r nucleus orbital electron wave λ /7/008 Quantum ( F.Robilliard) 1
Wat is a Quantum? A quantum is a discrete amount of some quantity. For example, an atom is a mass quantum of a cemical element te mass of a piece of pure Aluminium must be an integer multiple of te mass of te Aluminium atom. No piece of Aluminium can ave a mass less tan tat of an Aluminium atom. Te carge on an electron is a quantum of carge all electrons ave precisely te same carge. As te plate of a capacitor is carged, te total carge on te plate is increased in jumps, equal to te carge on te electron. It turns out, tat just about everyting (maybe even time) is quantised, but because te quanta are so small, we don t usually notice tem, on te macroscopic scale. Because te canges are so small, tings look continuous as if tey could be indefinitely divided up into smaller, and smaller fractions. On te atomic scale, and smaller, owever, te quantum nature of tings cannot be ignored. At its most fundamental level, our pysical world is quantised. Furtermore, quantisation forces us to reexamine some deep pilosopical issues. /7/008 Quantum ( F.Robilliard)
State Transitions: Pysics up to te 0t Century was deterministic. A pysical cause always produced a precisely predictable pysical effect. For example, if a known force acted on an electron in an atom, it would cause te electron to move continuously from one state to anoter. It s position at any instant would be precisely predictable. We now know tat an electron in an atom, can only exist at certain discrete energy levels, about te nucleus. It does not occupy levels between tese allowed levels. It s energy is quantised. Te electron cannot move continuously from one energy state to anoter, but must transition by a discontinuous quantum jump. And eac suc jump is accompanied by te emission (or absorption) of a quantum of energy, equal to te difference in energy of te electron, between te two levels. Experiment sows, tat we cannot, in principle, predict exactly wen an electron will transition from one allowed state, in an atom, to anoter. Nor can we specify a pat it takes between states. All tat is possible is to assign a probability to te transition. It s as if te electron dematerialises from te first state, and materialises in te second. Only te probability of te event taking place can be predicted by teory. Suc transitions can only be described by statistics. /7/008 Quantum ( F.Robilliard) 3
Quanta & Causality:. We can regard te Coulomb force acting on an electron in an upper atomic level, as causing its transition to a lower level. It is true, tat te force and te transition are in cause-effect order in time cause (force) precedes effect (transition). However, te force acts more like a precondition for te transition, rater tan te cause of it, in te classical sense. We cannot predict wen te transition will occur. Tere is no transition pat in spacetime from te upper state to te lower. We only know te probability, per unit time, tat te transition will occur. Tis means tat we need to reevaluate our classical notions of causality. A cause precedes an effect, but te link is probabilistic. Because of its quantised nature, canges in te pysical world are governed ultimately by cance. Tis is quite a profound and controversial pilosopical sift in our view of te pysical world, and was only accepted by mainstream Pysics, because of strong experimental evidence. /7/008 Quantum ( F.Robilliard) 4
Nature of Ligt: Te first particular quantum issue we look at ere, is te pysical nature of ligt. Since Newton s time, tere ave been two models for ligt. Newton tougt ligt consisted of a stream of particles, traveling troug space. Oters tougt of ligt as a wave composed of interacting electric and magnetic fields). In some experiments (eg. optical interference, diffraction), ligt beaves like an electromagnetic wave, and can be caracterised by a frequency (or wavelengt), and an amplitude. In oter experiments ( potoelectric effect, atomic spectra), ligt beaves like a stream of particles, called potons. Potons are tiny discrete lumps of electromagnetic energy. Te particle stream can be caracterised by te energy of particular potons, and te number of tose potons passing per square metre, per second. For example blue ligt would be an electromagnetic wave of wavelengt ~470 nm, on te wave model, but be composed of streams of potons, eac poton containing an energy about 4.x10-19 J, on te particle model. Te question arises: is ligt an electromagnetic wave, or a stream of particles? wic is te correct model? /7/008 Quantum ( F.Robilliard) 5
Wave-Particle Duality: Te only way to explain experimental results is to assume tat ligt, togeter wit oter forms of electromagnetic radiation, as a dual nature it sows bot wave and particle caracteristics. Ligt sows wave-particle duality! At longer wavelengts (suc as radio), te wave caracter tends to be more prominent; at sorter wavelengts (suc as x-rays), it is te particle caracter tat tends to dominate. It turns out, tat we eiter need to work in te wave model, or in te particle model we can t work in bot models simultaneously. Eiter model, by itself, is incomplete, but te two are mutually exclusive. Tis is called te principle of complementarity. However, we do need a link between te wave and particle models. We need a relationsip between wave frequency, and poton energy. Tis relationsip was first discovered by Max Planck, in about 1900, in is breaktroug teory of black-body radiation. About 5 years later, confirmation was given to Planck s idea, by Einstein, in is teory of te potoelectric effect. We will look at Planck s idea first, ten at its application to te potoelectric effect. /7/008 Quantum ( F.Robilliard) 6
