* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Quantum Physics
Bohr–Einstein debates wikipedia , lookup
Renormalization wikipedia , lookup
Schrödinger equation wikipedia , lookup
Dirac equation wikipedia , lookup
Wave function wikipedia , lookup
Renormalization group wikipedia , lookup
Double-slit experiment wikipedia , lookup
Quantum electrodynamics wikipedia , lookup
Ferromagnetism wikipedia , lookup
Molecular Hamiltonian wikipedia , lookup
X-ray fluorescence wikipedia , lookup
Particle in a box wikipedia , lookup
Rutherford backscattering spectrometry wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
X-ray photoelectron spectroscopy wikipedia , lookup
Tight binding wikipedia , lookup
Atomic orbital wikipedia , lookup
Hydrogen atom wikipedia , lookup
Matter wave wikipedia , lookup
Electron configuration wikipedia , lookup
Atomic theory wikipedia , lookup
Wave–particle duality wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Quantum Physical Phenomena in Life (and Medical) Sciences Quantum Physics Experiments and principles beyond the capacity of the classical (Newtonian) physics Péter Maróti Professor of Biophysics, University of Szeged, Hungary Suggested text: S. Damjanovich, J. Fidy and J. Szőlősi: Medical Biophysics, Semmelweis, Budapest 2006 New (quantum) physics is needed to describe and to understand • the chemical-physical nature of electrons in atoms and molecules (bonds, interactions, etc.), • the behavior of nucleus (radioactivity, nuclear reactions, radiation therapy, etc.), • the energetic changes in nuclea, atoms and molecules (spectroscopy) and • the image formation from human body and organs (iconography), because classical physics fails to work in these fields of the microworld. What is new that quantum physics offers? 1. Some physical quantities are quantized - the energy of electromagnetic fields (black body radiation) - the energy of the atoms (absorption and emission spectra of atoms, Franck-Hertz experiment) - the magnetic moments of particles, the spin (Einstein-de Haas experiment) 2. Wave-particle dualism - Wave properties of particles (Davisson-Germer experiment, Tunneling) - Particle properties of waves (photoelectric (Hallwachs) effect) 3. „Unusual” distribution - Spatial distribution: description using the term of probability (wave function) - Deviation from equipartition (temperature-dependence of the molar heat capacity of solids) 4. Uncertainty principle after Heisenberg Complementer quantities cannot be measured simultaneously with arbitrarily small precision. 1. The energy is quantized: the electromagnetic field and the atomic oscillators 1. Black body radiation and the UV catastrophe explained by quantized energy levels for light and emitting oscillators (Planck). Light emitted by a black body has a spectrum that depends on temperature. The “ideal black body” is a box with black walls, opening to the exterior through a small hole. Radiation escaping through the hole will be in thermal equilibrium with the temperature of the walls. Classical physics attempted to explain the shape of the curve of power (or energy density, ρ) as a function of wavelength through four laws. 1. Kirchhoff’s law: e/a = E(λ,T), for any substances, where e: emission and a: absorption. For black body: a = 1. 2. Wien’s displacement law: T·λmax= 2896 μm·K (constant) T is the absolute temperature and λmax is the peak of the curve. 3. Stefan-Boltzmann law: Etotal = σ·T4 Etotal is the emittance (total power per unit area), and σ = 56.7 nW·m-2·K-4. This is why bulb filaments are run hot! 4. Rayleigh-Jeans law: 8kT 4 The first three laws worked fine, but not the Rayleigh-Jeans law that led to „UV catastrophe” In the UV, the Rayleigh-Jeans curve diverges to infinity Ultraviolet catastrophe Rayleigh-Jeans curve Experiment black body radiation 8kT 4 All energy levels allowed, so fraction of energy density with high energy increases as temperature increases. The radiation escaping from a black body is determined by the energy loss from vibration of molecules (the electromagnetic oscillators) in the walls. These increase with T. The energy of the light emitted is determined by the oscillators. Planck’s curve Planck proposed that the energy of the electromagnetic oscillators was limited to discrete values, rather than continuous. Planck’s famous equation is E = nhn, where n = 0, 1, 2, …, n (= c/) is the frequency, and h is Planck’s constant, 6.626 10-34 Js. 8 hc 1 4 e hc / kT 1 The energy levels are quantized. At any temperature, the transitions at higher energy levels are less likely to occur, so contribution to energy density falls off in high energy range (UV). As the energy (E = hn = hc/) approaches kT, the term in brackets approaches 1, and the Planck equation becomes the same as the Rayleigh-Jeans equation. Conclusion: the energy of the oscillators that make up the electromagnetic field are QUANTIZED 2. Temperature dependence of molar heat capacities of solids Upon lowering the temperature, the observed heat capacity did not remain constant but approached to zero. The heat capacity (specific heat) is the proportionality factor relating temperature rise, DT, to the heat applied: q = C·DT. The classical view was that C was related to the oscillation of atoms about their mean position, which increased as heat was applied. If the atoms could be excited to any energy, then a value of C = 3R = 25 J·K-1 was expected, and this value, proposed by Dulong and Petit, was observed for many systems at ambient temperature. However, the expected behavior was a constant value as a function of T, and this was not seen, and some elements (diamond) had values way off. Temperature dependence of molar heat capacities of solids The Dulong and Petit’s law CV, m U m 3 R T V comes from the equipartition theorem of the classical physics. Significant deviations were observed at low temperatures. Einstein and Debye formula: The atoms oscillate with the same frequency (Einstein) or with the range of frequencies (Debye). Temperature dependence of molar heat capacities of solids Einstein showed that if Planck’s hypothesis of quantized energy levels was applied, the equation: 2hkTn hn e 2 C 3Rf , where f kT hkTn 1 e provided a good fit. This was later improved by Debye who allowed a range of values for n. Conclusion: the energy of atomic oscillators in matter are also QUANTIZED. Conclusions from Planck’s solution to the UV catastrophe and Einstein’s solution to the heat capacity problem 1. Heating a body leads to oscillations in the structure at the atomic level that generate light as electromagnetic waves over a broad region of the spectrum. This is an idea from classical physics. 