Waves and Particles. Photons. Summary. Photons. Photoeffect (cont d) Photoelectric Effect. Photon momentum: V stop

Transcription

1 Waves and Particles Today: 1. Photon: the elementary article of light.. Electron waves 3. Wave-article duality Photons Light is Quantized Einstein, 195 Energy and momentum is carried by hotons. Photon energy: E = hf Photon momentum: Planck s constant: h = Js Note: Because h is so small in SI units, we don t notice hotons in the light we see around us. Photon energy: Photon momentum: Photons Energy-momentum relation for an ordinary article (ch.37): But for a hoton we find: E = hf Equations agree if m=. ( c) ( mc ) E = + E = hf = hc / λ = c The hoton has zero rest mass and always moves with seed c. Summary Light is an electromagnetic wave. 8 f λ = c c = 3. 1 m / s Energy and momentum carried by hotons. E = hf = E / c h = Energetic hotons have short wavelengths. E = hf = hc / λ hc = 14 ev nm For examle a hoton with energy 3 ev has λ = 414 nm (viole while a hoton with energy ev has λ = 6 nm (red). Js Photoelectric Effect Light striking a solid target causes electrons to be emitted. Collect electrons to form a current. Measure current as function of voltage alied between target and collector. target surface ejected electron e - incident hoton γ Photoeffect (cont d) V sto collector surface Measure maximum electron energy by alying oosing voltage until current stos.

2 Photoeffect (cont d) Result: The maximum electron kinetic energy K is determined by the frequency f of the light used, not by its brightness. There is a direct roortionality between energy and frequency. Shows that electron gets its energy from a hoton of energy E = hf. Photoeffect (cont d) h sloe = e Exeriment: K V max sto = hf Φ = ev h = f V e sto Interaction of light with matter Detection of light: A hoton is absorbed by an electron, atom, or molecule. Photocell Photograhic film Digital camera Retina of the eye E = hf Production of light: A hoton is emitted by an electron, atom, or molecule. Scattering of light: A hoton can collide with an electron, exchanging energy and momentum, just like two billiard balls. Comton scattering In hoton collision with electron, use conservation of energy and momentum to solve for change in f,λ for given recoil angle θ. Calculate exactly like collision of two billiard balls in Ch. 9. Light waves and hotons We detect light by observing the hoton. So what is the wave doing? The electromagnetic wave intensity determines the robability that we will detect a hoton. Where the wave is strong, the light is bright, the eye will detect many hotons. Where the wave is weak, the light is dim, there are very few hotons. The double slit revisited Suose we decrease the intensity of the light in a double-slit exeriment until we are only detecting one hoton er second. Then each hoton hits somewhere on our screen, and we can record that hit. We cannot redict where the next hoton will hit. But if we kee this u for a long time, the redicted interference attern will gradually emerge!

3 Wave-Particle Duality So is it weird that light is both a article and a wave? Not really. Everything is both a article and a wave. A hoton is a quantum of a light wave. An electron is a quantum of an electron wave. etc Electron waves We have seen: A light wave determines the robability of detecting a hoton. Now we find: An electron wave determines the robability of detecting an electron! And the equation relating the wave and the article is the same: For historical reasons this wave is called a debroglie wave and λ is the debroglie wavelength. Particle Energies Note that λ = h/ alies to both hotons and electrons (and all other articles also). But the relation between momentum and energy is different, as we already know, deending on the mass of the article: Photons: Slow articles: E = c K = m Fast articles: ( ) ( ) E = c + mc Examle Electron microscoe: Problem 38-5 For studying very small structures (molecules, etc.) we need a wave with a very short wavelength. To get a wavelength the size of an atom we need λ =.1 nm, which means c = hc/λ = (14 ev-nm)/(.1 nm) = 1 kev. For light this means a hoton with energy E = c = 1 kev. But this is an Xray which has so much energy it will likely destroy the structure to be studied. But for an electron this means only 1% the kinetic energy: 3 1 ( c) (1 1 ev ) K mv = = = 14eV 6 m mc.51 1 ev Electron Interference Electron waves interfere just like any other wave. A double-slit exeriment with electrons works just like the Young s double slit with light. Each electron hits at a random sot, but as more electrons ass through the two-slit attern emerges. See text figure 38-8 on age 169. The electron wave intensity determines the robability of finding an electron. All other articles show the same robability-wave behavior (rotons, neutrons, even atoms). Double-slit Simulation Photons strike screen randomly. Next hit is more robable where light wave is more intense. Double-slit interference attern gradually emerges. This simulations is exactly the same for electrons, or any other article!

