What is Quantum Mechanics?

Transcription

1 What is Quantum Mechanics? QM is the theoretical framework that we use to describe the behaviour of particles, nuclei, atoms, molecules and condensed matter. Elementary particles are characterized by a set of numbers, e.g. charge and spin, whose values are discrete: these are called quantum numbers. Discreteness was not found first in the microcosm. Vibrating bodies, e.g. strings, have discrete frequencies, which seems to be the result of boundary conditions. Similarly, bound quantum systems, e.g. atoms, have discrete energy levels. 1

2 The quantum The values of quantized quantities are multiples of a minimum amount: the quantum of the related quantity. The quantum of energy of radiation of frequency v is E=hv All quanta are proportional to the Planck constant, h=6.62x10-34 Js. If h=0, quantum and classic physics would coincide. 2

3 Wave-particle duality The relation E=hv implies that waves behave like bunches of energy, i.e. they exhibit particle-like nature. Particles have also shown wave-like behaviour, e.g. electron beams hitting a double-slit give rise to interference phenomena. 3

4 Elementary Particles...they have: no structure (definition) mass electric charge other types of charges quantum numbers: - baryon number, B - lepton number, L (3) - spin, S antiparticles... which are they? antiparticles: u, ν, e + RAF211 - CZJ 4

5 Fundamental Interactions RAF211 - CZJ 5

6 The mediators e/m force: γ strong force: The interactions occurs by the exchange of mediators: strong: gluons, g (8) e/m: photon, γ weak: Z, W +, W gravity: graviton RAF211 - CZJ 6

7 n p+ e - + ν e A weak interaction means d u + W - and W - e - + ν e RAF211 - CZJ 7

8 Weak or electromagnetic? e + + e - γ c + c or e + + e - Z c + c RAF211 - CZJ 8

9 Fundamental interactions: properties interaction relative strength time range participating particles strong s m quarks (gluons) e/m s infinite electrically charged (γ) weak s m all (Z 0,W +,W - ) RAF211 - CZJ 9

10 The strong charge Quarks have colour, antiquarks have anticolour Gluons have colour and anticolour Gluons can interact with each other! RAF211 - CZJ 10

11 Quarks are not free particles Only colourless ( white ) objects can be free: confinement White hadrons: baryons (q q q) and mesons (q q) proton: p = uud, neutron: n = udd positive pion: π + = u d antiproton: p = u u d antineutron: n = u d d negative pion: π - = u d RAF211 - CZJ 11

12 What happens if we pull two quarks apart? hadronization RAF211 - CZJ 12

13 Quantum Numbers Quarks: B = 1/3, L = 0, S = 1/2, Q=2/3 (up) or -1/3 (down) Leptons: B = 0, L = 1, S = 1/2, Q = -1 or 0 (neutrinos) Antiparticles: opposite quantum numbers (but same spin) Conservation laws strong E/M weak energy momentum electric charge spin Baryon # Lepton # Parity RAF211 - CZJ 13

14 The muon decay µ e - + ν e + ν µ electron lepton number, L e : 0 = 1 + (-1) + 0 muon lepton number, L µ : 1 = tau lepton number, L τ : 0 = RAF211 - CZJ 14

15 The residual effects of the forces...which acts between elementary particles, but we see this: residual e/m force...and this: residual strong force RAF211 - CZJ 15

16 Special relativity It was assumed that light, like mechanical waves, propagates in a medium, called the ether, at the velocity c=3x10 8 m/s and has different velocities at other reference systems 1887: Michelson-Morley tried to measure the velocity of ether with respect to Earth but the experiment found no evidence of such a medium 1905: Einstein presents the postulates of the special theory of relativity: - the laws of physics are the same in all inertial reference frame - the speed of light in free space is c in all inertial reference frames 1

17 Time dilation Let us assume two reference frames (or observers), A and B, which are in relative motion. We assume that both observers use cartesian coordinate systems, (x A,y A,z A ) and (x B,y B,z B ), which coincide at t=0. The observer B then starts moving in the positive x direction at a velocity (u,0,0) with respect to the observer A. If something happens at rest in B and has a duration of in A will measure time t 0, the observer t t 0 = u 2 c 2 = γ t 0 2