2. From Planck’s hypothesis, the properties can be understood if the energy of the oscillations (and hence of the light) are constrained to discrete values, - 0, hn, 2hn, 3hn, etc. At any frequency value, the intensity of the light is a function of the number of quanta, n, at a fixed energy, determined by hn. This is in contrast to the classical view in which energy levels were assumed to be continuous, and intensity at any frequency was dependent on the amplitude of the wave. 3. A similar conclusion comes from Einstein's treatment of the heat capacity, but here the effect is seen from the oscillations generated in the substance on application of heat. Absorption of energy is therefore also quantized. 1. The energy is quantized: the atoms When a high voltage is discharged through a gas, the atoms or molecules absorb energy and from collision with the electrons, and reemit the energy as light. It is found that the light is not a continuous range at all frequencies, but is constrained to a few narrow lines (right). Explanations of the emission spectrum of atomic hydrogen played a critical role in development of a quantum mechanical understanding of the structure of atoms. Lines similar to those in the hydrogen emission spectrum were seen as absorption lines in the light from stars. Lines of hydrogen emission spectrum Balmer studied the emission spectrum in the near UV-visible region, and noticed that the distribution of the lines along the wavelength scale showed an interesting pattern that could be described by the Balmer formula: 1 1 1 ~ n 109678 2 2 n 2 where n is 3, 4, 5,… Later work revealed a more extensive set of lines in the UV and IR, which all followed the general pattern given by: 1 1 1 ~ n 2 2 n L nH where is the Rydberg constant, and nL and nH are lower and higher value integers. Term scheme of the H-atom Balmer series: Franck-Hertz experiment One of the most demonstrative experiments for the quantization of energy of atoms. The wavelegth of the emitted light is hc e U At the first step of U = 4.9 V, λ = 253.7 nm The steps of the currentvoltage characteristics describe the set of discrete energy levels (term scheme) of the mercury atom. 1. The magnetic moment is quantized: Spin; The Einstein- de Haas Experiment An iron bar (S) is hung from a string (R) in such a way that it can rotate around its axis. From the torsion vibrations of the string, the angular momentum N, which the bar gets as a whole when it is magnetized by external currents through the spoils (E), can be measured. Suppose that n electrons with their elementary angular momentum j contribute to the magnetization. m: mass of the electron, q =-e: elementary charge of the electron, c: speed of the light, μB: Bohr magneton q M bar n M electron ng j 2mc As n j N 0 q e M bar g N g N g B N 2mc 2mc By measuring the macroscopic quantities N and Mbar, we can determine the gyromagnetic ratio g of the elementary magnetic moments (the Landé factor): g = 2. The magnitude of two Bohr magnetons for the magnetic moment excludes the orbital magnetic moment as the source of the ferromagnetism and can be only explained by the existence of the electron spin. Deflection of Ag atom beam by a magnetic field: the Stern-Gerlach experiment Splitting of Na D-line emission by a magnetic field: the Zeeman effect. Classical physics: continuum of deflections No magnetic field, no splitting N S Quantum physics: two sharp peaks separated The atomic beam passes through an inhomogeneous magnetic field. One observes the splitting of the beam into two components. Consequences: 1) Experimental demonstration of directional quantisation and 2) The direct measurement of the magnetic moments of atoms. Some electron energy levels could be split by a magnetic field. This is the Zeeman effect. The electron spin quantum number, and Pauli’s exclusion principle. In order to account for the magnetic splitting of atomic lines, it was necessary to understand how the magnetic properties of the electron could be included in the grand scheme of the quantum physics. The interpretation of the spin by spinning the electron around the axis is a fiction only that makes the imagination easier (but it is not true)! spin up or a spin down or b Since the electron has a negative charge, it is clear that any electron moving in an orbital should also generate a magnetic field. Pauli realized that the problem could be inverted, - why do all atoms not show a magnetic response? The answer he proposed was that electrons normally came in pairs, so that their magnetic effects cancelled out. This would be the case if, in addition to movement in an “orbit”, the electrons were also spinning on their axes. Each electron would then be a magnet. A pair of electrons in the same orbit would cancel out, but only if they had opposite spins. More formally, Pauli’s exclusion principle states that two, but no more than two, electrons can occupy any orbit. As a result, the number of electron orbitals was doubled, by adding the fourth quantum number, ms, which can have one of two values, +½ and -½. With the addition of ms, the properties of elements in the Periodic Table could be accounted for. In addition to the atomic number, and the matching of electronic to protonic charge, the reactivities of the elements could be explained in terms of the need to fill the electronic orbitals by sharing electrons, thus explaining the valence properties, and their