* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Exam 3 review
Renormalization wikipedia , lookup
Elementary particle wikipedia , lookup
Quantum electrodynamics wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Tight binding wikipedia , lookup
Double-slit experiment wikipedia , lookup
Atomic orbital wikipedia , lookup
Molecular Hamiltonian wikipedia , lookup
Hydrogen atom wikipedia , lookup
X-ray photoelectron spectroscopy wikipedia , lookup
Electron configuration wikipedia , lookup
Particle in a box wikipedia , lookup
Rutherford backscattering spectrometry wikipedia , lookup
X-ray fluorescence wikipedia , lookup
Matter wave wikipedia , lookup
Population inversion wikipedia , lookup
Wave–particle duality wikipedia , lookup
Atomic theory wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
deBroglie Waves • But what is the wavelength of an electron?! • For photons, it was known that photons have momentum E= pc= hc/ p=h/ =h/p (wavelength) • deBroglie proposed that this is also true for massive particles (particles w/mass)! =h/p = “deBroglie wavelength” (He also proposed E = hf ). We’ll talk about this later.) (momentum) p A few more lengths • What would this make the wavelength of matter? – =h/p = “deBroglie wavelength” – Remember KE = ½ mv2 or gmc2 – h=6.6x10-34 J s Thing Energy (eV) Energy (J) Speed (m/s) Momentum (kg m/s) Wavelength (m) Photoelectron 1 1.6x10-19 6x105 5.4x10-25 1.2x10-9 Rutherford a (4 amu) 2x106 3.2x10-13 2x107 1.3x10-19 5x10-15 Billiard ball (0.2 kg) 6x1019 10 10 2 3x10-34 0.16 2.4x10-20 200 2.4x10-22 2.8x10-12 Thermal C60 molecule 1.2x10-24 kg Wave uncertainty • In general waves are a bit “fuzzy” • The exact location of a wave packet is somewhat uncertain. It has a distribution which has both an average value 𝑢0 and an uncertainty Δ𝑢 = 𝑢 − 𝑢0 2 : which is the standard deviation and means the average over that quantity. • This is a general statement about all waves • If a signal is made up of Ψ 𝑢 = 𝑛 𝐴𝑛 𝑒 𝑖𝑣𝑛 𝑢 • Then there is an uncertainty relationship: 1 Δ𝑢Δ𝑣 ≥ 2 But you already know this, maybe, from radio signals. Heisenberg Uncertainty Principle • In math: Δx·Δpx ℏ 2 • In words: Position and momentum cannot both be determined completely 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, not very weird if you think about waves. Heisenberg Uncertainty Principle Δx small Δp – only one wavelength Δx medium Δp – wave packet made of several waves Δx large Δp – wave packet made of lots of waves Δx·Δpx ℏ 2 What about time dependence? • They can be made up of different frequencies in time or wave-vectors (k or frequency in space) at the same time so: 1 Δ𝜔Δ𝑡 ≥ 2 hω 2𝜋 Remember for light : E = hf = = ℏ𝜔 ℏ ∆𝐸∆𝑡 ≥ 2 This means the uncertainty in time and energy are related. So a state with well defined energy has a long “lifetime” Review ideas from matter waves: Electron and other matter particles have wave properties. See electron interference If not looking, then electrons are waves … like wave of fluffy cloud. As soon as we look for an electron, they are like hard balls. Each electron goes through both slits … even though it has mass. (SEEMS TOTALLY WEIRD! Because different than our experience. Size scale of things we perceive) If all you know is fish, how do you describe a moose? Electrons & other particles described by wave functions () Not deterministic but probabilistic Physical meaning is in ||2 = * ||2 tells us about the probability of finding electron in various places. ||2 is always real, ||2 is what we measure LASERs! What’s so special about LASER light? A) It doesn't diffract when it goes through two slits. B) All the photons in the laser beam oscillate inphase with each other. C) The photons in the laser beam travel a little bit faster because they all go the same direction. D) Laser light is pure quantum light, and therefore, cannot be described with classical EM theory. E) Laser light is purely classical light, and therefore, it is incompatible with the photon picture. How light interacts with atoms e 2 out 1 spontaneous emission of light: Excited atom emits one photon. 