18 Length contraction Assuming again two observers in relative motion with velocity (u,0,0), length measurements in the rest frame, x 0, and in the frame in motion, x, are related as follows x = x 0 1 u 2 c 2 = x γ Length contraction occurs only along the direction of relative motion! In a given problem, the rest frame is not the frame at which we are at rest. The rest frame is the frame where the object whose properties we measure is at rest. 3

19 Relativistic momentum and energy The relativistic momentum of a particle with nonzero rest mass at a velocity v is given by m 0 v p = v 2 c 2 = γm 0 v m 0 moving The rest energy of a particle is determined by its rest mass E 0 = m 0 c 2 The total energy of a particle is given by E = mc 2 = γm 0 c 2 The kinetic energy of the particle is then =. K E E 0 A useful relation: E = pc + ( p = E c for a massless particle) ( ) 2 ( m 0 c 2 ) 2 4

20 How does the microcosm move? In classical mechanics, we have: - observables: p, v, x,... - parameter: t - Newton s law: F=ma - equation of motion: x(t) The observables take continuous values Is it so in the microcosm? RAF211 - CZJ 16

21 Quantum Mechanics (non-relativistic) basics operators momentum eigenstates energy eigenstates RAF211 - CZJ 1

22 Some basic notions object: what we study (particle, atom,...) observable: a measurable property of the object (E, p,...) parameter t: this time is not a property of the object state of an object: the values of its observables wavefunction: the mathematical expression of a state, e.g. ψ ( x ) =ksin( x ) We use the wavefunctions to find the state of an object, i.e. to find the value of its observables. How is this done? RAF211 - CZJ 2

23 Operators ( type 1 ) an operator is something that can be applied to a mathematical function and give a mathematical function e.g. A d 2 = is an operator: dx 2 Aψ = ( 1) ( k sinx) Postulate of Quantum Mechanics: observables are described by operators RAF211 - CZJ 3

24 Operators ( type 2 ) an operator can perform a transformation ( in space or in time or both) example: the transformation ( x, y, z) ( x, y, z) is performed by the operator P (definition): Pψ( xyz,, ) = ψ( x, y, z) the operator P is called the parity operator RAF211 - CZJ 4

25 Eigenstates and eigenvalues Assume an operator Q of type 1 or type 2. If Qψ = qψ, where q=constant, we say that ψ is eigenstate of the operator Q with eigenvalue q each eigenstate has only one eigenvalue degenerate eigenvalues (common eigenvalues) spectrum of an operator: the set of its eigenvalues Hermitian operator: has only real eigenvalues We use Hermitian operators to express observables RAF211 - CZJ 5

26 Example: parity eigenvalues We apply the parity operator (transformation) twice ( x, y, z) ( x, y, z) ( x, y, z) i.e. PPψ ( ) = ψ If Pψ = λψ, then PPψ ( ) = P( λψ) = λ( Pψ) = λλψ ( ) = λ 2 ψ = ψ i.e. λ= ± 1 = the parity of the state ψ ( λ = 1 : even parity, λ= 1: odd parity) RAF211 - CZJ 6

27 Measurements are eigenvalues Assume an observable A, represented by the operator A. We measure A in a state ψ. What will we get? case 1: ψ is eigenstate of A with eigenvalue a. result: all measurements are equal to a. case 2: ψ is not an eigenstate of A. result: a measurement will be any eigenvalue of A but we cannot predict which one. (We never use the same state to measure twice) RAF211 - CZJ 7

28 Probability interpretation wavefunctions are normalized to unity: ψ ( x ) 2 d x = 1 this allows us to say that P = ψ( x) 2 dx is the probability to find the object at (x,x+dx) RAF211 - CZJ 8

29 Indeterminacy in the position of an object it actually means that objects can be found where we don t expect them to be the proton is most likely here or it could be here or even here the electrons are here someplace these four particles might migrate outside the nucleus RAF211 - CZJ 9