periodic pattern. Filling of orbital shells, - the building-up (Aufbau) principle Spin-orbit coupling In addition to the angular momentum and magnetic properties arising from its spin (S), an electron in a molecular orbital has an orbital angular momentum, which also generates a magnetic field (L). The interaction between these two magnetic fields is called spin-orbit coupling. The antiparallel arrangement (bottom, right) has the lower energy, and is therefore favored. The electron magnetic field depends on the total angular momentum, J. In general, both electrons and nucleons spin-couple with other magnets in their environment. This coupling to neighboring spins can be measured in pulsed EPR and NMR applications which look at the kinetics of decay of spin states populated by a pulse of excitation. Analysis of the results provides structural information, - distance and types of atoms close enough to couple. Some nuclear magnets of importance in biological studies The gyromagnetic ratio (γ) determines the energy at which electromagnetic radiation will flip the spin of a nuclear magnet. A flip occurs when the energy matches (or is in resonance with) the energy of the transition. The energy needed (expressed in terms of frequency) depends on the applied magnetic field (the strength of the magnet), -4.7 Tesla in this case. What is the difference between the electron and the nucleus as magnets? 1. The spin quantum numbers for the electron, proton and neutron, all have the same value of ½. This is an intrinsic angular momentum, - every electron, proton and neutron has this property. 2. Because of the relation between angular momentum and mass, a spin ½ body has a frequency of rotation determined by the mass. Because nuclear particles (nucleons) have masses 1,836 that of the electron, the nucleus spins much more slowly. 3. Because the magnetic field depends on the rate of rotation, electrons have magnetic fields ~2,000 that of protons, which have higher fields than more massive nuclei. Hence magnetic resonance occurs at much higher energies for EPR than NMR. 2. Wave-Particle Dualism 1. Wave character of particles Diffraction of X-rays electron beams DavissonGermer experiment Beams of accelerated electrons were directed through thin metal foil and arrival of fast electrons were detected on a photographic plate. The diffraction pattern seen when the electron beam was accelerated to give a wavelength of 0.5 Å gave a pattern similar to that seen when a beam of X-rays of similar wavelength (0.71 Å) was used. barrier Application: Quantummechanical tunneling The barrier can be defeated by 1) thermal activation (Arrhenius) and ΔE A particle approaches a potential barrier 2) tunneling (Gamow) Δx k=k0·exp(-ΔE/kBT) ΔE T 1.0 Transmittance (Gamow’s expression) 2 8 m T exp 2 D E D x h2 The temperature-dependence of the rates of the two processes are highly different: 1) Arrhenius: strong temperature-dependence 2) Tunneling: no temperature-dependence Estimating the tunneling probability Estimate the relative probabilities that a proton and a deuteron can tunnel through the same barrier of height 1.0 eV and a length 100 pm when their energy is 0.9 eV! The mass of the proton is m = 1.673·10-27 kg, the Planck’s constant is h = 6.626·10-34 J·s the energy difference to overcome by tunneling is ΔE = 0.1 eV and the length of the barrier is Δx = 100 pm. Apply the Gamow-relationship of transmittance via tunneling: 2 8 m T exp 2 D E D x h2 Conclusions: 1) the tunneling probability of a proton (in the system specified) is Tp = 9.56·10-7 and is much (about 300 times) greater than that of a deuteron: Td = 3.07·10-9. 2) The ratio of the tunneling probabilities will be much greater when the barrier is twice as long and the other conditions remain unchanged: Tp/Td = 9·104. Quantum tunneling effect: Scanning Tunneling Microscope (STM) The electron passes through the gap between the surface of the sample and the tip of the scanning electrode via tunneling. By moving the electrode above the surface, the distance d can be measured and therefore the surface can be mapped. Application of tunneling in life sciences: Scanning Tunneling Microscope 1. Ultra-precise electromechanical instrumentation (piezoelectric scanner, precise electromechanic components for sample positioning) 2. Quantummechanical tunneling-effect: When approaching two biased conductors, the current increases exponentially by a factor of 10 for each 0.1nm reduction in distance before they finally make contact. Cross-section through tip and sample in STM (contour-lines indicate constant charge density) Because of the sharp increase in tunneling current with decreasing distance, even tips with moderate sharpness yield atomic resolution easily, because the front atom carries the lion´s share of the tunneling current! Electron tunneling in reaction center protein of photosynthetic bacteria LIGHT Upon lowering the temperature from ambient temperature, all electron transfer reaction rates are decreasing and they freeze finally except of the first and fastest reaction (3 ps halftime) which increases by a factor of about 3 at 4 K. Rate of electron transfer This clearly indicates for tunneling pathway instead of a temperature activated reaction of electron transfer between P and HA. Tunneling (e.g. P→BA→HA) k Thermal reactions (e.g. QA→QB) High temperature 1/T Low temperature Arrangement of the Rb. sphaeroides reaction center cofactors, representation of electron transfer pathways (arrows) and the corresponding time constants. Notations P: bacteriochlorophyll (BChl) dimer, B: monomeric BChl, H: bacteriopheophytine, Q: quinone, Fe: non-heme iron atom, A: photoactive branch, B: photoinactive branch. The circles in the middle of chlorins indicate the Mg atom in BChl. 2. Particle character of waves The photoelectric (Hallwachs) effect The limiting wavelengths at the red ends of the photoelectric spectra are indicated in brackets. The photoelectric effect When UV light shines on a metal surface, it induces the release of electrons, which can be detected as a current in a circuit such as that on the left. The released electrons are attracted by an applied voltage to an anode, and the resulting current detected, and used to measure the rate of electron release. The characteristics of this effect are as follows: 1. No electrons are ejected, regardless of intensity, unless the light is sufficiently energetic. In terms of Planck’s equation, they have to have a high enough frequency. The actual value (the work function) depends on the metal. 