2 in e 1 absorption of light: Atom absorbs one photon e in 2 out 1 stimulated emission: clone the photon -- A. Einstein 1916 Surprising fact: Chance of stimulated emission of excited atom EXACTLY the same as chance of absorption by lower state atom. Critical when making a laser. Laser: Stimulated emission to clone photon many times (~1020/s) Light Amplification by Stimulated Emission of Radiation Stimulated emission Identical phase Identical energy Identical momentum Legend: Photon Atom in ground state Atom in excited state e e Spontaneous emission Random phase Random direction Similar energy (as absorbed photon) Legend: Photon Atom in ground state Atom in excited state e e To increase the number of photons when going through the atoms, more atoms need to be in the upper energy level than in the lower. Need a “Population inversion” (This is the hard part of making laser, b/c atoms jump down so quickly.) Nupper > Nlower, more cloned than eaten. Nupper < Nlower, more eaten than cloned. How to get population inversion? e A) Use photons with hf < ΔE B) Use photons with hf = ΔE ΔE e excited C) Use photons with hf > ΔE D) Use very strong lamp with hf ≈ ΔE. E) Will never get population inversion in this system. not excited No population inversion in 2 level atom! e 2 out 1 spontaneous emission of light: Excited atom emits one photon. 2 in e 1 absorption of light: Atom absorbs one photon e in 2 out 1 stimulated emission: clone the photon Equal probability Population inversion means: More atoms are in the excited state than in the ground state. As soon as we have the same number of atoms in the excited state as in the ground state, the probability of creating an excited atom is same (or smaller, when considering spontaneous emission) as the probability of having stimulated emission! Can never reach population inversion in 2-level atom! Summary Mirror Half-silvered mirror out LASERs need: • Population inversion Gain • Optical feedback (optical resonator) Coherent light deBroglie Waves • Substituting the deBroglie wavelength (=h/p) into the condition for standing waves (2r = n), gives: 2r = nh/p • Or, rearranging: pr = nh/2 L = nħ • deBroglie EXPLAINS quantization of angular momentum, and therefore EXPLAINS quantization of energy! Three strings: Case I: no fixed ends (infinite length) y x y Case II: one fixed end (infinite length) x y Case III: two fixed end (finite length) x For which of these cases do you expect to have only certain wavelengths allowed… that is for which cases will the allowed wavelengths be quantized? A. I only B. II only C. III only D. more than one Three strings: Case I: no fixed ends y x y Case II: one fixed end x y Case III, two fixed end: x For which of these cases, do you expect to have only certain frequencies or wavelengths allowed… that is for which cases will the allowed frequencies be quantized. a. I only b. II only c. III only d. more than one Last class: quantization came when we applied 2nd boundary condition, bound on both sides (answer c). Electron bound in atom (by potential energy) Free electron PE Only certain energies allowed Quantized energies Any energy allowed Boundary Conditions standing waves, fixed wavelength No Boundary Conditions traveling waves, any wavelength allowed The Schrodinger Equation ( x, t ) ( x, t ) U ( x , t ) ( x , t ) i 2 2m x t 2 2 Once at the end of a colloquium Felix Bloch heard Debye saying something like: “Schrödinger, you are not working right now on very important problems…why don’t you tell us some time about that thesis of deBroglie, which seems to have attracted some attention?” So, in one of the next colloquia, Schrödinger gave a beautifully clear account of how deBroglie associated a wave with a particle, and how he could obtain the quantization rules…by demanding that an integer number of waves should be fitted along a stationary orbit. When he had finished, Debye casually remarked that he thought this way of talking was rather childish…To deal properly with waves, one had to have a wave equation. The Schrödinger equation in 1D ( x, t ) ( x, t ) U ( x , t ) ( x , t ) i 2 2m x t 2 Mass of particle 2 Potential space and time coordinates Complex i, with i2 = -1 The Schrödinger equation in 1D ( x, t ) ( x, t ) U ( x , t ) ( x , t ) i 2 2m x t 2 2 KE ( x, t ) PE ( x, t ) E ( x, t ) This is just the conservation of energy… And let’s see if it will work for non-plane wave solutions for different potentials. The answer is it does and it does extremely well. ( x, t ) ( x, t ) U ( x , t ) ( x , t ) i 2 2m x t 2 2 Most physical situations (e.g. H atom) no time dependence in V! Simplification #1 when U(x) only (or U(x,y,z) in 3D): (x,t) separates into position part dependent part (x) and time dependent part (t) =exp(-iEt/ħ). (x,t)= (x)(t) Plug in, get equation for (x) ( x) U ( x) ( x) E ( x) 2 2m x 2 2 “Time independent Schrödinger equation” Solving the Schrodinger equation for electron wave in 1D ( x) U ( x) ( x) E ( x) 2 2m x 2 2 time independent eq. 1. Figure out what U(x) is, for situation given. 