Energy and Frequency : Planck s Equation Planck treated ligt as a stream of potons (or quanta). He connected te particle and wave models of ligt, by proposing tat te energy, E, of an individual poton on te particle model, is proportional to te frequency, f, of te ligt, on te wave model. E f...(1) Planck s Equation. were and E energy of a single poton (energy quantum) f associated frequency of tat poton constant of proportionality called Planck s constant 6.66x10-34 J s If a poton as a frequency, f, it must also ave a wavelengt, λ, were tese must be related, troug te velocity of te wave, v, by - v c f...( ) (We will follow te usual convention for electromagnetic waves, and use c for wave velocity, rater tan v.) Tus: /7/008 Quantum ( F.Robilliard) 7
Energy, Frequency & Wavelengt : ()(1): E f c...(3) were λ Poton wavelengt c speed of ligt.998x10 8 m/s Te energy of individual potons is proportional to te frequency of te ligt. At a given frequency, te intensity of te ligt striking a surface, depends on ow many potons (ow muc energy) per second strike tat surface. Wen f is small (radio), te energy of an individual poton (E) is small, and as less impact on experiments; wen f is larger (x-rays), te larger poton energy as more impact. Plank s equation, (1), is te starting point of Quantum Pysics. It allows for te development of pysical models of te poto-electric effect, te emission spectra of atoms, and many oter penomena, wic cannot be modeled solely on a wave model. /7/008 Quantum ( F.Robilliard) 8
Some Calculations: Te following table sows typical values for wavelengt, λ, frequency, f, and poton energy, E, for some regions of te electromagnetic spectrum, using equation (3). radio microwave green ligt x - ray 50 m 1cm 500 nm 1nm f (Hz) 6 10 3 10 6 10 3 10 6 10 14 17 E (J) f 4 10 10 4 10 10 7 3 19 16 As can be seen by tese figures, te energy of a single poton is small, on te macroscopic scale. Quantum effects are not obvious until we get down to te scale of an atom. We tink of ligt as being a stream of potons, eac of wic is a quantum of energy, vibrating wit an associated wave frequency. E, f One of te first experiments in wic Planck s ideas were to prove critically important was te Potoelectric effect. We will next look at tis effect, and its quantum explanation. /7/008 Quantum ( F.Robilliard) 9
Potoelectric Effect : Metals contain electrons, tat are not strongly eld by individual metal atoms, and are tus free to move about in te metal. It is because of tese free electrons ( conduction electrons ), tat metals conduct electric currents. Wen ligt falls on a metal surface, te energy of te ligt is transferred to te electrons of te metal. Tis can cause tem to be ejected from te metal surface. Tis ejection of electrons from a metal surface by ligt is called te potoelectric effect. To study tis effect, te metal surface must be in a ligt vacuum, oterwise gas molecules would affect te PE tube e process. E C E & C are metal surfaces in a glass vacuum tube A (te Poto Electric ( PE) tube). i A variable power supply (PS) is connected in series V v wit te PE tube. E is negative. C is positive. Ligt strikes E, ejecting electrons e. Tese + electrons pass to C, tereby constituting a current. PS Te voltage across te PE tube, and te current troug it, are measured by voltmeter, V, and ammeter, A. /7/008 Quantum ( F.Robilliard) 10
Te Experiment : PE tube -V 0 i e E C V v + ligt PS A ig intensity low intensity fixed ligt frequency v i Te PE tube is illuminated by monocromatic (single-frequency) ligt. Current i is measured, as v is varied. Te resulting v-i caracteristic is sown below. Wen v is set to zero, a current flows. Tis is due to te kinetic energy of electrons carrying tem from E to C, and creating tis current. As v is increased more electrons are attracted from E to C, and te current increases, until all electrons are attracted. (Higer ligt intensity more potons more electrons greater final current.) To stop te current, a reverse voltage of ( V 0 ) (called te stopping potential) must be applied across te PE tube. If te intensity of te illuminating ligt is increased, te currents increase, but te stopping potential is unaffected. For a particular metal, te stopping potential is determined by te frequency, f, of te ligt. /7/008 Quantum ( F.Robilliard) 11
Summary of Experiment: 1. Below a certain ligt frequency, called te cutoff frequency, f c, no poto emission appens, no matter ow ig te ligt intensity. Above tis frequency, potoemission takes place no matter ow weak te ligt. Te cutoff frequency is caracteristic of te particular metal.. For a given metal, and for ligt wit frequency above te cutoff frequency, te stopping potential, v 0, required, increases in proportion to te ligt frequency, f. (It is independent of ligt intensity). i f 3 f f 1 given metal, fixed ligt intensity, different ligt frequencies Stopping potential is -V 01 for f 1 -V 0 for f -V 03 for f 3 -V 03 -V 0 -V 01 v Plotting stopping potential, v 0, against ligt frequency, for a given metal, produces a striking linear grap: /7/008 Quantum ( F.Robilliard) 1 V 03 V 0 V 01 V 0 f c f 1 f f 3 f
Einstein s Explanation: Ligt is composed of a stream of discrete potons. Tese potons interact wit te conduction electrons of te metal. One poton interacts wit one electron, in te metal surface. Te poton disappears, and its energy becomes te kinetic energy of te electron. f e Poton gives its energy, f, to an electron, e. Tere is a minimum amount of work to extract an electron from a metal surface. Tis is called te work function, φ, of te surface, and is caracteristic of a particular metal. If te kinetic energy of an electron exceeds te surface work function, ten te electron can escape te metal surface. e e KE φ Electron loses energy, φ, getting out of metal surface, and as (f φ) energy left as KE /7/008 Quantum ( F.Robilliard) 13