2. The kinetic energy of the ejected electrons varies linearly with the frequency of the incident light, but is independent of intensity. 3. Even at low intensity, electrons are ejected immediately if the frequency is high enough. According to the classical view, the energy of radiation should be proportional to the amplitude squared. It should therefore be related to intensity, which is in contradiction to the result observed. Einstein suggested a solution to this dilemma, by invoking Planck’s hypothesis. The electron is ejected if it picks up enough energy from collision with a photon. However, the energy of the photon is given by the Planck equation, and so is proportional to frequency, and quantized. From the 1st law of thermodynamics, energy has to be conserved. We can therefore write an equation in which the kinetic energy of the electron is equal to the energy picked up from the photon, minus the energy needed to dislodge the electron (the work function, f): ½mev2 = hn – f This is Einstein’s photoelectric law. Dual nature of matter – wave and particle properties apply to subatomic particles Einstein’s relation between energy and mass and the de Broglie equation Einstein suggested in the context of special relativity that energy and mass are equivalent, and related through the famous E = mc2. De Broglie realized that, if all matter was quantized, this implied a general relation between the momentum of a particle and its energy as expressed in terms of frequency. By combining the Planck equation, E = hn = hc/, and the relationship for the momentum of an electromagnetic wave (given by p = mc), p = E/c, we get p = h/λ or, rearranging: h h p mv The de Broglie relationship implies that any particle of mass m moving with velocity v will possess wavelike properties. In view of the value of the Planck constant, the effect will be appreciable for particles of low mass. Conclusion: on atomic scale, the concept of particle and wave melts together, particles take on the characteristics of waves, and waves the characteristics of particles. Direct application of the de Broglie relation to resolution limit of microscopes The resolution limit of an optical microscope is defined as the shortest distance between two points on a specimen that can still be distinguished by the observer or camera system as separate entities: 0.61·λ/(n·sinα), where n·sinα is the numerical aperture. The wavelength is given by the de Broglie relationship: h mv The resolution limit of the microscope is inversely proportional to the mass and velocity of the particle. The higher is the speed of the particle, the smaller will be the resolution limit. E.g. for electron of v = c/50 in the electronmicroscope, the limit is λ = 0.12 nm (X-ray). The larger is the mass of the particle, the smaller will be the resolution limit. E.g. the (hypothetical) neutronmicroscope would offer about 2000 times smaller resolution limit than the electronmicroscope with the same velocities of the particles because mneutron ≈ 2000·melectron The atomic de Broglie microscope is an imaging system which is expected to provide resolution at the nanometer scale using neutral He atoms as probe particles. 3. Unusual spatial distribution and discrete energy levels Examples: Particles in potential wells • Electron in hyperbolic potential well (spectroscopy of H-type atoms) • Electrons in rectangular potential well (spectroscopy of dyes in the visible range) • Atoms/molecules in parabolic potential well (spectroscopy of vibrations in the infrared spectral range) 1. Electron in hyperbolic potential well (quantization of the H atom) Semiclassical treatment: the forms of the wave functions and the corresponding energy leveles can be anticipated in terms of the de Broglie relation („standing waves”). The Coulomb potential around the nucleus U (r ) 1 4 0 e r where ε0 is the dielectric constant of the vacuum, e is the elementary charge of the nucleus (proton), The potential energy of the electron Epot r is the distance of the electron from the nucleus. e2 4 0 r 1 The shape of the function (potential barrier) is hyperbolic. The bound electron should be inside the hyperbolic walls. The electron is free to move inside the segment but reflected at the boundary. The hyperbolic wall is the constrain that will quantize the energy of the electron. In lack of this condition (i.e. the electron is not bound, it is free), the energy of the electron can take any values, it is continuous. Each wavefunction is - a standing wave, - fits into the segment and - successive functions must possess one more half-wavelength. The permitted wavelengths n 2 n 2 rn where n 1,2,3,... The average speed of the electron can be estimated by the de Broglie relation vn h h n n me 4 rn me Here me is the mass of the electron. The electron has only kinetic energy inside the well thus the kinetic energy of the electron Ekin 1 h2 2 2 me v n n 2 32 rn2 me The total energy of the electron as a function of r En (r ) Ekin,n Epot,n h2 2 e2 n 2 4 0 rn 32 rn me where n 1,2,3,... 