30 Example: a particle in a box ψ ψ 2 x x RAF211 - CZJ 10

31 The representation of observables...some simple formulae for operators RAF211 - CZJ 1

32 The recipe for cooking up operators 1. take classical variable 2. replace as follows: x xψ = xψ p x p x ψ i h ψ = π x 3. do the same for y and z 4. same applies to relations between variables RAF211 - CZJ 2

33 A vector* operator The position operator(s) where xψ R = xu x + yu y + zu z = xψ, yψ = yψ and zψ = zψ operators act only on wavefunctions are the component operators * Operators of this form are not ordinary 3d vectors! RAF211 - CZJ 3

34 The linear momentum operator(s) A vector operator where p = p x u x + p y u y + p z u z p x ψ i h ψ = -----, py ψ i h ψ = and p 2π x 2π y z ψ = are the component operators What is the operator for p 2? i h ψ π z RAF211 - CZJ 4

35 Multiplication of operators e.g. gives us p x 2 ψ What does it mean to multiply operators? p x 2 ψ It means that we apply the operator to a wavefunction more than once: = px ( p x ψ) where p x ψ = i----- h ψ 2π x = i h px ψ = i----- h 2 ψ = h ψ 2π x 2π x x 2π x 2 2 RAF211 - CZJ 5

36 The operator for orbital angular momentum The classical representation (variable) is L = r p, so the quantum mechanical representation (operator) will be L = r p, which gives L x = ( yp z zp y ), L y = ( zp x xp z ), L z = ( xp y yp x ) For example, L x ψ = ( yp z zp y )ψ = yp z ψ zp y ψ = yp ( z ψ) zp ( y ψ) L x ψ = i----- h 2π y ψ ψ z z y RAF211 - CZJ 6

37 The kinetic energy operator We use the formula of classical mechanics: 1 K p ( p 2m 2m x + p y + p z ) h = = = 2m 2π where 2 = 2 x y z 2 RAF211 - CZJ 7

38 The total energy operator (when E=K+V) H We assume an object in motion (K) in a region of a potential V: = K+ V( x, y, z) = h m 2π Vxyz (,, ) Observe that the potential operator is given by the classical variable because it depends on x, y, z The operator H is called the Hamiltonian operator ( the Hamiltonian of the system ) + RAF211 - CZJ 8

39 The operator E E i h = π t It gives the time-dependence of energy eigenstates By combining with the expression E=K+V, we obtain: i h Ψ h = Ψ + VΨ 2π t 2m 2π the time-dependent Schrödinger equation RAF211 - CZJ 9

40 The commutator operator Can we multiply two operators A and B in any order? We can, if their commutator [ A, B] is zero. Definition of the commutator operator: [ AB, ] = AB BA Physical meaning of [ AB, ] = 0: the operators A and B have common (simultaneous) eigenstates RAF211 - CZJ 10

41 Examples of commutators 1. position and momentum in the same direction: [, xp x ] = i----- h (same relation for y and z) 2π 2. angular momentum in different directions: [ L x, L y ] = i----- h L (and cyclic permutations) but 2π z [ L 2, L x ] = [ L 2, L y ] = [ L 2, L z ] = 0 RAF211 - CZJ 11

42 Common eigenstate problem in atoms Assume that we have an object (atom) in a potential V(r) and we are interested in states of constant total energy, E. The following relations hold: [ x, H] 0, [ y, H] 0, [ z, H] 0 [ Hp, x ] 0, [ Hp, y ] 0, [ Hp, z ] 0 [ L 2, H] = [ HL, z ] = [ L 2, L z ] = 0 So we can measure simultaneously: E, L 2, L z. RAF211 - CZJ 12

43 Mean value and Expectation value We measure the observable A n times: a 1, a 2,..., and we calculate the mean value as a = n 1 -- a n i i = 1 a n Postulate of Quantum Mechanics: a = O = O = ψ OψdV where O is the expectation value of the operator O in the state ψ. RAF211 - CZJ 13

44 The standard deviation Gives the spread of the measured values around the mean value O = w k ( a k a) 2 k where w k = relative frequence of value Quantum mechanically, we calculate O = ψ ( O O ) 2 ψdv a k RAF211 - CZJ 14