2. Guess or look up functional form of solution. 3. Plug in to check if ’s, and all x’s drop out, leaving equation involving only bunch of constants; showing that trial solution is correct functional form. 4. Figure out what boundary conditions must be to make sense physically. ∞ 5. Figure out values of constants to meet |(x)|2dx =1 boundary conditions and normalization -∞ 6. Multiply by time dependence (t) =exp(-iEt/ħ) to have full solution if needed. Ψ(x,t) STILL HAS TIME DEPENDENCE! For a U(x) where does an electron want to be? Electron wants (will fall to) to be at position where a. U(x) is largest b. U(x) is lowest c. KE > U(x) d. KE < U(x) e. where elec. wants to be does not depend on U(x) Electron wants to be at position where a. U(x) is largest b. U(x) is lowest c. Kin. Energy > U(x) d. Kin. E. < U(x) e. where elec. wants to be does not depend on U(x) U(x) x electrons always want to go to position of lowest potential energy, just like ball going downhill. c. and d. not right, because actual value of U(x) is arbitrary, so can choose bigger or smaller than KE. Example: simplest case, free space U(x) = const. ( x) U ( x) ( x) E ( x) 2 2m x 2 2 ( x) E ( x) 2 2m x 2 Smart choice: constant U(x) U(x) 0! 2 Solution: ( x) A cos kx, or: ( x) B sin kx 2 2 with: k E 2m No boundary conditions not quantized! So we found the solution for the time-independent Schrödinger equation for the special case with U(x) = 0: ( x) A cos kx , 2 2m k E, 2 p k k, and therefore E, can take on any value. So almost have solution, but remember still have to include time dependence: ( x, t ) ( x) (t ) (t ) e iEt / A bit of algebra, and identity ei = cos + isin gives: (x,t) Acos(kx t) Ai sin(kx t) (Solution of time-independent Schrödinger eqn. with U(x)=0) Which free electron has more kinetic energy? a. 1., b. 2., c. same big k = big KE 1. 2. x small k = small KE x If U=0, then E = Kinetic energy. So first term in Schröd. Eq. is always just kinetic energy! 2 2 (x) E (x) 2 2m x More Negative Curvature KE. Bending tighter = more KE • Review for exam 3 part 2. Let’s do an estimate in real life! 0.5 0.0 -0.5 -1.0 -1.5 0 Small “non-rigid box” accidentally made by jamming an STM tip into a surface. 2 4 6 8 10 12 14 Profile of the box (along the blue line) all heights are in nm. Using the uncertainty relationship estimate the energy of a bound state (if any) a) 10 eV b) 1 eV c) 0.1 eV d) <0.01 eV 0.7 0.6 0.5 0.4 -0.2 -0.1 0.0 0.1 Moore’s Law The end of Moore's law? As feature density goes up, device and line sizes must get smaller and smaller. Semiconductor chips are made with optical lithographic techniques. Current linewidths are 28-32 nm. These are essentially at the diffraction limit for optical techniques using visible light. Problem making features much smaller. As device sizes get smaller and smaller, then intrinsic quantum effects will get more and more important. This may be good or bad. Quantum dots, nanotubes, and all of “nanotechnology”. Nanotechnology: how small does a wire have to be before movement of electrons starts to depend on size and shape due to quantum effects? How to start? Need to look at? a. size of wire compared to size of atom b. size of wire compared to size of electron wave function c. Spacing between wires compared to wavelength of ed. Energy level spacing compared to thermal energy, kBT. e. something else (what?) or more than one of the above. Nanotechnology: how small does a wire have to be before movement of electrons starts to depend on size and shape due to quantum effects? How to start? Need to look at Energy level spacing compared to thermal energy, kBT. kB=Boltzmann’s constant Typically focus on energies in