1 The permitted energies are En me e 4 2 2 02 h 2 1 n 2 The rigorous solution from the Schrödinger equation (see below) En me e 4 8 02 h 2 1 n 2 where n 1,2,3,... It differs from the semiclassical solution by a factor of π2/4 ≈ 2 only. Conclusions: 1)The energy of the electron in the H atom is quantized, 2) the quantization arises from the boundary conditions that ψ must satisfy and 3) the subsequent energy levels relate as the inverse squares of the neighboring integer numbers: E1 : E2 : E3: ... = 1 : 2-2 : 3-2: ... The Table on the right shows wave functions, in x, y and z coordinates, for hydrogenic atoms (H, He+, Li2+, etc., atoms with only 1 electron; Z is the atomic number, giving the charge of the nucleus). The Figures below show the “shapes” (boundary surfaces) of the corresponding orbitals. s The subshells are named s, p, d, f, g, and h for electrons with l = 0, 1, 2, 3, 4 and 5. Molecular orbitals The energy due to atomic interaction can be calculated as a function of R. The equilibrium bond length is at the minimum. Formation of the bonds in N2. The 2pz orbitals overlap and coalesce into a s molecular orbital; the px and py orbitals form orbitals perpendicular to each other. 1s orbitals of H-atoms coalesce to form a s bond of H2, with cylindrical symmetry Bond wave functions are formed by summing atomic wave functions. Bonds formed by combing antiparallel spins are favored. For H2, the molecular orbital has a lower energy than the atomic orbitals, so is stable. Bonds of carbon Figs. from Atkins, The Elements of Physical Chemistry In methane, each sp3 forms a s bond with a 1s atom of H Carbon has one empty, and two 1-electron filled 2p orbitals, giving four valence electrons. In order to explain the symmetrical chemical behavior of molecules like methane, we have to have four identical orbitals in tetrahedral symmetry. In order to provide these, a mixing of orbitals occurs, to give four sp3 hybrid orbitals (left). This involves promotion of 1 2s electron to the vacant 2p orbital, hence sp3. In ethylene (ethene), the double-bond is made up of a s and a bond. Take-home message on bonds. 1. Bonding orbitals can take up quite complicated shapes. Extended -bonding occurs in molecules like cytochromes, chlorophylls, flavins, nucleic acid bases, tryptophan, etc. The electrons in these extended orbitals roam over the entire coordinated system (see later the quantumphysical treatment of these systems). 2. Transitions between energy levels in molecules can occur between orbitals of different type. The change in electron distribution results in an electrical dipole difference between the ground and excited states. When the orbital is asymmetric, as in an extended system, the excited state can remain in the -orbital, but will have a different eigenfunction and eigenvalue, and so a different orbital “shape”. This also gives rise to an excited state dipole. 3. Normally, molecular orbitals are at lowest energy when they are filled by two electrons of antiparallel spin. If an electron is removed (for example, by oxidation), or an extra electron added (for example, by reduction), the lone electron is not spin-coupled, and therefore acts as a magnet. The magnetic effect arises from the unpaired spin, and the magnetic dipole results from the angular momentum of the electron in its orbital. 2. Electrons in rectangular potential well (electronic energy levels) 2π-electrons in conjugated chains Metallic model for the π-electrons: They can move freely along the chain (in the well) but cannot escape (from the well). The shape of the potential well is rectangular and infinitely deep. The double and single bonds between neighboring carbon atoms are alternating. The length of an elementary group is l (≈ 3 Å) Bond Length (Å) Energy (kcal/mole) C-C 1.54 83 C=C 1.34 122 C-H 1.08 101 the number of elementary groups is N, each group has 2 π-electrons, the whole chain has 2N π-electrons and the total length of the chain is L = N·l Electrons in the box: rectangular potential well There are 2·N π-electrons in the well. Because of the bundary conditions, the electrons cannot have arbitrary (continuous set of) energies but well defined discrete values only. We are looking for the possible energy levels using 1) semiclassical physics (based on de Broglie’s standing waves of the electrons) and 2) quantum physics (Schrödinger equations). Semiclassical treatment Only those energy levels are allowed along which standing n L where n 1,2,3,... waves can be created with nodal points at the potential 2 walls. h The speeds of the electrons are determined from the de vn Broglie’s relationship. m vn nh 2mL 2 1 h 2 En m v 2n n 2 8 m L2 Replacement of the wavelength in the expression of speed. The total energy consists of kinetic energy term only. Energies of electrons in the box: rectangular potential well The energy of the electrons are quantized: - The lowest energy (zero-point energy) is not zero. It comes from the uncertainty principle that requires a particle to possess kinetic energy if it is confined to a finite region. - The energy levels are dense on the bottom and become more diluted upon getting out of the well. The heights of the energy levels are related as the squares of the integer numbers. - According to the Pauli principle, there can be two electrons with antiparallel spins at every level of energy. As there are N groups, the 2·N π-electrons occupy the lowest N energy levels. En h2 8 m L2 n2 - The separation between adjacent energy levels decreases as the length of the container (L) increases and becomes very small when the where n 1,2,3,...container has macroscopic dimensions. In laboratory-sized vessels the energy is not quantized. The lowest energy band (the most red part) of the absorption spectrum The absorption band with the longest wavelength (smallest energy, the center of the farest red absorption band) h n red abs EN 1 EN n red abs 1 ( N 1) 2 h 2 N 2 h 2 2 2 h 8 m L 8 m L n red abs If N >>1, then n red abs h 1 2 N 4ml h 8ml 2 2N 1 N2 or red abs The wavelength of the absorption band is proportional to N. 4 m c l2 N h Discussion 4 m c l2 red abs N h The wavelength of the most red absorption band is proportional to N, the length of the conjugated chain. With elongation of the chain, its absorption band shifts toward the longer wavelengths: It shifts to red („red shift”). N (length of chain) λred abs spectral range 3 445 nm visible 4 593 nm visible 5 741 nm near IR 6 889 nm near IR molar absorption 3. Particles (atoms and molecules) in parabolic potential well The energy profile of a harmonic oscillator is parabolic. The particle is freely moving inside the parabolic potential well. Semiclassic treatment The total energy of the particle is E 1 k x02 1 m v 2 2 2 L 2 x0 which is equal to the kinetic energy between the walls. Those energy levels are allowed which can be covered with standing waves n n h Possible velocities of the particle Possible energies of the particle 1 n2 h2 k 2 E n m vn 2 8m 2En E n mn 4 n hn nh nh k 4 x0 m 4 m 2 E n vn where n 0,1,2,3,... 