45 The Heisenberg principle Assume that we measure P and Q in the state ψ (which is not an eigenstate of their operators). The following holds for their standard deviations: ( P) 2 ( Q) [ ψ ( PQ QP)ψdV 4 ] 2 If the operators do not commute, the right-hand side is not zero. Examples: p x x = h ( 4π), E t = h ( 4π) RAF211 - CZJ 15

46 Conserved Observables An observable is conserved (i.e. its expectation value does not change with time) when its operator commutes with the Hamiltonian (of the system we study) d dt 2πi O = [ H, O] h Application: the angular orbital momentum of atoms is a constant of the motion RAF211 - CZJ 16

47 Momentum eigenstates linear momentum eigenstates orbital angular momentum eigenstates spin angular momentum eigenstates addition of angular momenta the generic angular momentum operator We will work mainly with quantum numbers in the applications RAF211 - CZJ 1

48 Linear momentum eigenstates We will consider motion in one dimension (x): p x i h d = πdx By solving the eigenvalue equation p x ψ = p x ψ (where both ψ and are unknown), we find p x ψ( x) = c e 2πi xpx h RAF211 - CZJ 2

49 Eigenstates of orbital angular momentum We will look for the simultaneous eigenstates of the operators L 2 and L z ψ L z ψ = L z ψ, L 2 ψ = L 2 ψ We will use spherical coordinates: (r,θ,φ) x z φ θ r y RAF211 - CZJ 3

50 L z eigenstates and eigenvalues The eigenvalue equation i h dψ = L gives 2π dφ z ψφ ( ) = ce 2πi φlz h quantized Condition: ψφ ( ) = ψφ ( + 2π) L z = m----- h 2π m = 0, ± 1, ± 2, is the magnetic quantum number RAF211 - CZJ 4

51 where Eigenstates of L 2 and L z Spherical harmonics Y lm ( θφ, ) ( 1) m ( l + 1) ( l m)! 1 2 m = P 4π( l + m)! l ( cosθ)e imφ m P l ( cosθ) are the associated Legendre functions L 2 h = ll ( + 1), l = 0, 1, 2, the orbital quantum 2π number the magnetic quantum number is restricted by l: m = l, l+ 1,, 0,, l 1, l RAF211 - CZJ 5

52 Parity eigenstates The spherical harmonics are also eigenstates of the parity operator PY lm = ( 1) l Y lm Application: transitions in atoms RAF211 - CZJ 6

53 The generic angular momentum operator J = J x u + x J y u + y J z u z J x J y J 2 = h 2 jj ( + 1) L 2π z m h 2π, [, ] = i----- h J and cyclic permutations 2π z, = -----, j = 012,,,, m = j j, j + 1,, j By solving the eigenvalue equations for J 2 and J z simultaneously and assuming n eigenstates, we find that there are two types of angular momentum: integer j half-integer j RAF211 - CZJ 7

54 The spin angular momentum Spin does not involve a rotation in space S 2 = h ss ( + 1) 2π where s is the spin quantum number S z h = m s -----, m 2π s = s, s + 1,, s 1, s integer s: bosons half-integer s: fermions RAF211 - CZJ 8

55 Addition of angular momenta Let us assume that we want to add the quantum numbers and to find the quantum number of the total angular momentum, j, and the quantum number of the z-component, m j The rules are: 1. j = l s,, l + s 2. for each j, we have 2j + 1 m j s: m j = j, j+ 1,, j Application: L-S coupling and J-J coupling in atoms l s RAF211 - CZJ 9

56 Spin eigenfunctions Spin states are described by spin wavefunctions, which are functions in spin space. Spin space is a mathematical space which can be treated separately from the ordinary space (x,y,z). If a spinless object is described by a wavefunction ψ( x, y, z, t) and we assign it a spin state described by a spin wavefunction χ, the total wavefunction describing the object will be Ψ( ordinary space, time, spin space) = ψ( x, y, z, t)χ The wavefunctions χ describing the spin state of an object are defined by the quantum numbers and of the object: s m s χ sms, 1