QM. Electrons, atoms, etc. hopping around with random energy kBT. kBT >> than spacing, spacing irrelevant. Quantum does not play a big role. Quantum effects = notice the discrete energy levels. Quantum effects not critical Quantum effects critical PE + 1 atom + + + + + + + + + many atoms but lot of e’s move around to lowest PE repel other electrons = potential energy near that spot higher. as more electrons fill in, potential energy for later ones gets flatter and flatter. For top ones, is VERY flat. PE PE for electrons with most PE. “On top” + + + + + + + + + + + + + + ++ As more electrons fill in, potential energy for later ones gets flatter and flatter. For top ones: U(x) is VERY flat. Approximation: the infinite square well Exact Potential Energy curve (U) “Finite square well” or “non-rigid box” small chance electrons get out of wire (x<0 or x>L)~0, but not exactly 0! U(x) U(x) 0 0 L Good Approximation: “Infinite square well” or “rigid box” Electrons never get out of wire (x<0 or x>L) =0. x (OK when Energy << work function) NOTE: Book uses “rigid box” for “infinite square well” Last class: “infinite square well” U(x) U(x) = Energy (x) n=3 For 0<x<L: ( x) E ( x) 2 2m x 2 n=2 n=1 0 ∞, x ≤ 0 0, 0<x<L ∞, x ≥ L 0 Solution: 2 With B.C.: (0)=(L) =0 L x ( x, t ) ( x ) (t ) 2 nx iEnt / sin( )e L L p 2 n 2 22 kn = n / L, En , (n=1,2,3,4 …) 2 2m 2mL ( x, t ) 2 nx iEnt / sin( )e L L Quantized: kn = n/L n=2 Quantized: En n What you expect classically: Electron can have any energy Electron is localized If it has KE, it bounces back and forth between the walls 2 2 2 2mL2 n 2 E1 What you get quantum mechanically: Electron can only have specific energies. (quantized) Electron is delocalized … spread out between 0 and L Electron does not “bounce” from one wall to the other. 4.7 eV mathematically U(x) = 4.7 eV for x<0 and x>L U(x) = 0 eV for 0<x<L 0 eV 0 L x What does that say about boundary condition on (x) ? A. (x) is about the same everywhere B. (x) 0 everywhere, except for 0<x<L C. (x) = 0 everywhere, except for 0<x<L D. (x) 0 for 0<x<L, and (x)=0 everywhere else. E. Can’t tell anything yet. Need to find (x) first. The Finite Square Well Need to solve for exact potential energy curve: Finite U(x): small chance electrons get out of wire, so: (x<0 or x>L)~0, but not exactly 0! Energy U(x) 0 ~Work function L Thin wire along x-axes x Outside well (E<V): (Region I) On HW Outside well (E<V): (Region III) d III ( x) d II ( x) 2 2 a III ( x) k II ( x) 2 2 dx dx II ( x) C sin(kx) D cos(kx) III ( x) Be ax 2 2 Energy 4.7 eV Inside well (E>V): (Region II) Eelectron U=0 eV 0 L x ψ(x): smooth & continuous! Regions I and III: d 2 ( x) 0 2 0 (curves upward) dx or: or: x 2 d ( x) 0 2 0 (curves downward) dx Boundary conditions Energy 4.7 eV Eelectron U=0 eV 0 L 1) ψ(x) has to be continuous: II ( L) III ( L) 2) ψ(x) has to be smooth: ' II ( L) ' III ( L) 3) ψ(|x| ) 0 (required for normalization) (similar conditions for x=0) x d 2 ( x) 2m 2 (U E ) ( x) 2 dx “Classically allowed” (ψ(x): sinusoid) Energy 4.7 eV Eelectron U=0 eV 0 L “Classically forbidden” regions (ψ(x): exponentially decaying) Electron is delocalized … spread out. Some small part of wave is where total energy is less than potential energy! x (L) ( L) *1 / e 1/a Eelectron 0 L d 2 ( x) 2m 2 ( U E ) ( x ) a ( x) 2 2 dx wire How far does wave extend into this “classically forbidden” region? ( x) Be a ( x L ) 2m a (U E ) 2 Big a -> quick decay: Small a -> slow decay: Measure of penetration depth: 1/a when decreases by factor of 1/e For U-E = 4 eV, 1/a ~ 0.1 nm (very small ~ an atom!!!) Consider the ground state wavefunction for an electron in an infinite square well of length L and a finite square well of length L (same L for both cases). What can you say about the ground state energies? U(x) Energy a)They are the same b) Finite well has lower energy. c) Infinite well has lower energy. 0 0 L x Remember for the infinite square well: E1 22 2mL2 Consider the ground state wavefunction for an electron in an infinite square well of length L and a finite square well of length L (same L for both cases). What can you say about the ground state energies? V(x) Energy a)They are the same b) Finite well has lower energy. c) Infinite well has lower energy. 