Stopping Potential: Since φ is te minimum energy tat can be lost by an electron, on exiting te surface, te KE, for suc an electron, must be te maximum possible KE max. From te conservation of energy - Max KE wit wic Energy given to Min energy lost an electron can leave - electron by poton in escaping te surface te surface KE f -φ max...( 4) Te electron emerges from te metal surface, E, of te PE tube, wit kinetic energy no greater tan KE max. PE tube KE max e E C + - For an electron to be just stopped from getting across gap EC, by te stopping potential, V 0, its KE must just equal te work it must do against te electric field, to cross te gap. KE max ev 0...(5) were: e te carge on an electron. V 0 /7/008 Quantum ( F.Robilliard) 14
KE max f -φ...( 4) KE max ev 0...(5) Te PE Equation: (5) (4): e V 0 f -φ φ Gives V0 f -...( 6) e e Te Potoelectric (PE) Equation V 0 f c f Equation (6) sows tat V 0 is proportional to f, and explains te corresponding experimental grap. Gradient At V 0, e wic allows for te experimental determination of Planck's constant c wic allows for te determination ( 6) gives fc of work function by measurements of f c c 0 φ were: and: λ c is te cutoff wavelengt corresponding to f c c speed of ligt. /7/008 Quantum ( F.Robilliard) 15
KE max Gives e V 0 V 0 f -φ e f - φ e...( 6) Different Metals: For different metals, te stopping potential curves will be as follows: V 0 1 3 f c1 f c f c3 f All curves ave te same gradient, but different intercepts on te f-axis, corresponding to te different tresold frequencies for te different metals.. For metals 1,, & 3. Note: Te usual unit used for work functions is te non-si unit ev (rater tan J). 1 ev 1 electron Volt te work to move an electron troug a potential difference of 1 V 1.6 x 10-19 J /7/008 Quantum ( F.Robilliard) 16
Example: Sodium as a work function of.46 ev. Wat is its cutoff wavelengt? Note: 1 ev 1 electron Volt te work to move an electron troug a potential difference of 1 V 1.6 x 10-19 J c ( 34 )( 8 6.63 10 3.0 10 ).46 ( 19 1.6 10 ) 505 c φ 5.05 10 nm -7 /7/008 Quantum ( F.Robilliard) 17 m Cutoff wavelengt is in te green part of te visible spectrum. As a comparison, Cu, Ag, Fe, ave work functions of 4.70, 4.73, 4.50 ev, respectively.
Te Bor Model of te Hydrogen Atom: (an example in quantisation) Wen atoms of ydrogen are excited, tey emit ligt. For example, wen an electric current is passed troug ydrogen gas, at low pressure, te gas glows. Under appropriate conditions, only certain discrete wavelengts are present in te emitted ligt. We call suc a distribution of wavelengts, a line emission spectrum. Hydrogen atoms always emit te same caracteristic line spectrum. In 1913, Niels Bor developed a model of te ydrogen atom to explain tis spectrum. Tis model became te starting point for our understanding of te structure of te atom. Bor started from Ruterford s model of te atom, in wic low-mass negative electrons circulate about a eavy positive atomic nucleus, in circular orbital pats, because of te attractive Coulomb force between (+) & (-) carges. He ten quantised te angular momentum of te electrons. Angular momentum is te angular analogue of linear momentum, for particles tat rotate about some central point. Before we can proceed, we need to define angular momentum. /7/008 Quantum ( F.Robilliard) 18
Angular Momentum, L : Consider a particle, of mass, m, moving around a circle, of radius, r, wit tangential velocity, v. r m v Linear momentum, p, of te particle is defined by: p m v Its angular momentum, L, is defined by: L r p r mv (cross product) L is a vector directed along te symmetry axis of te circle. Magnitude of L: L r p r mv...(a) SI units of L: kg m s -1 /7/008 Quantum ( F.Robilliard) 19
Bor s Assumptions : 1. electrons move in circular orbits about te atomic nucleus under te action of te attractive Coulomb force between te negative electron and te positive nucleus.. Only certain orbits are stable; oters are forbidden. Te angular momentum of te electron in stable orbits is an integer multiple of ( / π.) L r mv n ( n 1,, 3, 4,... )...(b) Quantisation of angular momentum. were: r radius of te allowed orbit m mass of electron v tangential velocity of electron in its orbit 3. Wen an electron jumps from a iger orbit to a lower, a poton is emitted, wose energy, δe, is equal to te energy difference between te two orbits. E f E i - E f...(c) were: E i energy of electron in initial orbit E f energy of electron in final orbit /7/008 Quantum ( F.Robilliard) 0
Essential Pysics of te Atom: Consider te ydrogen atom, consisting of a single negative electron, in orbit about te positive atomic nucleus ( a single proton). Coulomb s Law: For electron in circular pat ΣF Coulomb force e k r m a ( mass) m v r centripetal. acceleration...( d) r v m (-e) (+e) k v (+e) r m (-e) radius of circular orbit velocity of electron around its orbit mass of electron carge of electron carge on proton Coulomb const. Quantisation of angular momentum of electron: L r mv n ( n 1,, 3, 4,...)...(b) /7/008 Quantum ( F.Robilliard) 1
from from ' ' ( b) ( d) r r n Allowed Orbits: Quantising te angular momentum of te electron, constrains it to exist in only certain stable orbits, wit certain discrete radii. ' ( d) : k m v...( d) ' ( b) : v n...( b) n e r 1 rm e : k m n r 4 mke 1 rm ( n 1,,3... )...( e) e k r L m r mv n ( n 1,, 3, 4,... )...(b) (+e) v r r...( d) m (-e) v Te electron can exist only in orbits wit radii, r n, wic satisfy (e). It cannot exist between tese orbits /7/008 Quantum ( F.Robilliard)
Te Bor radius, a 0 : r n 4 mke n a 0 4 mke 5.9 10 0.059 nm ( n 1,,3... )...( e) Te electron s smallest radius as n1, and is called te Bor radius, a 0. Te Bor radius is a measure of te size of te ydrogen atom. -11... m r n a 0, 4a 0, 9a 0,16a 0 ( f ) Using (e), allowed orbits for te ydrogen atom ave radii:...( g) n3 n n1 + a 0 4a0 9a 0 n is called te principal quantum number of te electron. /7/008 Quantum ( F.Robilliard) 3