2 L Energy levels of the harmonic oscillator Uniform ladder of spacing h·ν Solution from the Schrödinger equation: En = h·ν(n + ½), where n = 0, 1, 2, ... Conclusions: (a) The energy levels of the oscillator are equidistant: ΔE = h·ν. (b) The selection rule: Δn = ± 1. Transitions are allowed between adjacent states only (in first order). Transitions among not neigboring levels are forbidden. (c) The minimum energy of the oscillator (n = 0) differs from zero: E0 = ½ hν. The zero-point energy is not zero, because of For typical molecular oscillator, the zerothe uncertainty principle: the particle is point energy is about confined, its position is not completely E0 = 3·10-20 J = 100 meV uncertain, and therefore its momentum, The Boltzmann-energy at room temperature and hence its kinetic energy, cannot be exactly zero. is ½ kBT = 25 meV 4. Heisenberg’s uncertainty principle Simultaneous estimation of the precision of the space coordinate and the impuls of a particle Wave packet The uncertainty (Δx) of the position (x) of the particle is characterized by the extension of the wave packet: Dx n n h p λ n·λ The precision of the determination of the impuls (p) depends on the precision of the determination of the wavelength. The longer is the wave packet (i.e. the larger is the number n), the more precise is the determination of the wavelength: Dp Dx h Dp D 1 p n From the two equations: The product of the uncertainties of the simultanous determination of the space and impuls coordinates cannot be smaller than a definit limit determined by the Planck’s constant. The measuring device inherently perturbes the state of the microsystem under physical observation. The perturbation causes the uncertainty: the energy and impuls of the absorbed and/or emitted photon are commeasureable with those of the observed (micro)particle. Similar complementer quantity-pair is the time (t) and energy (E). The product of the uncertainties of the lifetime (Δt) and the excited state (ΔE) in simultaneous measurement is limited by the Planck’s constant: DE Dt h Problems for Seminar 1. What is the resolution power of the electron microscope using electrons accelerated to one hundreds of the light speed in vacuum (c/100)? 2. What is the ratio of protons at room temperature that are able to absorb photon from the 1 GHz microwave radiation to provide NMR signal in the spectrometer? 3. What is the magnetic induction in the 1 GHz NMR speψctrometer? Make reasonable estimate for the required current! 4. How many 1000 nm photons does a 1 mW monochromatic infrared rangefinder emit in 0.1 s? 5. Based on the spectral lines of mercury (Hg) emission, construct the approximate current-voltage characteristics of the Franck-Hertz experiment! The most intense spectral lines (in nm): 184.45, 253.7, 300, 312, 334, 365.4, 404.656, 435.835, 546.074 and 579.065 Problems for Seminar 6. Calculate the wavelength of an electron in a 10 MeV particle accelerator. 7. The wave function of an electron in the lowest energy state of a hydrogen atom is ψ =(π·a03)-1/2·exp(-r/a0), with a0 = 52.9 pm, and r the distance from the nucleus. Calculate the probabilities of finding the electron inside a small volume of magnitude 1 pm3 located at a) the nucleus, (b) a distance a0 from the nucleus. The total volume of a hydrogen atom is of the order of 107 pm3. How many times are more probable to find the electron at the nucleus than in the same volume element located at a distance a0 from the nucleus? 8. Determine the surface temperature of the star Sirius when the wavelength of the radiation of the most intense electromagnetic radiation emitted from the surface is 263.27 nm. 9. A glow-worm of mass 5.0 g emits red light (650 nm) with a power of 0.10 W entirely in the backward direction. To what speed will it have accelerated after 10 years if released into free space and assumed to live? 10. In an X-ray photoelectron experiment, a photon of wavelength 150 pm ejects an electron from the inner shell of an atom and it emerges with a speed of 2.14·107 m/s. Calculate the binding energy of the electron. Appendix Introduction of advanced studies in quantum physics Somewhat advanced level Precaution! The Schrödinger equation but not detrimental to your health The refinement of the Bohr model by de Broglie (we will see later in discussion of the principles of spectroscopy of H-atom) paved the way for the more formal description by Schrödinger. What Schrödinger added was a more powerful formalism for the description of the wave function. Unfortunately, this involves some elegant math, - a thing of beauty to the physicist, but perhaps not to life scientists. Fortunately, we don’t need to understand the math in detail to appreciate the results, so the approach here will be non-mathematical. It will help to appreciate a few points: 1. The new approach makes it possible to describe wave functions that are three-dimensional, and more complex in shape, while maintaining the constraints required by quantization, and the need to form a “standing wave”. This made it possible to make the Schrödinger equation time-independent. E H Here H is the Hamiltonian (energy term operator) of the system, ψ is the wave function and E denote the allowed (actual) energy values of the system. 