0 0 L x Remember for the infinite square well: E1 22 2mL2 Review: ‘Penetration depth’ E<V: Classically forbidden region Eelectron 0 L ( x) Be a ( x L ) Penetration depth: Distance 1/a over which the wave function decays to 1/e of its initial value at the potential boundary (here (L)): 2m (U E ) , with a 2 (L) ( L) / e 1/a x For U-E = 4 eV, 1/a ~ 0.5 nm (very small ~ an atom!!!) ( x) Be ax 2m (U E ) with: a 2 0 L What changes would increase how far the wave penetrates into the classically forbidden region? A. Decrease potential depth (= work-function of metal) B. Increase potential depth C. Decrease wire length L D. Increase wire length L E. More than one of the above Wide vs. narrow finite potential well U U-E E1 2 2 L 2mL2eff 2m a (U E ) 2 Smaller U-E smaller a larger penetration depth STM (picture with reversed voltage, works exactly the same) end of tip always atomically sharp Application of quantum tunneling: Scanning Tunneling Microscope 'See' single atoms! Use tunneling to measure very(!) small changes in distance. Nobel prize winning idea: Invention of “scanning tunneling microscope (STM)”. Measure atoms on conductive surfaces. Measure current between tip and sample Current vs. Position. • • • • Tuneling Controversy STM now Superconductors STM (picture with reversed voltage, works exactly the same) end of tip always atomically sharp Radioactive decay (Quantum tunneling – George Gamow) Nucleus is unstable ejects alpha particle (2 netrons, 2 protons) Typically found for large atoms with lots of protons and neutrons. Polonium-210 84 protons, 126 neutrons Proton (positive charge) Neutron (no charge) Nucleus has lots of protons and lots of neutrons. Two forces acting in nucleus: - Coulomb force .. Protons really close together, so very big repulsion between protons due to coulomb force. - Nuclear force (attraction between nuclear particles is very strong if very close together) … called the STRONG Force. Potential energy curve for the α particle + KE New nucleus Nucleus Alpha particle (Z-2 protons, (Z protons, (2 protons, & bunch of neutrons) bunch of neutrons) 2 neutrons) Strong attractive force (Nuclear forces) U(r) Look at this system… as the distance between the alpha particle and the nucleus changes. As we bring the a particle closer to the core, what happens to potential energy? r Coulomb repulsion: Nucleus: (Z-2) protons kq1q2 k ( Z 2)(e)(2e) U (r ) r r V=0 for r ∞ Energy What’s the kinetic energy of this particle inside the nucleus? U(r) D C B x A E: Something else Energy What would the kinetic energy of that particle be after it tunneled out from the nucleus? U(r) D C B x A E: Something else Energy So we found that the particle has less kinetic energy outside than inside the nucleus. Did it loose energy? U(r) KEoutside x KEinside A) Yes. B) No. C) Impossible to tell. Need to solve Schröd. equ. first. Energy Wave function picture: U(r) ~100MeV of KE inside the nucleus Exponential decay in the barrier ~1-10MeV of KE outside Wave function of the free particle: ‘small’ KE Large wavelength Wave function of the particle inside the potential well: Large KE small Wavelength Observe a particles from different isotopes (same # protons, different # neutrons), exit with different amounts of energy. 2m a (U E ) 2 U(r) Energy 30 MeV 1. Less distance to tunnel 2. U-E is smaller (smaller a) Wave function doesn’t decay as much before reaches other side … more probable! 9MeV KE 4MeV KE x The 9 MeV electron more probable… Isotopes that emit higher energy alpha particles, have shorter lifetimes!!! Solving Schrodinger equation for this potential energy is hard! V(x) Square barrier is much easier… and get almost the same answer! V(x) The “classically forbidden regions” are where … a. a particle’s total energy is less than its kinetic energy b. a particle’s total energy is greater than its kinetic energy c. a particle’s total energy is less than its potential energy d. a particle’s total energy is greater than its potential energy Answer is c. E=hc/… A. B. C. D. …is true for both photons and electrons. …is true for photons but not electrons. …is true for electrons but not photons. …is not true for photons or electrons. c = speed of light! E = hf is always true but f = c/ only applies to light, so E = hf ≠ hc/ for electrons.