Total KE 1 Energy of te Electron: Next we will work out te energy of te electron in its various allowed orbits. From mv energy of + ( d) PE k e r e : k r 1 e k r ( i) ( ) E - 1 k electron...( ) v m r 1 mv : e r E...( i)...( j) Coulomb PE (negative) (+e) /7/008 Quantum ( F.Robilliard) 4 r (-e) m r radius of circular orbit v velocity of electron around its orbit m mass of electron (-e) carge of electron (+e) carge on proton k Coulomb const. Total energy, E, of an electron in an orbit of radius r. v
From Quantised Energy of te Electron: But te electron can only exist in certain orbits. ( e) and ( f ) : a (n 1,,3...)... ( ) rn n 0 a 0 Bor radius. E - 1 k e r...( j) ( ) ( j) : ke 1 En -... a n 0 ( n 1,,3... )...( k) Allowed electron energies for ydrogen atom Te energy of te electron is negative, since we would ave to do work on te electron to extract it to infinity, were its energy is defined to be zero. Using numerical values, and converting to electron-volts: 13.6 - n En ( ev) ( n 1,,3... )...( m) Te energy of te lowest orbit is 13.6 ev. Tis is called te ydrogen ground state. It is te maximum work tat needs to be done to strip an electron from te atom ionise te atom. /7/008 Quantum ( F.Robilliard) 5
Hydrogen Spectra: n - ( n 1,,3... )...( k) E Ei - Ef f...( c) ke a 1 n E 0 Electrons usually occupy te ground state of te atom. However, if energy is transferred to te atom ( te atom is excited ), an electron can rise to a iger energy state. It will subsequently fall back to a lower level. Wen tis appens, te energy difference between te initial and final states, is emitted as a poton. Te frequency, f, of tis poton is given by Bor s tird postulate: E i energy of initial state ( n i ) Ef energy of initial state ( n f ) Tis allows us to compute te frequencies, and wavelengts, of te potons in te spectrum of atomic ydrogen. c ke ke E f E i - E f - - - a 0ni a0nf 1 ke 1 1 -... n a 0c nf ni ( ) using (k). λ wavelengt of emitted poton n i, n f quantum numbers of initial and final levels of te electron. /7/008 Quantum ( F.Robilliard) 6
Experimental Hydrogen Spectra: Te ydrogen spectrum ad been measured experimentally, prior to te Bor teory. Te wavelengts found ad been fitted to te following empirical relationsip: H 1 R H 1 n f...( o) ( 7 ) -1 1.097373 10 m R te Rydberg constant - n 1 i -... n a0c nf ni were R H is an experimental constant, called te Rydberg constant, and n i and n f were integer constants, wose significance was not clear. Compare tis expression wit te teoretical Bor expression for λ.. 1 ke 1 1 ( ) Wen compared, te teoretical constant in Bor s expression, (n), agreed wit te experimental Rydberg constant (o), wic confirmed te Bor teory, and gave significance to te integer constants in te experimental expression. ke a0c R witin 1% H /7/008 Quantum ( F.Robilliard) 7
0 Details of Hydrogen Spectra: -0.85-1.51-3.40 Te ydrogen energy levels can be represented grapically. Balmer (visible) Pascen (IR) 4 3 -... n a 0c nf ni 1 ke 1 1 A transition of an electron from an upper level to a lower, is represented by an arrow, and corresponds to an emission line in te spectrum. ( ) -13.6 E (ev) Lyman (UV) n 1 Tere is a series of lines, tat terminate on te n1 level, called te Lyman series, wic are in te ultra-violet (UV) part of te spectrum. Te visible Balmer series terminates on te n, and te infra-red (IR) Pascen series terminates on n3. /7/008 Quantum ( F.Robilliard) 8
Example: Calculate te wavelengts of te two longest wavelengts of te Balmer series, in te spectrum of atomic ydrogen. Te Balmer series ends on n f. Tus, wavelengts of te Balmer series are given by - - - a 0c ni 1 ke 1 1 n i can ave values 3, 4, 5, 6,... for te Balmer spectrum. Hence (using accepted values for te constants k, e, and a 0 ) te two longest wavelengts are - 1 1 1 1 3 ( 7 1.0973 10 ) - 653.3 nm called te H-α line (red) 1 1 1 4 ( 7 1.0973 10 ) - 486.1nm called te H-β line (green) Tese are te two most prominent visible lines in te spectrum of atomic ydrogen. /7/008 Quantum ( F.Robilliard) 9
Summary So Far : Let us summarise te key aspects of electromagnetic (EM) radiation, so far. EM radiation (suc as ligt) consists of streams of particles called potons. Potons are made of energy. Potons exibit bot particle, and wave caracteristics tey sow wave-particle duality. Te energy of an individual poton is proportional to its wave frequency. E f c...(3) Tis idea can be extended in a very fundamental way, but before we can do tis, we need to review some ideas tat come from te Teory of Relativity. /7/008 Quantum ( F.Robilliard) 30
Mass-Energy Equivalence: One of te profound revelations of te teory of Relativity is tat energy and mass are two different forms of te same, very fundamental, pysical entity. Te energy equivalent, E, of a mass, m, is given by - E mc...(7) Were: c 3.0x10 8 m/s is te speed of ligt. A particle of mass m, can be created from, or converted to, an amount (mc ) of energy. Example: an electron (mass 9.11x10-31 kg) can be made from 8.0x10-14 J of energy. (For symmetry reasons, electrons are always created in pairs, so twice tis energy is actually required. ) Because of its simplicity, and its profound implications, equation (7) is probably te most famous equation in Pysics. Next, let us now follow up an important logical consequence of te equivalence of mass and energy. /7/008 Quantum ( F.Robilliard) 31