57 Eigenvalue equations for spin The wavefunctions and : S z χ sms, are simultaneous eigenstates of the operators S 2 As the quantum number S 2 χ sms, m s = S z χ s ms, ss ( + 1) h 2 χ 2π s, ms = m h s χ 2π sms, can assume the (2s+1) values m s = s, s + 1,, s 1, s there are 2s + 1 different spin states for such an object. 2

58 Spin space For each value of the quantum number s, there is a spin space of (2s+1) dimensions for which the eigenfunctions χ sms, form a basis. value) can be written as a super- Any spin wavefunction (with the same position of the states. χ sms, s χ sms, = unit basis vectors of the (2s+1)-dimensional spin space (compare with u x, u y, u z, the unit basis vectors of ordinary space) 3

59 The spin eigenfunctions States of spin 1/2 χ sms, for a particle with spin 1/2 are denoted by χ 1 2 -, 1 2 -, χ 1 2-1, - 2 and the sec- where the first function describes the spin state with ond function describes the spin state with m s = 1 2. m s = 1 2 These two functions are the basis in a two-dimensional spin space. Any function ( vector ) in that space can be written as χ = c 1 χ 1 2 -, c 2 χ 1 2-1, - 2 where the constants c 1 and c 2 can be thought of as the coordinates of the vector in the two-dimensional spin space. 4

60 The matrix representation The operators S 2, S x, S y and S z are represented by the (2s+1)x(2s+1) matrices S x h h = S 4π y = π 0 i i 0 S z h = S 2 3h 2 = π π 2 01 S 2 The simultaneous eigenfunctions of the operators and S z are given by the column vectors χ 1 2 -, = and χ , 1-2 = 0 1 5

61 Pauli spin matrices We define the Pauli vector operator s = s x u x + s y u y + s z u z 4π S h We then obtain the Pauli spin matrices s 01 x = s 0 i y = s z = i

62 Energy eigenstates We look for the states of constant energy, Ψ( x, y, z, t) h m 2π i h Ψ( x, y, z, t) = EΨ( x, y, z, t) 2π t + Vxyz (,, ) Ψ( x, y, z, t) = EΨ( x, y, z, t) We can separate the time- and space-dependence of the wavefunction Ψ( x, y, z, t) = ψ( xyz,, )φ() t RAF211 - CZJ 1

63 Energy eigenstates (cont d) i h d φ() t = Eφ() t 2π dt h ψ( x, y, z) + Vψ( x, y, z) = Eψ( x, y, z) 2m 2π (time-independent) Schrödinger equation The stationary states (energy eigenstates) are Ψ( x, y, z, t) = ψ( xyz,, )e 2πiEt h RAF211 - CZJ 2

64 Properties of the stationary state 1. there is always a ground state 2. there may be excited states 3. some energy eigenstates can be degenerate 4. discrete eigenvalues = finite motion = bound state 5. constant probability distribution Ψ 2 6. constant expectation values Terminology energy levels = energy eigenvalues energy spectrum = the set of energy eigenstates RAF211 - CZJ 3

65 Application: The free particle We want to find the stationary states of a particles that moves in the x-direction, V=0, E=K The Schrödinger equation h d ψ 2m 2π dx 2 = Eψ has the solution: ψ( x) = c 1 e ikx + c 2 e i kx where k = 2π 2mE h RAF211 - CZJ 4

66 Application: The free particle (cont d) We include the time-dependence Ψ( xt, ) = ( c 1 e ikx + c 2 e ikx )e 2πiEt h non-relativistic motion: E = p 2 ( 2m) Combining with k = 2π 2mE h, we obtain Ψ( xt, ) = ( c 1 e 2πipx h + c 2 e 2πipx h )e 2πiEt h RAF211 - CZJ 5

67 Interpretation of free particle wavefunction 1. superposition of a particle s momentum eigenstates indeterminate: direction of motion, position degenerate energy eigenvalues E = p 2 ( 2m) 2. superposition of two plane waves Ψ 1 ( xt, ) = c 1 e ikx e iωt and Ψ 2 ( xt, ) = c 2 e i kx e iωt of angular frequency ω = 2πf = 2πE h E = and wavelength λ = 2π k λ = h p hf RAF211 - CZJ 6

