Photons are the particles of light. Matter is made of atoms, and atoms are made protons, neutrons and electrons. These are not macroscopic particles. Typical atomic dimensions are on the order of 10-10 m, nuclear dimensions are on the order of 10-15 m, and the electron seems to be a point particle with no size at all. How do these particles behave? If a wave equation describes the behavior of photons, maybe a wave equation also describes the behavior of other microscopic particles. In 1924, Luis deBroglie (Nobel Prize in Physics in 1929) proposed that a wave function is associated with all particles. Where this wave function has nonzero amplitude, we are likely to find the particle. The standard interpretation is that the intensity of the wave function of a particle at any point is proportional to the probability of finding the particle at that point. The wavelengths of the harmonic waves used to build the wave function let us calculate the most likely momentum of the particle and the uncertainty in the momentum. The wave function for a material particle is often called a matter wave. The relationship between momentum and wavelength for matter waves is given by p = h/λ, and the relationship energy and frequency is E = hf. The wavelength λ = h/p is called the de Broglie wavelength, and the relations λ = h/p and f = E/h are called the de Broglie relations. These are the same relations we have for the photon, but for particle E = (1/2)mv2 = p2/(2m), so E = ћ2k2/(2m), λ = h/√(2mE). The relationship between λ and E is different for particles than for photons. A spread in wavelengths means an uncertainty in the momentum. The uncertainty principle also holds for material particles. The minimum value for the product ∆x ∆p is on the order of ħ. ∆x∆p ~ ħ. For any particle, we cannot predict its position and momentum with absolute certainty. The product of the uncertainties is on the order of h/2π = ħ or greater. A free particle of mass m moving with exactly determined velocity v in the positive x-direction has momentum p = mv, pointing into the positive x-direction and kinetic energy E = p2/(2m). Its wave function, which often is denoted by ψ(x,t), is a plane wave. ψ(x,t) = Acos(kx - ωt + φ) [The wave function is actually a complex function, and Acos(kx - ωt + φ) is the real part of this function. We will learn about complex functions in the next section.] The wave function has a well defined deBroglie wavelength λ = h/p which determines k = 2π/λ. Problem:
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If a particle is described by a plane wave ψ(x,t) = Acos(kx - ωt + φ) we know the particles momentum and wavelength exactly, but its position is completely unknown. The plane wave extends to infinity, the particle could be anywhere. The wave function for a particle for which we have some position information is a wave packet, and the particle's momentum now has some uncertainty. Link: Wave packet Explorer The more massive the particle, the larger is the magnitude of the momentum p of the particle when it is moving with speed v, and the smaller is the deBroglie wavelength λ. Wave packets build from harmonic waves with smaller wavelength can be smaller in size. The smallest wave packet we can build has a size on the order of the deBroglie wavelength λ of a free particle moving with the same speed v. Example: When we synthesized a square pulse with a width of 100 units using harmonic waves, the longest wavelength we used λ = 200 units. We can know the position of more massive particles moving with approximate speed v more precisely than the position of less massive particle moving with the same speed since the more massive particles have shorter deBroglie wavelengths. Problem:
DispersionPhotons always move with the speed of light c. Particles can move with any speed. The plane matter waves of a free particle moving with speed v has the form ψ(x,t) = Acos(kx - ωt + φ). For any wave λf = ω/k = vwave. Using the dispersion relation for a free particle we find vwave = ω/k = ħk/(2m) = p/(2m) = mv/(2m) = v/2. This may seem surprising. Our classical intuition suggests that a wave describing a particle moving with speed v should move with speed v itself. How else can it track the particle? But remember! If a particle is described by a plane wave ψ(x,t) = Acos(kx - ωt + φ) we know the particles momentum and wavelength exactly, but its position is completely unknown. It is impossible to track a particle whose momentum is exactly known. The dispersion relation for free particles implies that the plane matter waves of particles do not all have the same speed. The wave function for a particle for which we have some position information is a wave packet. We have to superimpose plane waves describing particles with slightly different momenta and energies, or with slightly different wavelengths and frequencies. But these component waves now all move with slightly different speeds. So what happens to the shape of a wave packet as time goes on? Let us investigate using a spreadsheet. We have previously built a square wave packets at t = 0 from sinusoidal waves with different wavelengths λl or different wave numbers k. p = ħk, E = p2/2m = (ħk)2/(2m), ω = E/ħ = ħk2/(2m) We will now let each of these waves move with speed ω/k = vwave. Spreadsheet: A moving square pulse The speed of the plane waves making up a wave packet is called the phase velocity, vp = ω/k. The speed of a wave packet build using those plane waves is called the group velocity and is calculated using vg = dω/dk. Electron Diffraction
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