Rest Mass & Relativistic Mass: Consider a particle tat is at rest wit respect to an observer. Te measured mass of suc a particle is called its rest mass, m 0. Tis rest mass, m 0, as an equivalent rest mass energy, E 0, given (using equation (7)) by - E 0 m c 0...(8) Let s say, tat te particle accelerates, so tat it now as motion relative to an observer. Because of its relative motion, te particle as gained kinetic energy. But kinetic energy is equivalent to mass. Tus te total mass of te particle must ave increased, relative to te observer. Te new, greater mass of te moving particle is called its relativistic mass, m. As te velocity of te particle increases, its measured ( relativistic ) mass, m, also increases, by an amount tat is proportional to te increase in its kinetic energy. (total particle mass) (mass equivalent of its kinetic energy) + (rest mass) Or equivalently - ( total particle energy) ( kinetic energy) + ( rest mass energy) E KE + m c...( 9) /7/008 Quantum ( F.Robilliard) 3 0 m 0
Momentum: As te particle accelerates, relative to an observer, it must also gain momentum. At any stage, te momentum of te particle, p, is defined by - p mv...(10) were, m is its relativistic mass and v is te velocity of te particle, relative to te observer. Since kinetic energy and momentum increase simultaneously, it sould be possible to express one in terms of te oter. Wen tis is done, and te results incorporated into equation (9), togeter wit equations (7) and (8), we get te following important result (te detailed derivation of wic, will not be given ere)- were E ( ) m c...( 11) p c + m 0 E total energy of a moving particle p (relativistic) momentum of te particle m 0 particle rest mass c speed of ligt 3.0x10 8 m/s /7/008 Quantum ( F.Robilliard) 33 p If we know te total energy, E, and te rest mass, m 0, of a particle, ten its relativistic momentum, p, can be found from (11)
Poton Momentum: E ( ) m c...( 11) p c + m We ave spoken so far about a mass particle. But a poton is a particle, wic, because it is made of electromagnetic energy, as a zero rest mass. Terefore, equation (11) can be used to find te momentum of a poton. Putting m 0 0, in (11), gives: 0 E pc...(1) p Hence: E f p c c...( 13) using c f λ using E f Te momentum of a poton is proportional to its energy. Consequently, in terms of wave properties, poton momentum is proportional to its frequency, or inversely proportional to its wavelengt. (Tere ave been some serious proposals to use te momentum of potons from te sun, falling on reflective sails, for interplanetary travel.) /7/008 Quantum ( F.Robilliard) 34
Angular Momentum: We ave been treating te poton as a particle. In fact, it is an elementary particle one tat cannot be decomposed into a system of smaller, simpler, particles. Elementary particles ave caracteristic quantum properties. For te poton, we ave looked at its energy, rest mass, and momentum, all of wic are quantised. Tere is anoter important quantum property for te poton angular momentum. Angular momentum, wic we defined, for a mass particle, earlier in tis section, is te rotational analogue of linear momentum. For elementary particles, angular momentum is an intrinsic quantum property, and is not associated wit any rotation of te particle. It turns out, tat te angular momentum of any elementary particle, is given by - angular momentum s π were s 0,,1, ( s + 1)...( 14),... were s is called te spin of te particle, and Planck s constant. Eac type of elementary particle as a caracteristic value of s. For potons, s 1. For electrons, s 1/ /7/008 Quantum ( F.Robilliard) 35 1 3,
Properties of te Poton: Here is a list of te main properties of te poton, as a particle. is a quantum of electromagnetic energy, wit frequency, f, and wavelengt, λ. is an elementary particle travels at te speed of ligt, c. as zero carge as energy E f as zero rest mass as momentum p E/c as angular momentum (spin 1) /7/008 Quantum ( F.Robilliard) 36
Compare te Properties of te Electron: As a comparison wit te poton, ere is a list of te main particle properties of te electron. is a carged elementary particle travels at any speed, v, below te speed of ligt, c. as negative carge e -1.6x10-19 C as rest mass m 0 9.109x10-31 kg as any momentum, p mv, between 0 and infinity as energy, E, given by E (pc) + (m 0 C ) as angular momentum (spin 1/) /7/008 Quantum ( F.Robilliard) 37
Particles & Waves: We ave found it necessary to treat ligt as a stream of potons, wic are particles of energy, aving wave caracteristics, suc as frequency and wavelengt. But E m c tells us tat energy and mass are equivalent. Ten souldn t any mass particle, suc as an electron, or a proton, also ave wave caracteristics, suc as frequency and wavelengt. We are using te symmetry between mass and energy, to suggest tat bot sould exist as particles wit wave properties. Tis is te principle of wave-particle duality. Tis principle as been confirmed by numerous experiments. In fact, it is one of te most fundamental principles of quantum pysics. One immediate consequence of tis idea, is tat particles are not localised, at a point in space, at a given instant, but are distributed over a region of space like a wave. We can only describe a particle as aving a certain probability of being present in a given region of space, during a given time interval. If a particle canges position, we cannot predict causally, wen suc a cange will take place. We can only assign a probability to suc a cange. Tis reality is also confirmed by experiments, too numerous to detail. /7/008 Quantum ( F.Robilliard) 38
Wavelengt of a Mass Particle: In 193 Louis de Broglie first suggested te principle of wave-particle duality. He postulated tat, any mass particle, suc as an electron or a proton, as an effective wavelengt, λ. For a poton, we found: From 1 E pc /7/008 Quantum ( F.Robilliard) 39 and E f c c...... ( 1) ( 1) ( )& ( 1) pc...( 15) wic expresses poton wavelengt, λ, in terms of its momentum, p. By wave-particle duality, tis sould also work for a mass particle, suc as an electron. p mv...( 16) Te wavelengt of a particle, λ, is inversely proportional to its momentum, p. p m particle s relativistic mass v particle s velocity Note, tat if te velocity of a mass particle is less tan about 0.5 c, ten te relativistic mass, m, can usually be taken as equal to te rest mass, m 0.