4 Schrödinger s Equation Just as a light wave is governed by Maxwell s Equations, there is a differential equation that governs the roagation of the debroglie wave. The function that describes the wave, and satisfies Schrödinger s Equation, is called the wave function (for lack of a better name). It s a scalar function, simler than the electromagnetic case. E ( x, = E sin( kx ω B( x, = B sin( kx ω ψ ( x, sin( kx ω Or in general ψ ( x, y, z, Schrödinger s Equation Just as a light wave is governed by Maxwell s Equations, there is a differential equation that governs the roagation of the debroglie wave. The function that describes the wave, and satisfies Schrödinger s Equation, is called the wave function for lack of a better name. As with any wave, the intensity is the square of the amlitude, and the robability of finding the electron at oint x is given by the square of the wavefunction: Probability = P( x) = Wave-Particle Duality Everything is both a article and a wave. A hoton is a quantum of a light wave. An electron is a quantum of an electron wave. etc Wave gives robability of article showing u. Universal relation between wavelength of the wave and momentum of the article: h λ = Planck s constant: h = = J s ev s The Uncertainty Princile If all articles are reresented by waves, then we never know exactly where a article is at any articular time. Heisenberg raised this fact to the level of a fundamental ostulate of quantum mechanics. There is a relation between the uncertainty in each osition coordinate of a article, and in the corresonding momentum, and also between the energy and the time. x x y y z z t E h = = π 14 ev s Examle When the structure of the nucleus was not understood, one idea was that it contained rotons and electrons. The uncertainty rincile shows that to be unlikely. Because of the small size of the nucleus, the uncertainty in the electron s momentum would be very large. x = 1 6 nm = / x hc 14 ev nm c = = MeV 6 π x π 1 nm E = = ( c) + ( mc ) () + (.51) MeV MeV More about uncertainty There is sometimes some misunderstanding, even among hysicists, about what this means. It is sometimes said that it is just the fact that it is hard to measure small things. But it is more than that. Quantum mechanics, which has been reeatedly verified by exeriments, asserts that these uncertainties are fundamental. The electron actually does not have a well-determined osition. Only the robability of its being at a articular lace is determined.

5 Barrier enetration (tunneling) One consequence of Schrödinger s Equation is that an electron (or any other article) can move through a region where it s forbidden to go by energy conservation! Energy conservation can be violated, but only for a brief time, given by the uncertainty relation. Does uck go over the hill? Newton: v Schrödinger: t E If ½mv > U b, yes. Otherwise, no. If ½mv > U b, yes. Otherwise, maybe! Tunneling (barrier enetration) electron Negative V means ositive otential energy U for electron in barrier region. Suose E = kinetic energy is less than U = barrier height. What is robability of electron tunneling through? T = transmission coefficient =? Very thin wire (nanotube). Central section given negative otential. All this on a solid surface (silicon chi). Tunneling Result T = transmission coefficient =? Here b m = [ U E] b So a small change in U b or L makes a big difference in the current. For e in allowed region (x< or x>l) we have ψ =ψ For e in forbidden region (<x<l) we have ψ e bx This means robability of getting through is ψ ( L) T () = e bl Simlified Schrödinger Consider simle case, wave function ψ as a function of only x: Schrödinger s Equation is then a second-order ordinary differential equation, similar to the equations we have studied for simle harmonic motion or oscillations of an LC circuit, excet the indeendent variable is x, rather than t. d ψ Just take the second derivative ψ of sin(kx) and see that it s right. π Now we know k = and λ = λ So x = and x = h so = k Plane wave Monochromatic Monoenergetic ψ d ψ Simlified Schrödinger This works for free article with fixed momentum But what if article is subject to a otential energy function U(x), so that is not constant? k Kinetic energy is K = = = E U( x) m m So if E>U(x), we have k >, and we have a sine wave as before. But if E<U(x), we have k <, and solution is an exonential! d ψ = b ψ gives ψ e bx b m = = k [ U( x) E] Simlified Schrödinger Summary b d ψ ψ Classically allowed regions, E>U(x): = k Classically forbidden regions, E<U(x): ψ e bx = m = ) [ U( x E]

6 Back to Tunneling For e in forbidden region (<x<l) we have ψ e bx This means robability of getting through is T = transmission coefficient =? m Here b = [ Ub E] So this means a small change in U b or L makes a big difference in the current. ψ ( L) T () = e bl