68 2 ψ t j The probability flux From the time-dependent Schrödinger equation = i h ψ h = ψ + Vψ 2π t 2m 2π + j = 0 we find the continuity equation for probability h ( ψ ψ ψ ψ ) 4πmi is the probability flux RAF211 - CZJ 7

69 Properties of the Schrödinger equation 1. ψ is single-valued 2. ψ and dψ dx are continuous 3. objects cannot penetrate regions of infinite potential 4. only negative potentials that vanish at infinity can give bound states (application: Coulomb potential) RAF211 - CZJ 8

70 Matrix elements They are used to 1. represent operators (and find their eigenstates) 2. study transitions between energy eigenstates Definition: f nm () t = Ψ n fψ m dv is the matrix element which corresponds to the transition from the stationary state m to the stationary state n RAF211 - CZJ 9

71 Transitions Why do we study transitions? Because that s when radiation is produced How about E t = h ( 4π)? RAF211 - CZJ 10

72 Quasi-stationary states Transition matrix element V fi = Ψ f V Ψ i dv The quasi-stationary state has a width τ The transition (or decay) probability is = Fermi s Golden Rule: h πΓ and a lifetime RAF211 - CZJ 11 λ = Γ 1 λ = -- τ 4π V h fi ρ ( Ef )

73 The atom Stationary states for central potentials Solutions for the hydrogen atom Magnetic dipole moment of hydrogen The independent electron model Ground state of atoms The Zeeman effect Spin-orbit coupling (X-rays and Auger electrons) (Fluorescence) RAF211 - CZJ 1

74 Spherically symmetric potentials We will look for stationary states describing the 3d motion of an object in a central potential V(r) We will not consider the time dependence of the wavefunctions (measurements) Our stationary states will be used to describe the electron states of the hydrogen atom. They are also eigenstates of L 2 and L z. RAF211 - CZJ 2

75 The Schrödinger equation Stationary states: ψ( r, φ, θ) h m 2π r ψ r r 2m r 2 ψ + Vr ( ) ψ = Eψ but we already have the spherical harmonics so we can write ψ( r, φ, θ) = Rr ( )Y lm ( θ, φ) and solve the radial equation: h d 2 d 1 2m 2π dr r dr r 2 ll ( + 1) + Vr ( ) Rr ( ) = ER( r) RAF211 - CZJ 3

76 Application: the hydrogen atom V H ( r) = e πε 0 r r = distance between the electron and the proton The radial equation can be use for V H ( r) provided that m µ = m p m e + m p m e RAF211 - CZJ 4

77 The solution RAF211 - CZJ 5

78 Hydrogen atom: the results 1. The energy levels: where E n = E R, E R = 13.6 ev n = 12,, n 2 is the principal quantum number 2. The radial wavefunction: R nl ( r) depends on nl, probability density for electron Pr ( ) = r 2 R nl ( r) 2 average position varies roughly as n 2 (application: shells) RAF211 - CZJ 6

79 Hydrogen atom : the results (cont d) 3. The spatial dependence of the wavefunction: ψ nlm ( r, φ, θ) = R nl ( r)y ( θ, φ) but ψ lm nl ( r, θ) 2 4. The parity of hydrogen: Pψ nlm = ( 1) l ψ nlm 5. Orbitals instead of orbits 6. Spin: the complete wavefunction is ψ nlml m s 7. Degeneracy of energy levels: 2n 2 RAF211 - CZJ 7

80 Magnetic dipole moment of hydrogen In general, µ is due to angular momentum The hydrogen atom has: (a) orbital angular momentum, (b) spin orbital momentum, s l We thus have two magnetic dipole moments: eh eh µ Lz, = m = and 4πm l m l µ µ B S, z = m = 2πm s 2m s µ B e e RAF211 - CZJ 8