Examples: Te wavelengt of a mass particle is called its de Broglie wavelengt. de Broglie Wavelengt p...( 16) mv Find te wavelengt of a rock of mass 1 kg, tat travels at a velocity of 1 m/s. mv 6.63 10 1.0 1.0 /7/008 Quantum ( F.Robilliard) 40-34 6.6 10 Tis wavelengt is so small, tat te rock sows no measurable wave caracteristics Find te wavelengt of an electron (mass m 9.11 x 10-31 kg), tat travels at a velocity of v 1.0x10 5 m/s. mv 6.63 10-34 ( -31) ( 6 9.11 10 1.0 10 ) -34 7.8 10 Tis wavelengt is comparable to te wavelengt of x-rays. m -10 m 0.78 nm Only wen te mass m is small (as for sub-atomic particles), are wave properties significant.
De Broglie Wavelengt & te Atom: We ave noted earlier (te Bor teory of te atom), tat te orbital electrons of an atom exist only in certain discrete levels, about te nucleus. Tey do not exist between tese levels. Below is a diagram of suc a level (an orbital). Regarding electrons as aving a wavelengt gives a sudden insigt into ow tis works. Te electron wave must fit precisely around te orbital. Tat is, te circumference of te orbital must equal an integer number of electron wavelengts. If tis condition is not satisfied, te r orbital is forbidden. nucleus Te circumference of te illustrated orbital is five times te electron wavelengt. orbital electron wave λ Te electron wave is actually a standing wave around te circumference of te orbital. Te electron will create an interference pattern wit itself around an allowed orbital a standing wave. Between allowed orbitals, te electron wave would interfere wit itself destructively, and produce a null interference pattern a standing wave of zero amplitude! Tat is no electron! /7/008 Quantum ( F.Robilliard) 41
de Broglie & Bor: r nucleus Condition for allowed orbitals (de Broglie): If r radius of an orbital and λ electron wavelengt ten, only orbitals wic satisfy orbital electron wave ( 16) (*) r n m v r n mv λ But te de Broglie wavelengt of ( n 1,, 3,...) π r n λ (n 1,,3,4,5...)...( ) are allowed. te electron is...( 16) Tis is precisely te quantisation p mv assumption made arbitrarily by : Bor, in is model of te ydrogen atom. n is clearly identified wit te principle quantum number of te electron. /7/008 Quantum ( F.Robilliard) 4
orbital electron wave r nucleus λ 1 3 4 N Te Quantised Atom: Tus we see tat te de Broglie wavelengt of te electron justifies Bor s quantisation of its angular momentum in te ydrogen atom. It makes some sense, of wy te electron cannot exist between allowed orbitals. Because of its wavelike caracter, te electron is not localised in space, but can only be described as aving a particular probability of existing in a particular volume witin a given orbital region. Eac orbital is caracterised by a principle quantum number, n 1,, 3,... Because of te wavelike nature of te electron, transitions between orbitals are non-causal. Tey are governed by te laws of probability. /7/008 Quantum ( F.Robilliard) 43
Uncertainty in Sampling a Wave: Finally, in tis introduction to Quantum Pysics, we look at te profound, and fundamental, consequence tat follows from te wavelike nature of mass particles. We firstly need to consider an important, and fundamental, question in wave teory. How can we measure te frequency of a sinusoidal wave? We need to sample te wave over an interval of time. To sample te wave at one instant only, gives us a single value of te wave variable, y, at tat instant, but gives us no knowledge at all, of ow te wave is varying, from time to time, and consequently, no knowledge of te wave s frequency. To determine te wave frequency, we need to sample te wave variable, y, over a finite interval of time, t. y Te wave may ave started well before tis time interval, and it may ave continued long after it, but we only ave knowledge of ow te wave canged during te interval of widt t. t t t To be consistent wit our laboratory program, we will use alf te widt, t, of te time window, as a measure of an uncertainty in time, t. /7/008 Quantum ( F.Robilliard) 44
Uncertainty in te Frequency: y a t t f 0 f f Because we ave an incomplete knowledge of ow te wave varies over all time, tere is an intrinsic, and inescapable, uncertainty in our determination of its frequency te frequency could ave any value, over a range of values. Te only way tat we could be sure of te precise frequency, is to sample te original wave for an infinite time, wic is impossible. We cannot assume tat te vibrations of te wave, outside our sampling window, are consistent wit wat appened during te sampling interval. Te wave could vibrate faster, or slower, outside te interval, tereby affecting te frequency. In fact, te uncertainty in te frequency depends on ow long we sample te wave te longer we sample, te smaller will be te uncertainty in te frequency. Say tat we measure te wave over a time interval of ± t, and tat te frequency we determine is witin te range ( f ± f ). 0 /7/008 Quantum ( F.Robilliard) 45
