics” by Tipler and Llewellyn, others from a variety of sources (PowerPoint clip art, Wikipedia encyclopedia etc), and contributions are noted wherever possible in the PowerPoint file. The PDF handouts are intended for my Modern Physics class, as a study aid only.

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The de Broglie Hypothesis Measurement of Particle wavelength Wave Motion and Wave Packets The Probabilistic Interpretation of the Wave Function The Uncertainty Principle Some Consequences of the Uncertainty Principle The Wavelike Properties of Particles I thus arrived at the overall concept which guided my studies: for both matter and radiations, light in particular, it is necessary to introduce the corpuscle concept and the wave concept at the same time. - Louis de Broglie, 1929 Louis de Broglie (1892-1987) CHAPTER 5

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De Broglie Waves In his thesis in 1923, Prince Louis V. de Broglie suggested that mass particles should have wave properties similar to electromagnetic radiation. The energy can be written as: hf = pc = p f Thus the wavelength of a matter wave is called the de Broglie wavelength: Louis V. de Broglie (1892-1987) If a light-wave could also act like a particle, why shouldn’t matter-particles also act like waves?

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Bohr’s Quantization Condition revisited One of Bohr’s assumptions in his hydrogen atom model was that the angular momentum of the electron in a stationary state is nħ. This turns out to be equivalent to saying that the electron’s orbit consists of an integral number of electron de Broglie wavelengths: Circumference electron de Broglie wavelength

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Bohr’s Quantization Condition revisited Exercise1: What is the de Broglie wavelength for a very small but microscopic object of mass 10 -9 g moving with speed 3x10 -8 m/s? Exercise2: What is the de Broglie wavelength of a 10-eV electron?

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First measurement of the wavelength of electrons were made by Davisson and Germer in 1927 Measurement of Particle Wavelength Electrons scattered at an angle φ from a nickel crystal are detected in an ionization chamber

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Exercise 3: Calculate the wavelength of scattered electrons by using Bragg diffraction condition. Measurement of Particle Wavelength

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G. P. Thomson and son J. J. Thomson share the Nobel prize in 1937 with Davisson. Diffraction of x-rays to produce Laue patterns Another Demonstration a)Diffraction from Al target b)Diffraction pattern for x- rays of Al c)Diffraction pattern for electron of Al. fig. a and b show similarities θ satisfies the Bragg condition. 2θ is the scattered beam

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O. Stern and I. Estermann 1930 demonstrated diffraction of beams of helium atoms and hydrogen molecules from lithium fluoride crystals Diffraction of Other Particles a)Diffraction of He atom from LiF crystal b)Experimental setup c)Intensity as a function of detector angle

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Diffraction of Other Particles Top left: Diffraction pattern produced by neutrons Top right: Laue pattern of NaCl produced by neutrons Bottom right: Diffraction pattern produced by protons from oxygen nuclei

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In Ocean it is the water that “waves”, for sound it is the air molecules, for light it is the E and the B field. So for matter “What is waving?” De Broglie matter waves should be described in the same manner as light waves (no ether). The matter wave should be a solution to a wave equation like the one for electromagnetic waves: Wave Motion and Wave Packets In order to understand matter waves we need to review properties of classical waves first Exercise 4: Consider a stretched string displaced from its normal position, derive the wave equation and explore the possible solutions to this wave equation.

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What is a wave? A wave is anything that moves. A disturbance or a pulse. To displace any function f(x) to the right, just change its argument from x to x-a, where a is a positive number. If we let a = v t, where v is positive and t is time, then the displacement will increase with time. So represents a rightward, or forward, propagating wave. Similarly, represents a leftward, or backward, propagating wave. v will be the velocity of the wave. f(x)f(x) f(x-3) f(x-2) f(x-1) x 0 1 2 3 f(x - v t) f(x + v t)

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The one-dimensional wave equation In the previous exercise we derived the wave equation. Here it is in its one-dimensional form for scalar (i.e., non-vector) functions, f : Matter waves will be a solution to this equation. And v will be the velocity of de Broglie waves.