81 The independent electron model We describe many-electron atoms by using an effective potential in the Schrödinger equation - wavefunction: as in hydrogen atom but ψ nlml m s - energy levels: depend on nl, l = 0, 1, 2, 3,... name of the level: s, p, d, f,... - notation for electronic energy levels: nl, e.g. 1s - energy ordering: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s,... RAF211 - CZJ 9

82 The independent electron model E Energy levels 5p 5s 4d 4p 4s 3d 3p 3s s and p orbitals go closer to the nucleus 2p 2s 1s RAF211 - CZJ 10

83 Application of the independent electron model RAF211 - CZJ 11

84 The Pauli principle For each set of values for the quantum numbers n, l, m l, m s, we can have only one electron RAF211 - CZJ 12

85 Applications of the model (i.e.m) - electronic structure of atoms - determine the chemical and electromagnetic properties of the atom - calculate angular momentum of atom (L-S coupling) - explain optical transitions - ground state for light atoms cannot be used for the study of X-rays (heavy atoms) RAF211 - CZJ 13

86 (Optical) transitions frequency of emitted/absorbed photon: ν = 1 -- ( E h i E f ) selection rule: l = ± 1 RAF211 - CZJ 14

87 The ground state of atoms Hund s rules 1. add spins of valence electrons so that the total spin is maximized 2. add the orbital angular momenta of valence electrons so that L = z m h l = h ml is max. 2π 3. S and J are added to calculate J 2π RAF211 - CZJ 15

88 The Zeeman effect Assume spinless valence electron with n=2, l=1 in a magnetic field (along z-axis) Interaction energy between the magnetic moment and the magnetic field: U = = m l µ B B field off µ L, z B field on E + µ B B E 2 2 E 2 - µ B B RAF211 - CZJ 16

89 Spin-orbit coupling The nucleus produces a magnetic field that interacts with the spin magnetic dipole moment of the electron U = = 2m s µ B B = ± µ B B µ S, z B Electron states: eigenstates of J 2, L 2, S 2, J z (i.e.m.) notation: 2s + 1 A j RAF211 - CZJ 17

90 The nucleus The Geiger-Marsden experiment (1909) Scattering of a particles by gold and silver foils showed that some of the particles were backscattered. The observed scattering angles could only be explained by a charge concentrated over a range of a few times m. Atoms are empty! With an atom occuping a room of sides of 10 m, the nucleus would hardly stretch across 1 mm. The nucleus carries about 99.95% of the atomic mass and has an extremely high density, of the order of times the density of water CZJ 1

91 The forces in the nucleus There are two types of interaction acting in the nucleus: a strong interaction: attractive force between nucleons. It behaves similarly for neutrons and protons because of their quark composition (udd and uud) an electromagnetic interaction: repulsive force between protons By nuclear forces we denote the first type. They hold the nucleus together, which also explains the size of the nuclei. The theory of strong interactions (QCD) can only be solved analytically for quarks and gluons which are not confined in hadrons because of the magnitude of the coupling constant of the strong interactions. Therefore, we only have models to describe the potential in the nucleus CZJ 2

92 The mass of the nucleus A nucleus with N neutrons and Z protons has a mass equal to m A = Nm n + Zm p B c 2 B is the binding energy of the nucleus, i.e. the energy required to break up the nucleus to its constituents. B is always positive because the nucleus is a bound system. Typical value for B: 8 MeV per nucleon. The most tightly bound nucleus is 62 Ni with B=8.795 MeV per nucleon Compare B with the mass of a nucleon: ~1000 MeV and atomic energies, i.e. the binding energy of H: ~14 ev CZJ 3

93 Calculating binding energy m atom c 2 = m nucleus c 2 + Zm electron c 2 + B electron B electron is the binding energy of the electrons in the atom (at least 4 orders of magnitude smaller than the mass of the nucleus). The binding energy of the nucleus, B = Zm p c 2 + Nm n c 2 m nucleus c 2, is B = ( Zm p c 2 + Zm e c 2 ) + Nm n c 2 ( m nucleus c 2 + Zm e c 2 ) B = Zm1 c 2 + Nm n c 2 m atom c 2 1 H 0 The calculation is done using atomic masses, c 2 = 931 MeV u CZJ 4