y t & f: a of t 1 f t k...( 17) t were k is a constant proportionality. t It turns out (toug we will not prove tis ere), tat te uncertainty in te determined frequency, f, of a wave, is inversely proportional to te time interval, t, during wic we sample te wave. But tis is a teoretical minimum uncertainty Tere may be oter measurement factors tat contribute to a greater uncertainty in f. Tus we sould write equation (17) as or f f t f 0 k t k... f f ( 18) Our measurement of frequency can not be any more accurate tan given by (18). /7/008 Quantum ( F.Robilliard) 46
Enter te Poton: Let us now apply tese ideas to a poton. Te energy of te poton is proportional to its frequency, according to te Planck equation. E E f f terefore f E... E t ( 19) ( 19) ( 18) : E t k...( 0) Te uncertainty in poton frequency translates into an uncertainty in poton energy. By wave-particle duality, tis relationsip must also apply to mass particles, since tey ave an associated de Broglie wave. Tus, for bot waves and particles - k...( 0) Te product of te uncertainty in te energy of a particle, wit te uncertainty in te time at wic tis energy was measured, is never less tan k(planck s constant). Te longer we sample te energy of a particle, te more accurately we can determine tat energy f t k...( 18) /7/008 Quantum ( F.Robilliard) 47
E t k...( 19) Te Value of k: Te value of te constant, k, depends on ow we define te uncertainties, E and t. Assume tat te possible values of energy are distributed about a mean value, E 0 N Two possible measures of te uncertainty in E are δe -- Standard deviation, δe -- Maximum error, E. Detailed teory sows tat if standard E 0 E deviation values are used, for E and t, in equation (19): E E energy value (scaled in standard deviations from te mean) N number of times a particular energy value occurs (scale is normalised) Taking E E andt t... ( ) ( 1) : k...( ) ( ) /7/008 Quantum ( F.Robilliard) 48 k In consistency wit te laboratory work, we will use maximum error values, rater tan standard deviations, for uncertainties. 1 1 4...( 1)
Heisenberg Uncertainty Principle: () (19): E t...( 3) were E, and t are te uncertainties in energy, E, and time, t. (maximum error values) Te longer we can observe a particle, te more precisely can we determine its energy; te sorter te time of observation, te more uncertain are we of te energy Te more precisely we know te energy of a particle, te less precisely do we know te time of observation of tat energy. Tis relationsip was first derived by Werner Heisenberg in 197, and is called te Heisenberg Uncertainty Principle. /7/008 Quantum ( F.Robilliard) 49
Alternate Version: Tere is an alternate version of te uncertainty principle. Consider a poton moving along te x-axis. t t x ± x ± E ± E p ± c x p x x t x-axis We observe te poton during a time interval ( t), during wic it moves a distance ( x). Say tat te poton is at position at a time x ± x t ± t Te poton moves at te speed of ligt, c, as momentum and energy p ± x p x E ± E /7/008 Quantum ( F.Robilliard) 50
E t...( 3) Momentum & Position : Planck cf λ de Broglie Definition of velocity (4), (5) (3): E E c f c x t p x x p c c... x t x c p x......( 6) tus ( 4 )...( 5) wic is te alternative form for te uncertainty principle. We derived tis for a poton, but because of wave-particle duality, it applies equally to mass particles /7/008 Quantum ( F.Robilliard) 51
p x x...( 6) p x and x : x p x Tis relationsip [(6)] means tat tere is a fundamental related uncertainty in a particle s position and its momentum - tat te product of te uncertainty in position wit te uncertainty in momentum is always greater tan a certain constant. Te more accurately we specify position, te more uncertain is te particle s momentum; te more accurate te momentum, te more uncertain te position. /7/008 Quantum ( F.Robilliard) 5 p x ± x ± x-axis Tis follows from te wave nature of particles. Momentum is inversely proportional to te de Broglie wavelengt. To determine tis wavelengt, we need to sample te wave over a region of space. Tis is analogous to te need to sample a wave over a period of time to determine its energy. Te greater te distance over wic we sample te wave, te more accurately can we determine te momentum. An example of tis, is in te concept of electron orbitals. Te position of an electron about an atomic nucleus, is intrinsically uncertain. We regard te electron as aving a probability distribution about te nucleus as aving existence over te volume of an atomic orbital. Its exact location is teoretically meaningless. We empasize again, tat equation (6) gives te teoretically smallest possible uncertainties ( quantum uncertainties ) Total uncertainties may be greater tan tis.