2. Heisenberg had introduced his Uncertainty Principle, which showed that the momentum and position of the electron could not be determined simultaneously with certainty. Schrödinger therefore used a term (wave function) related to the probability of the electron occupying a particular volume. Example: the Schrödinger equation for H-like atoms The Hamiltonian for H-like atoms consists of kinetic and potential energy terms: h2 2 Ze H 2 2 4 0r 8 m and the Schrödinger equation: h2 2 Ze E 2 2 4 0r 8 m Let’s compare this to the classical expression for the energy of the electron: 1 Ze2 2 E me v 2 40r The similarity arises from the need to consider the two balancing forces that determine the electron energy, - the kinetic term (the “centrifugal force” if we consider a planetary model), and the constraining electrostatic term for the potential energy. The first term on the right (the kinetic term) is now quantized, and all terms are modified by the wave function, . What is , the wave function and what is that odd symbol, 2 („del squared” or nabla operator)? h2 Ze 2 2 E 2 8 m 40 r Wave function of an electron in Htype atom Probability of finding the electron between x and x+ δx Radial distribution of the electron Born’s interpretation of the wave function It’s difficult to form a solid picture of what represents, but 2 is easier to grasp; it represents the probability of finding the electron at a particular point in space. This is more formally described by the Born approximation, which, for a one-dimensional system, states that the probability of finding the particle between x and x + dx is proportional to 2dx. In three dimensions the probability is proportional to 2dV. The term2 is a probability density, and we get the probability by multiplying the density by the volume of the region of interest. Examples of a particular wave function, , in onedimension, and its square, 2, plotted against distance are shown on the left. Note that for 2, all values are positive even when goes negative. The shaded bar at the bottom is gray-scale encoded to show probability. The Laplace-operator The odd symbol, 2 (pronounced del-squared) is a Laplace operator. In the one-dimensional case (particle-in-a-box treatment, see later), the term 2 is replaced by the second derivative of with distance 2 d2 dx 2 For the three dimensional case, in its simplest form as applied to Cartesian coordinates, 2 is an abbreviation for the term 2 2 2 d d d 2 2 2 2 dz dx dy It tells us that we have to perform the second derivative operation on the wave function in a three dimensional coordinate system. It takes a more complicated appearance for other coordinate systems, but serves the same function. Because trigonometric functions provide a rich field for investigation of potential wave functions, 2 is often used in the context of spherical polar coordinates. 1 2 1 2 1 2 r sin 2 2 r r r r sin f 2 r 2 sin 2 The energy operators h2 Ze 2 2 2 E 8 m 40 r As noted above, the three operators on all have energy units. They can be expressed in various forms, which makes for some confusion, but basically they represent the same energies as appear in the classical equation. The form of the equation above is that usually presented (this is just the previous equation rearranged). The Table below shows the equation for the 1-D case in different forms. To cut the clutter, Planck’s constant, h, has been replace by ħ, defined by ħ=h/2π. Note that in the last line, the kinetic and potential energy terms have been lumped together into a single operator. This term also has energy units, and is called a Hamiltonian operator, H. p x2 2m, The kinetic energy operator, Tx is shown in classical form, Tx = ½mv2 = and in the quantized form above. The potential energy operator is represented by U(x). Finding appropriate forms for Solution of the Schrödinger equation involves the second derivative of the wave function, . There are many solutions that can be found, and the problem is to know which ones are appropriate. We have to recognize constraints that relate to the physical reality of the system. These can be summarized as follows: 1. Acceptable solutions must be single-valued at all points in space. This is because there cannot be more than one answer to the question “What is the probability of finding the particle at a particular point in space?”. 2. Similarly, must be continuous and finite. This constraint is necessary to provide the “standing-wave” character of the function. 3. The total value of 2 summed over all space must be 1. This is because the particle has to be somewhere, but there’s only one of them. Because energy levels are quantized, the wave function can take different forms, depending on the quantum number. The different quantum numbers in the Bohr model represent different energy levels for orbitals in the ground state and excited states of the H atom. The effect of this last property has interesting connotations when we consider the shapes of the orbitals defined by the wave function (see later the next lecture). 1. Applying the Schrödinger equation to the hydrogen atom h2 The time-independent Schrödinger equation is: 2 2 V E 8 m e2 where the potential energy term is: 40 r V After substitution, we bring all terms to one side and rearrange them. h2 e2 2 2 E 8 m 40 r We substitute into the leftmost term using 2 in its spherical polar coordinate form because of the spherical symmetry of the H atom: 2 2 1 1 2 8 2m e2 2 (E ) 0 sin dr 2 rr r 2 sin r 2 sin 2 f 2 h 40r Fortunately, this fearsome-looking equation has already been solved, so we need be concerned only with the result. The solutions are given in a form appropriate for the spherical polar coordinates chosen. The solutions requires that we use three quantum numbers to describe each energy level. The nomenclature for these is not very helpful. The three quantum numbers are: n, the principal quantum number; l, the angular (azimuthal) momentum quantum number; ml, the magnetic quantum number. n = 1, 2, 3, …n l = 0, 1, 2, …(n-1) ml = 0, ±1, ±2, …, ±l The resulting orbitals are grouped into subshells and shells, as nicely summarized on the left. The solutions of the equation give the energy of the electrons in the orbitals, and their distribution in space. The energy of the electron is given by: mee4 1 En 2 2 2 n 1, 2, 3, ... 