* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download fizika kvantum
Schrödinger equation wikipedia , lookup
Quantum dot wikipedia , lookup
Atomic orbital wikipedia , lookup
Identical particles wikipedia , lookup
Aharonov–Bohm effect wikipedia , lookup
Atomic theory wikipedia , lookup
Wave function wikipedia , lookup
Bell test experiments wikipedia , lookup
Wheeler's delayed choice experiment wikipedia , lookup
Quantum fiction wikipedia , lookup
Quantum field theory wikipedia , lookup
Quantum computing wikipedia , lookup
Orchestrated objective reduction wikipedia , lookup
Erwin Schrödinger wikipedia , lookup
Renormalization wikipedia , lookup
Measurement in quantum mechanics wikipedia , lookup
Density matrix wikipedia , lookup
Quantum entanglement wikipedia , lookup
Quantum group wikipedia , lookup
Quantum machine learning wikipedia , lookup
Bohr–Einstein debates wikipedia , lookup
Delayed choice quantum eraser wikipedia , lookup
Particle in a box wikipedia , lookup
Coherent states wikipedia , lookup
Matter wave wikipedia , lookup
Quantum teleportation wikipedia , lookup
Many-worlds interpretation wikipedia , lookup
Renormalization group wikipedia , lookup
Bell's theorem wikipedia , lookup
Copenhagen interpretation wikipedia , lookup
Hydrogen atom wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Wave–particle duality wikipedia , lookup
Quantum key distribution wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
History of quantum field theory wikipedia , lookup
Probability amplitude wikipedia , lookup
EPR paradox wikipedia , lookup
Quantum state wikipedia , lookup
Double-slit experiment wikipedia , lookup
Interpretations of quantum mechanics wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Path integral formulation wikipedia , lookup
Canonical quantization wikipedia , lookup
PETER PAZMANY CATHOLIC UNIVERSITY SEMMELWEIS UNIVERSITY Development of Complex Curricula for Molecular Bionics and Infobionics Programs within a consortial* framework** Consortium leader PETER PAZMANY CATHOLIC UNIVERSITY Consortium members SEMMELWEIS UNIVERSITY, DIALOG CAMPUS PUBLISHER The Project has been realised with the support of the European Union and has been co-financed by the European Social Fund *** **Molekuláris bionika és Infobionika Szakok tananyagának komplex fejlesztése konzorciumi keretben ***A projekt az Európai Unió támogatásával, az Európai Szociális Alap társfinanszírozásával valósul meg. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 1 Peter Pazmany Catholic University Faculty of Information Technology www.itk.ppke.hu PHYSICS FOR NANOBIO-TECHNOLOGY Principles of Physics for Bionic Engineering (A nanobio-technológia fizikai alapjai ) Chapter 6. Quantum Mechanics – I (Kvantum mechanika) Árpád I. CSURGAY, Ádám FEKETE, Kristóf TAHY, Ildikó CSURGAYNÉ TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 2 Physics for Nanobio-technology: Chapter 6. Quantum Mechanics – I www.itk.ppke.hu Table of Contents 6. Quantum Mechanics I 1. A Glimpse of the Quantum Story 2. Experimental foundation 3. Feynman‟s Path Integral 4. Schrödinger Equation 5. Measurements and Operators 6. Dirac Formalism 2011.05.31. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 3 Physics for Nanobio-technology: Chapter 6. Quantum Mechanics – I www.itk.ppke.hu 1.A Glimpse of the Quantum Story In the late 1890‟s and early 1900‟s new technologies (vacuum technology, optical spectroscopy) and new “layers” of matter were discovered: 1890 1895 1897 1900 1902 1905 1911 1915 2011.05.31. Cathode Rays (J. J. Thomson) Röntgen‟s X-ray Electron (J. J. Thomson) Planck‟s Law – Black-body Radiation Photoelectric Effect Einstein‟s – Photon, Special Relativity, Brown Motion Rutherford: Nucleus of the Atoms Bohr‟s Model of the Hydrogen Atom TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 4 Physics for Nanobio-technology: Chapter 6. Quantum Mechanics – I www.itk.ppke.hu Attempts to explain the new experiments with classical laws failed, e.g. “Ultraviolet Catastrophe” ( Lord J. W. S. Rayleigh, James Jeans ) “Thermodynamic Paradox“ ( Ludwig Boltzmann ) The laws of classical mechanics and classical electrodynamics could not explain the outcome at the new experimental frames (deeper layers) Why does the electromagnetic energy not change in a cavity continuously? Why does the electron not fall into the nucleus? Why the atomic spectrum is discrete? Why the chemical “forces” are so mysterious?, etc. 2011.05.31. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 5 Physics for Nanobio-technology: Chapter 6. Quantum Mechanics – I www.itk.ppke.hu 2. Experimental foundation For a human eye it takes only ~(5 – 6) photons to activate a nerve cell and to send a signal to the brain. Single photons can be detected by photomultipliers. Photocathode Incident photon Electrons Anode Scintillator Light photon 2011.05.31. Focusing electrode Dynode TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 Photomultiplier tube (PMT) 6 Physics for Nanobio-technology: Chapter 6. Quantum Mechanics – I www.itk.ppke.hu When a photon strikes the photocathode of a photo-multiplier, an electron is knocked loose and attracted to positively a charged plate, knocking more electrons loose. This is repeated many times, until significant number of electrons is loosed, the anode is reached, and through an amplifier a click can be heard. Clicks of uniform loudness are heard each time a photon hits the photocathode Amplifier Speaker one photon Photocathode Richard P. Feynman, QED – The Strange Theory of Light and Matter, Princeton University Press, 1985 2011.05.31. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 7 Physics for Nanobio-technology: Chapter 6. Quantum Mechanics – I www.itk.ppke.hu If we put a lot of photomultipliers around and let the intensity of light to be very dim, then the light goes into one multiplier or another and makes a click of full intensity. There is no splitting of light into “half particles” that go to different sensors: we experience particles of light. If we do reflection and transmission experiments with very dim light, we learn that only probabilities can be predicted. 2011.05.31. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 8 Physics for Nanobio-technology: Chapter 6. Quantum Mechanics – I www.itk.ppke.hu Partial reflection of light by a single glass surface We send 100 photons from a light source toward a glass surface. Two detectors, A and B, are set to count photons. We hear a click either in A or in B. Only probabilities can be predicted, in this example we find 4 %. 2011.05.31. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 A 100 4 glass 96 B 9 Physics for Nanobio-technology: Chapter 6. Quantum Mechanics – I www.itk.ppke.hu Partial reflection of light by two surfaces of glass The probability of reflection depends on the thickness of glass! Instead of 4 %, the experiment gives 0 to 16% probs. A 100 0 to 16 Percentage of reflection 16% 8% 100 to 84 0% B Thickness of glass 2011.05.31. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 10 Physics for Nanobio-technology: Chapter 6. Quantum Mechanics – I www.itk.ppke.hu Fundamental question of quantum mechanics: Given that a particle is located at x1 at a time t1, what is the probability that it will be at x2 at time t2? The experimental results of partial reflection can be predicted by assuming that the photon explores all paths between emitter and detector, paths that include single and multiple reflections from each glass surface. The hand of an imaginary „quantum stopwatch‟ rotates (phase) as the photon explores each path. 2011.05.31. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 11 Physics for Nanobio-technology: Chapter 6. Quantum Mechanics – I www.itk.ppke.hu Rotation rate for the hand of the photon quantum stopwatch: the frequency of the corresponding classical wave. (Classical wave optics can predict the result) (Interference). A stopwatch 0,2 front reflection arrow Probability is predicted from the sum over paths. The absolute square-value of the sum gives the probability. 2011.05.31. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 12 Physics for Nanobio-technology: Chapter 6. Quantum Mechanics – I www.itk.ppke.hu Calculation of the probabilities The probability of an event is always the absolute square of the complex probability amplitude. Each path is equally possible, nature explores all possibilities, thus we have to calculate the complex probability amplitude for each path. If an event could happen in two alternative ways, we add the complex amplitudes. If an event could happen in two consecutive ways, we multiply the complex amplitudes. The virtual stopwatch determine the phase changes of the individual complex amplitudes. 2011.05.31. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 13 Physics for Nanobio-technology: Chapter 6. Quantum Mechanics – I www.itk.ppke.hu Single electrons can be detected by electron-multipliers. Slit experiments show that also in case of electrons only probabilities can be predicted. Given that an electron is located at x1 at a time t1, what is the probability that it will be at x2 at time t2? The similarity between electron interference and photon interference suggests that the behavior of the electron may also be correctly predicted by assuming that it explores all paths between emission and detection. 2011.05.31. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 14 Physics for Nanobio-technology: Chapter 6. Quantum Mechanics – I www.itk.ppke.hu Exploration along each path is accompanied by the rotating hand of an imaginary stopwatch. From the interference experiments we can learn that the number of rotations that the quantum stopwatch makes as the particle explores a given path is equal to the action S along that path divided by Planck‟s constant h (quantum of the action). Complex probability amplitude A~e 2011.05.31. j S S 2π f TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 15 Physics for Nanobio-technology: Chapter 6. Quantum Mechanics – I www.itk.ppke.hu 3. Feynman’s Path Integral Totality of experiments suggests that small particles (e. g. electrons, protons, atoms) behave as particle-like and as wave-like objects. R. P. Feynman calls them wave + particle = “wavicle”. Interactions with themselves and with their environment is „wavelike”, however, as particles, they move in a probabilistic way (only probabilities can be known). 2011.05.31. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 16 Physics for Nanobio-technology: Chapter 6. Quantum Mechanics – I www.itk.ppke.hu A „wave-icle” state is represented by a probability complex amplitude r,t . Probability that a wavicle at time t is at position x P(x, t ) x, t 2 dx 1. A single electron moves in one direction: ( , ) ( x, t ). 1. Let us draw every path x(t ), 2. Calculate the action for each path S[ x(t )] A e j S [ x ( t )] j 1 U ( , , x, t ) e Θ path S [ x ( t )] ; Θ : constant of normalization 2011.05.31. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 17 Physics for Nanobio-technology: Chapter 6. Quantum Mechanics – I www.itk.ppke.hu The propagator: S [ x ( t )] 1 j U ( , , x, t ) e Dx (t ), Θ The complex amplitude of the wavicle at ( x, t ) can be calculated from the complex amplitude at ( , ) as ( x, t ) U ( x, t , , ) ( , ) d . If we increase the size of a small particle (nano → micro → macro), then the movement of a wavicle gradually approaches the trajectory of a classical particle (correspondence principle). 2011.05.31. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 18 Physics for Nanobio-technology: Chapter 6. Quantum Mechanics – I www.itk.ppke.hu In the propagator every path, including the classical one, gets the same weight. Modulus of every path is 1. If S / 1, the classical path prevails. In the neighborhood of the classical path the action is close to stationary. In the vicinity of the classical path the paths contribute „coherently‟. In macro case Sclass / ~ 1025 , for an electron Sel / 1. Consider a free particle that leaves the origin at (t 0) and arrives at ( x 1 cm, t 1 s). 2011.05.31. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 19 Physics for Nanobio-technology: Chapter 6. Quantum Mechanics – I www.itk.ppke.hu x cm Example 1: Free particles 1 Consider a free particle that leaves the origin at t = 0 and arrives to x = 1 cm, at t = 1 s. The classical path is x t , Consider another path x t 2 . Phase change S x 0.01t 2 S x 0.01t 1 t s 1 1 1 2 2 1 / m 0.02t d t 0.01 m d t / 0.16 104 m / . 2 0 0 2 For a macroscopic particle For an electron m 1g m 9.11031 kg 0.14 rad 1.6 1026 rad 2011.05.31. 1,1 TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 20 Physics for Nanobio-technology: Chapter 6. Quantum Mechanics – I www.itk.ppke.hu Mechanics S / 1 0 Classical Mechanics S r(t ) 0 The probability of the classical path is 1. The probability of all other paths is 0. 2011.05.31. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 S / ~1 1.05 1034 Ws 2 Quantum Mechanics The “weight” of each path is equal. The phase is determined by S/ . 21 Physics for Nanobio-technology: Chapter 6. Quantum Mechanics – I www.itk.ppke.hu C60 Prof. A. Zeilinger http://www.quantum.univie.ac.at/zeilinger/ 2011.05.31. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 22 Physics for Nanobio-technology: Chapter 6. Quantum Mechanics – I www.itk.ppke.hu 4. Schrödinger Equation In classical mechanics, from the principle of least action we get the local differential equations of motion (Newton). In quantum mechanics, if a single particle of mass m moves in a potential field V(x), then it can be shown from the Feynman path-integral we can get a local partial differential equation of the probability complex amplitude (Schrödinger equation). ( x, t ) U ( x, t , , ) ( , ) d , 2011.05.31. 2 2 j V . 2 t 2m x TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 23 Physics for Nanobio-technology: Chapter 6. Quantum Mechanics – I www.itk.ppke.hu ( x, t ) j H ( x, t ), t 2 2 H V . 2 2m x In three dimensions 2 ( x, t ) 2 ( x, y, z, t ) 2 ( x, y, z, t ) 2 ( x, y, z, t ) 2 2 2 x x y z 2 ( x, y, z, t ) (r, t ). Time-dependent Schrödinger equation for a single particle (r, t ) j H (r, t ), t 2011.05.31. 2 H Vpot (r ). 2m TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 r, t. 24 Physics for Nanobio-technology: Chapter 6. Quantum Mechanics – I www.itk.ppke.hu Time-independent Schrödinger equation for a single particle Let us look for the solution as ( x, t ) Ψ(r) (t ). (t ) jΨ(r ) (t )HΨ(r ) t 1 (t ) 1 j HΨ(r ) constant E (t ) t Ψ(r) HΨ(r) EΨ(r) (t ) E j (t ) t E j t (t ) E j (t ) (t ) e t 2011.05.31. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 25 Physics for Nanobio-technology: Chapter 6. Quantum Mechanics – I www.itk.ppke.hu Eigenvalue problem Eigenvalues E1 , Eigenfunctions Stationary solutions HΨ(r) E Ψ(r) E2 , ... Ei , Ψ 2 (r), ... Ψi (r) 2 (r, t ) ,..., Ψ1 (r), 1 (r, t ) Ψ1 (r ) e j E1 General solution: Discrete (bounded electron ) Continuous (free electron) t , Ψ 2 (r) e Ei , j E2 t ,..., i (r, t ) Ψ i (r) e c Ψ (r) e i j j Ei t , Ei t i i Erwin Schrödinger, Quantisierung als Eigenwertproblem, Annalen der Physik, 361 – 376, 1926 2011.05.31. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 26 Physics for Nanobio-technology: Chapter 6. Quantum Mechanics – I www.itk.ppke.hu Example 2: Single electron in a one-dimensional configuration space V pot (x) d 2 ( x) E ( x) if 2 2m dx ( x) 0 if 2 0 x a, x 0, a x . x0 d 2 ( x) k 2 ( x), 2 dx (0) 0 B, k 2mE x ( x) A sin kx B cos kx, (a) A sin ka 0 ka nπ. nπ 2 2 h2 2 ( x ) A sin x, En kn n , n 2 a 2m 8ma 2011.05.31. a TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 A2 sin 2 nπ 2 xd x 1 A . a a 27 Physics for Nanobio-technology: Chapter 6. Quantum Mechanics – I www.itk.ppke.hu Epot E Inside the box: j n t 2 nπ n ( x, t ) sin xe a a E2 Outside the box: n ( x, t ) 0 h2 E1 8ma 2 Ground-state energy E1 En n E1 2 a 106 m 1mikron, E1 0.376 106 eV, a 109 m 1 nm, E1 0.376 eV, a 1010 m 0.1 nm, E1 37.6 eV, ~ atom a 1015 m 106 nm, E1 376 GeV, ~ nucleus 2011.05.31. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 x0 a x 28 Physics for Nanobio-technology: Chapter 6. Quantum Mechanics – I www.itk.ppke.hu 5. Measurements and Operators The expectation value of the measurement of the position of a wavicle is 2 x x ( x) d x , because the probability that it is at position (x, x + dx) is ( x) d x. 2 We would like to know the expectation values of the measurements of the momentum, angular momentum, energy, etc. of the wavicle. If we knew the complex probability amplitude for the momentum p, we could calculate it as p p ( p) d p. 2 2011.05.31. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 29 Physics for Nanobio-technology: Chapter 6. Quantum Mechanics – I www.itk.ppke.hu Let us assume that we are measuring observable „a‟. Let us also assume that we can get the complex probability amplitude for „a‟ from x as a (a) U a ( x) x d x. The expectation value of the measurement of „a‟ a a a (a) da a a (a) a (a) d a 2 U ( x) xd x U ( x) xd xd a ( x) aU ( x)U ( x' ) d a ( x' ) d x' d x a a a a a Aˆ ( x, x' ) ˆ Aˆ ( x, x' ) x' d x' A 2011.05.31. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 30 Physics for Nanobio-technology: Chapter 6. Quantum Mechanics – I www.itk.ppke.hu ˆ d x, a A ˆ Aˆ ( x, x' ) x' d x'. A If we know the operators  we can calculate the expectation value of „a‟ from the prob. amplitude of the position. Without proofs we present the results: Observable Operator ˆ x ( x) X ˆ x x x dx X x x dx, 2 p 2011.05.31. Pˆ x ( x) Pˆ x j j TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 x dx j d dx. j dx p 31 Physics for Nanobio-technology: Chapter 6. Quantum Mechanics – I www.itk.ppke.hu Observables and Operators x ˆ x ( x) X ˆ x X px ˆP ( x) Pˆ x x j j Vpot ( x) ˆ V ( x) ( x) V ˆ V ( x) V pot pot pot 2 p E Vpot ( x) 2m 2 ( x) H Vpot ( x) ( x) 2 2m x 2 2 ˆ H Vpot ( x) 2 2m x 2 2011.05.31. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 32 Physics for Nanobio-technology: Chapter 6. Quantum Mechanics – I www.itk.ppke.hu 6. Dirac Formalism The state of a mechanical system is represented by the complex probability (wave) function (q1 ,..., q f , t ) Dirac introduced a shorthand notation for the state, and he called it “ket” (q1 ,..., q f , t ) , x The complex conjugate of the state function is called “bra”. (q1 ,..., q f , t ) , x . The integral form of the expectation value is called “bra-ket” A dx A a . The eigen-functions are frequently represented by their index un un , n 2011.05.31. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 33 Physics for Nanobio-technology: Chapter 6. Quantum Mechanics – I www.itk.ppke.hu The time-dependent Schrödinger equation j t Time-independent Schrödinger equation H E H E h The total energy of a stationary state Frequency, period in time e j t e Momentum p k , p k E j t h Wavenumber, wavelength, period in space 2011.05.31. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 k 2π 34 Physics for Nanobio-technology: Chapter 6. Quantum Mechanics – I www.itk.ppke.hu The expectation value change in time as d L L L d V L L L dt t t V d L j j j H H L L H t dt d L j j j HL LH HL LH dt The operators can change in time Commutator of operators H and L 2011.05.31. dL j H, L dt H, L HL LH TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 35 Physics for Nanobio-technology: Chapter 6. Quantum Mechanics – I www.itk.ppke.hu Example 2: Electron in a one-dimensional box For an electron in a one-dimensional ideal “box” of size a calculate a) the expectation values of the electron‟s position and momentum; p x,t j x,t dx x x x,t x x,t d x b) the expectation values of the squares of position and momentum; x 2 x,t x 2 x,t d x p 2 2 2 x,t d x x,t 2 x c) check the validity of the Heisenberg‟s uncertainty principle. x p 2011.05.31. x2 x 2 p2 p 2 TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 36 Physics for Nanobio-technology: Chapter 6. Quantum Mechanics – I www.itk.ppke.hu a x a 2 2 2 πx x sin d x x a0 a a0 a x 2 0 a 3 3 3 π x 2 a 1 2π x 2 a a 2 2 2 x sin d x x cos dx 2 0, 2833 a 2 a a 6 20 a a 6 4π a 2 πx p sin j a0 a x x2 x 2 π πx cos dx 0 a a p2 2 2 x,t d x x,t 2 x 0,2833 0,25 a 0,18 a x p 2011.05.31. 2π x a dx a 2 2 1 cos x2 x 2 p2 p TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 p 2 p2 p 0,57 2 π a 2 37