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The solution to the one-dimensional wave equation where f (u) can be any twice-differentiable function. The wave equation has the simple solution:

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Solution of a Wave equation Exercise 5: Show that the 1-D wave equation has a simple solution of the form f(x,t)=f(x±t), where f(u) can be any twice differentiable function

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Proof that f (x ± vt) solves the wave equation Write f (x ± vt) as f (u), where u = x ± vt. So and Now, use the chain rule: So and Substituting into the wave equation:

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Definitions: Amplitude and Absolute phase y(x,t) = A cos[(k x – t ) – ] A = Amplitude = Absolute phase (or initial phase)

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Definitions Spatial quantities: Wave number= number of radians Corresponding t a wave train 1m long Temporal quantities: ν=f

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The Phase Velocity How fast is the wave traveling? Velocity is a reference distance divided by a reference time. The phase velocity is the wavelength / period: v = / Since f = 1/ : In terms of the k-vector, k = 2 /, and the angular frequency, = 2 / , this is: v = f v = / k

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Human wave A typical human wave has a phase velocity of about 20 seats per second.

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Complex numbers So, instead of using an ordered pair, ( x, y ), we write: P = x + i y = A cos( ) + i A sin( ) where i = (-1) 1/2 Consider a point, P = (x,y), on a 2D Cartesian grid. Let the x-coordinate be the real part and the y-coordinate the imaginary part of a complex number.

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Euler's Formula exp (i ) = cos( ) + i sin( ) so the point, P = A cos( ) + i A sin( ), can be written: P = A exp(i ) where A = Amplitude = Phase

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Proof of Euler's Formula Use Taylor Series: exp(i ) = cos( ) + i sin( ) If we substitute x = i into exp(x), then:

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Complex number theorems

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More complex number theorems Any complex number, z, can be written: z = Re{ z } + i Im{ z } So Re{ z } = 1/2 ( z + z* ) and Im{ z } = 1/2i ( z – z* ) where z* is the complex conjugate of z ( i –i ) The "magnitude," | z |, of a complex number is: | z | 2 = z z* = Re{ z } 2 + Im{ z } 2 To convert z into polar form, A exp(i ) : A 2 = Re{ z } 2 + Im{ z } 2 tan( ) = Im{ z } / Re{ z }

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We can also differentiate exp(ikx) as if the argument were real.

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De Broglie Velocity Exercise 6: What is the phase velocity of de Broglie waves? Justify your result. Let v p =f λ be the de Broglie phase velocity, then the de Broglie wave function can be written as where k is a propagating constant The amplitude of the de Broglie wave that correspond to a moving body reflects the probability that it may be found at a particular place at a particular time. The above equation is an indefinite series with same amplitude y o We expect that matter waves travel with different amplitudes in a wave packet or a wave group

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De Broglie Group Velocity Example of waves that can travel in a group is Beats Beats are two waves of same amplitude, but different frequencies that are produces simultaneously. Tuning forks of 440 Hz and 442 Hz sounded together produces two loudness peaks per second. How to find de Broglie group velocity?

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Classical Uncertainty Relations Figure b - ∆x~1/12 Figure c - ∆k~4π~12

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Classical Uncertainty Relations Example: Standing in the middle of a 20-m long pier, you notice that at any given instant there are 15 wave crest between the two ends of the pier. Estimate the minimum uncertainty in the wavelength that could be computed from this information.

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Particle Wave Packets Exercise 7: Show that for an electron the phase velocity of the wave is half its particle velocity and that the group velocity is the same as its particle velocity.

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Probabilistic Interpretation of the Wave Function C. Jönsson of Tübingen, Germany, succeeded in 1961 in showing double-slit interference effects for electrons by constructing very narrow slits and using relatively large distances between the slits and the observation screen. This experiment demonstrated that precisely the same behavior occurs for both light (waves) and electrons (particles).