8h 0 n Not surprisingly (because it worked), this is the same as the equations Bohr and de Broglie developed to account for the emission lines for the H-atom. Where have the l and ml quantum numbers gone? The answer is straightforward. All three quantum numbers characterize the electron distribution, but for any shell, all the wave functions use the same value for radius. The radius enters into the energy through the potential energy function, - the coulombic force holding the electron to the nucleus, which depends only on r, and is spherically symmetrical. Since this balances the centrifugal energy represented by the angular momentum of the electron, all electrons in the shell have the same potential energy function, determined by n, as in the Bohr equation. This simple case applies only to the H-atom. Because n uniquely determines the energy, the value of E for a particular n is called its characteristic energy, or eigenvalue. The wave function for a particular n is called the characteristic wave function, or eigenfunction. The relation between and probability of distribution depends on the quantum number. Choice of polar coordinates gives as a function of r, , and f, so that ( r, , f ) R( r )Q( )F (f ) The terms on the right are called the radial part, (R(r)), and the angular part, (Q()F(f)), of this expression. As the equation shows, the radial part is dependent on r, so we can think of n as characterizing the distance dependence, with each shell having a different radius. The solution for each orbital differs, depending on the values of the quantum numbers. For orbitals with l and m equal to zero, the solution depends only on r, the radius. As a consequence, these orbits are spherical. For non-zero l, and for m quantum numbers, the “shape” of the orbit is not spherical, and the wave function is determined by x, y and z, the parameters of its position in Cartesian space. For example, of the four electrons in the L shell, one has l = 0, and hence m is also zero, and the orbital is spherical. The other three are arranged along the x, y, and z axes, and are not spherical. 2. Particle in a rectangular box, - the Schrödinger equation in 1-D If we constrain our electron to a one-dimensional box, we can simplify things. In 1-D, we don’t have to worry about the potential energy term of the planetary model, the electron is constrained by the walls of the box, not by the attraction of the nucleus, so we only have to worry about the kinetic energy. And we don’t have to worry about 2, because we’re in 1-D. The Schrödinger equation then becomes: h2 d 2 2 E 2 8 m dx To solve this equation, we have to find an appropriate wave function, . The wavelength of a harmonic oscillator in one-dimension is a sine function, and this provides an appropriate starting point for our search for a wave function. Let’s try = A·sin kx where x is distance. We will take a look at what happens if we set k = n/L, with L as the length of our box. Because sin = sin 2 = sin 3 = 0, we expect this function to generate a curve that oscillates. In the next slide this function (with A =1) is plotted against distance (x) with different values of n, and with distance scaled to units of L. n=1 n=2 n=3 n=4 2 1.0 (sin(nx/L)) sin(nx/L) 1.2 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 -1.2 n=1 n=2 n=3 n=4 0.8 0.6 0.4 0.2 nodes 0.0 0.0 0.2 0.4 0.6 x/L 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 x/L Note that if we change the value of n in this equation, we change the number of beats per unit distance as measured by the number of times the curve crosses the zero-line at a node. The nodes define the distance at which we will form a “standing wave”. At distance x=L, for n = 1 we have 1 node; for n = 2, 2 nodes; n = 3, 3 nodes; etc. The plot on the right shows the square of the function versus the same distance scale. Each node generates a peak, so there are n peaks for each curve. The curves are all positive, so (remembering the 2 gives probability) they look suitable. Our choice of wave function looks pretty good, - it has the sorts of property we’re looking for. We now need to plug it into our Schrödinger equation to see if it satisfies all the constraints. Starting with = Asinkx, the critical steps are: 1. Find the second derivative of this function (it’s given by -k2) 8 mE 2. Substitute from this into the Schrödinger equation to find that k , 2 1 h 8 2 mE 2 and that, by back substitution, A sin 2 x. 2 h 3. Satisfy the boundary condition that is zero at the walls. As we saw in previous slides, sin n = 0, so this is satisfied when x =1 0 or L, the length of our box. From this, our function can work if substitution, n A sin n x L 8 2mE 2 n 2 L h and (n is integer). Then by back n2h2 En 8mL2 4. Satisfy the constraint that the probability is 1. For this we need to adjust the amplitude of our sine wave, using the factor A. Tables of integrals give us a normalization constant such that A 2 L . We finally end up with: 2 n n sin x L L 1 2 We have already seen plots of this function, and a version is shown on the left below, in the context of energy levels. We can now say that the energy of an electron in a one-dimensional box has different levels that are quantized, and that at each level the wave function, , is a sine wave, with the number of nodes equal to the quantum number. The probability is given by 2, and this is shown in the right panel below. 3. Solution of the Schrödinger equation for harmonic oscillator 2 d 2 1 2 k x E 2 2 m dx 2 The Scrödinger equation The wavefunction n 8 4 mk h / 2 2n n! H n e y 2 /2 The permitted energy levels mk where y 4 2 x 1 En (n ) hn 2 with n 0,1,2,... and n 1 2 The factor Hn is a Hermite polynomial, it depends on the vibrational quantum number n and some of which are listed in the Table. n Hn(y) 0 1 1 2y The wavefunction of the ground state (the lowest energy state) is (n = 0) 2 4y2 – 2 3 8y3 – 12y 4 16y4 -28y2 +12 5 32y5 – 160y3 + 120y 0 and the probability density is a bell-shaped Gaussian function 02 8 mk 4 h/2 4 mk h/2 e y 2 e y /2 2 The Hermite polynomials k m