Summarising: Conjugate Variables: E t...( 3) were E, and t are te conjugate uncertainties in te energy of a particle, E, and te time, t, at wic tat energy was measured. Te more precisely te energy of a particle is determined, te less precisely te time of observation is known, and vice versa. x p x...( 6) were p x, and x are te conjugate uncertainties in te momentum of a particle along te x-axis, p x, and te position, x, at wic tat momentum was measured. Te more precisely te momentum of a particle is determined, te less precisely is its position known, and vice versa. Energy & time are called conjugate variables, as are position & momentum /7/008 Quantum ( F.Robilliard) 53
Significance of te Uncertainty Principle: E t...( 3) x p x...( 6) Te uncertainty principle as profound implications for Pysics, and reality. It says tat tere is an intrinsic teoretical uncertainty in simultaneously determining a particle s position and momentum. In classical Pysics, once bot te position and momentum of a particle are known, te position and momentum of te particle at any future moment could be calculated precisely tat is, te future pat of te particle is calculable. But if te initial position and momentum are intrinsically uncertain, so too are any future values, and te pat. Tis destroys te precise causal relationsip between present and future. All we can do is state probabilities. Causality fails! Te uncertainty principle is not about uncertainties due to equipment limitations in measurements, wic could, in principle, be improved wit better equipment. It is a statement about te fundamental teoretical limits on precision in determining pysical quantities. /7/008 Quantum ( F.Robilliard) 54
Et...( 6) Example: x p x...( 3) Wat is te uncertainty in te momentum of an electron tat is confined to an atomic nucleus of radius of 1.0 x 10-15 m? Position of electron ( x ± r), were r nuclear radius r x r ( 6) p x : 1.0 10 p ( ) x x x -15 m ( -34 6.63 10 ) ( ) ( -15 1.0 10 ).1 10 19 kg ms -1 x nucleus x-axis Quantum uncertainty in electron momentum is.1 x 10-19 kg m/s To limit a single electron to te nucleus results in it aving quite a large uncertainty in momentum, relatively speaking. Te smaller te region to wic a particle is restricted, te more frenetic will it become. /7/008 Quantum ( F.Robilliard) 55
Uncertainty in Atomic Spectra: As discussed earlier, te electrons of an atom can exist only in certain orbitals about te atomic nucleus. An electron in a particular orbital as a particular energy. Tus orbitals represent particular electron energy levels. Et p x x Energy...( 3)...( 6) Te figure sows two suc atomic energy levels, E 1 and E (E >E 1 ). Wen an electron falls from an upper level, suc as E, to a lower level, suc as E 1, te energy difference, E, between te levels is emitted from te atom, as a poton of energy E (E E 1 ) f. /7/008 Quantum ( F.Robilliard) 56 E E 1 Ε t t 1 energy of electron So a particular transition, E -> E 1 produces ligt of a caracteristic frequency, f. We migt expect tis to be a precise frequency, but because of intrinsic uncertainties in te energies E and E 1,tere is an uncertainty in te frequency. Te uncertainty in te energy of an energy level, is connected to te uncertainty in te time tat we ave available to measure it. Tis is determined by te lifetime, τ, of te level (tat is, te time tat te electron is in te level before it transitions to anoter energy level). Let te lifetime of level E 1 be τ 1, and of level E be τ.
Et...( 3) Example: p x x...( 6) Te 589 nm emission line of Sodium is produced by an electron transition from an upper energy level, of.109 ev ( 3.37 x10-19 J ), tat as an average lifetime of 16 ns, to te ground state. Wat is te quantum uncertainty in te frequency of tis emission line? E E 1 Energy Ε t t 1 Te orbital electron transitions from energy level E, to a lower energy level E 1. Te energy of te emitted poton is E (E E 1 ). energy of electron Let t 1 time uncertainty in wic we can measure te electron s energy in state E 1 and t time uncertainty in wic we can measure te electron s energy in state E Let E 1 and E be te quantum uncertainties in energy levels E 1 and E. Because te electron remains in te ground state E 1 for an indefinitely long time, t 1 is large, and, using equation (3), E 1 small. Te uncertainty in te energy of tat state is negligible. Te uncertainty, E, in te energy, E, of te emitted poton, is terefore equal to te uncertainty in level E E. Tis uncertainty in E is connected by te uncertainty principle, to te time, t, we ave to measure it, namely te average lifetime of te state, τ. Terefore t τ /7/008 Quantum ( F.Robilliard) 57
Et But E...( 3) E f f E f x p x f...( 6) ( 3) : Et Et E ( ) E ( ) ( )( ) -34 ( 6.63 10 ) -8 ( ) ( 1.6 10 ) ( Planck) ( -6 1.3 10 ) ( 34 6.63 10 ) 0 MHz Examples: 1.3 10 1.99 10 A finite state lifetime results in quantum uncertainty in poton energy, and a corresponding uncertainty in te frequency (and wavelengt) of te poton. Te emission line is not just one frequency, but a narrow continuous band of frequencies. Tis is te quantum bandwidt of te emission spectral line, and is, teoretically, te narrowest widt possible. Lines are usually broader tan tis. /7/008 Quantum ( F.Robilliard) 58 6 7 J Hz N E E 1 Energy of state t Ε τ t t 1 t lifetime of E state N occurrence of t t
Summary: In tis topic we ave introduced some of te fundamental ideas of Quantum Pysics. We firstly introduced te idea tat, on te atomic scale, quantities are not continuous, but packaged into small discrete amounts called quanta. We saw tat ligt is composed of quanta called potons, and tat te energy of a poton is given by Planck s equation E f. We saw tat mass and energy are equivalent, according to te Einstein equation E (pc) + (m 0 c ), and can be interconverted. We ten introduced te concept of wave-particle duality, te wavelengt of a mass particle being given by te De Broglie equation λ /p Finally we looked at te Heisenberg Uncertainty Principle Et /7/008 Quantum ( F.Robilliard) 59 p x x
r nucleus orbital electron wave λ /7/008 Quantum ( F.Robilliard) 60