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Probabilistic Interpretation of the Wave Function

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Okay, if particles are also waves, what’s waving? Probability The wave function determines the likelihood (or probability) of finding a particle at a particular position in space at a given time: Probability, Wave Functions, and the Copenhagen Interpretation The total probability of finding the particle is 1. Forcing this condition on the wave function is called normalization. The probability of the particle being between x 1 and x 2 is given by:

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Uncertainty Principle: Energy Uncertainty The energy uncertainty of a wave packet is: Combined with the angular frequency relation we derived earlier: Energy-Time Uncertainty Principle:. The factor ½ accounts for the fact that in Gaussian distribution the product of the uncertainty are 1/2

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The same mathematics relates x and k : k x ≥ ½ So it’s also impossible to measure simultaneously the precise values of k and x for a wave. Now the momentum can be written in terms of k : So the uncertainty in momentum is: But multiplying k x ≥ ½ by ħ : And we have Heisenberg’s Uncertainty Principle: Momentum Uncertainty Principle

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How to think about Uncertainty The act of making one measurement perturbs the other. Precisely measuring the time disturbs the energy. Precisely measuring the position disturbs the momentum. The Heisenbergmobile. The problem was that when you looked at the speedometer you got lost.

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Kinetic Energy Minimum: Since we’re always uncertain as to the exact position,, of a particle, for example, an electron somewhere inside an atom, the particle can’t have zero kinetic energy: Consequences of the Uncertainty Principle so: The average of a positive quantity must always exceed its uncertainty:

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A particle (wave) of mass m is in a one-dimensional box of width ℓ. The box puts boundary conditions on the wave. The wave function must be zero at the walls of the box and on the outside. In order for the probability to vanish at the walls, we must have an integral number of half wavelengths in the box: The energy: The possible wavelengths are quantized and hence so are the energies: Particle in a Box

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Exercise 8: Use uncertainty principle to find the spread in spectral lines of an electron jumping to the ground state in an atom. Width of Spectral Lines

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Wave-particle duality It’s somewhat disturbing that everything is both a particle and a wave. The wave-particle duality is a little less disturbing if we think in terms of: Bohr’s Principle of Complementarity: It’s not possible to describe physical observables simultaneously in terms of both particles and waves. When we’re making a measurement, use the particle description, but when we’re not, use the wave description. When we’re looking, fundamental quantities are particles; when we’re not, they’re waves.

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Waves or Particles? Dimming the light in Young’s two-slit experiment results in single photons at the screen. Since photons are particles, each can only go through one slit, so, at such low intensities, their distribution should become the single-slit pattern. Each photon actually goes through both slits!

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Can you tell which slit the photon went through in Young’s double-slit exp’t? When you block one slit, the one-slit pattern returns. Two-slit pattern One-slit pattern At low intensities, Young’s two-slit experiment shows that light propagates as a wave and is detected as a particle.

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Which slit does the electron go through? Shine light on the double slit and observe with a microscope. After the electron passes through one of the slits, light bounces off it; observing the reflected light, we determine which slit the electron went through. The photon momentum is: The electron momentum is: Need ph < d to distinguish the slits. Diffraction is significant only when the aperture is ~ the wavelength of the wave. The momentum of the photons used to determine which slit the electron went through is enough to strongly modify the momentum of the electron itself—changing the direction of the electron! The attempt to identify which slit the electron passes through will in itself change the diffraction pattern! Electrons also propagate as waves and are detected as particles.

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Probability of the particle vs. position Note that E 0 = 0 is not a possible energy level. The concept of energy levels, as first discussed in the Bohr model, has surfaced in a natural way by using waves. The probability of observing the particle between x and x + dx in each state is

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The Copenhagen Interpretation Bohr’s interpretation of the wave function consisted of three principles: Heisenberg’s uncertainty principle Bohr’s complementarity principle Born’s statistical interpretation, based on probabilities determined by the wave function Together these three concepts form a logical interpretation of the physical meaning of quantum theory. In the Copenhagen interpretation, physics describes only the results of measurements.

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