94 Separation energy Neutron (proton) separation energy is the energy required to remove a neutron (proton) from a nucleus and is equal to the difference of the binding energies of the two nuclei: A Z N A 1 Z N 1 S n = B( X ) B ( X ) A Z S p = B( X ) B ( X ) N A 1 Z 1 N Separation energies in nuclear physics are analogous to ionization energies in atomic physics: they refer to the binding of the outermost nucleon. Separation energies also show shell structure in the nucleus like ionization energies show shell structure in the atom CZJ 5

95 Binding energy per nucleon A<62: light nucleus + light nucleus heavy nucleus (fusion) A>62: heavy nucleus light nucleus + light nucleus (fission) CZJ 6

96 Mass distribution and nuclear radius From electron scattering experiments, we know that the protons are distributed in the nucleus as these curves show. The distance at which the curve has half of its central value is defined as the radius, R, of the nucleus. R r 0 A 1 3, r 1.07 fm 0 density (nucleons/fm 3 ) Neutron distributions are similar. distance from centre of the nucleus (fm) CZJ 7

97 Spin and parity Protons and neutrons move inside the nucleons, so they have both intrinsic angular momentum and orbital angular momentum. The total angular momentum of the nucleus, J, is calculated using J-J coupling on the momenta of the nucleons. J is usually called the spin of the nucleus! The total orbital angular momentum, l, of the nucleus, determines the parity, λ, of the nucleus (under space reflection): λ = ( 1) l In decay diagrams, the nuclei are denoted by J λ, i.e. J + or J CZJ 8

98 Multipole moments A nucleus is a distribution of charges and currents, which produce electric and magnetic fields. The space-dependence of these fields is described in terms of the multipole moments of the nucleus. They are labeled by their order, L: L=0 (0th or monopole moment): the electric field varies as r -2 L=1 (1st or dipole moment): the electric field varies as r -3 L=2 (2nd or quadrupole moment): the electric field varies as r We have the same moments for the magnetic field (no monopole). The moments of the nucleus reflect its shape: a spherical nucleus has only a monopole electric moment CZJ 9

99 The parity of the moments The multipole moments of the nucleus are quantum mechanically described by operators, which are functions of the coordinates x, y, and z, and therefore have a parity eigenvalue. The parity of electric multipole moments is (-1) L The parity of electric multipole moments is (-1) L+1 (L=order of the moment) Only even-parity moments exist: electric monopole, magnetic dipole, electric quadrupole, etc. (Krane, Introductory nuclear physics, 3.5) CZJ 10

100 Excited states The nucleus, being a bound quantum mechanical system, can have excited states, like atoms have. The density of excited states increases as the excitation energy does. Excited states are reached as a result of a nuclear reaction. Examples: 1. Single-particle states: excited states of nuclei with filled shells + one valence nucleon that can be excited without disturbing the core. 2. Rotational collective states: nuclei sufficiently deformed from the spherical shape have excited states that are described by a rotational motion of the entire nucleus. 3. Vibrational collective states: visualized as harmonic oscillations in shape about a spherical mean CZJ 11

101 Example of single-particle excited states: CZJ 12

102 Example of rotational and vibrational nuclei: CZJ 13

103 Times and relativity Typical kinetic energy of a nucleon within a nucleus: 30 MeV. This gives a velocity ~ 10 7 m/s and for a nuclear circumference of 10 fm, the nucleon would make a complete orbit in ~ s. This is the characteristic time for nucleon motions in the nucleus. (compare with the time it takes an electron to complete one orbit and which is of the order s.) Considering the typical nucleon kinetic energy above (30 MeV), we conclude that the motion of the nucleons is non-relativistic. Most nuclear phenomena can be described by non-relativistic quantum mechanics. Exceptions: (a) beta decay, where the electron and neutrino must be described relativistically using Dirac s equation, (b) nuclear scattering with projectile energies of several hundred MeV per nucleon require a relativistic description of the kinematics of the collision CZJ 14