Summary of quantum numbers Course evaluations Please evaluate Prof. Rzchowski today. Start lecture ~ 8:55 am This week’s honors lecture: Prof. Ron Wakai, “Biomagnetic imaging” Final Exam: Monday May 12 12:25 - 2:25 pm Ingraham B10 Wed. Apr 30, 2008 For hydrogen atom: • n : describes energy of orbit • ℓ describes the magnitude of orbital angular momentum • m ℓ describes the angle of the orbital angular momentum Phy208 Lecture 27 Wed. Apr 30, 2008 1 Modified Bohr model • Different orbit shapes A • Directions of ‘orbital bar magnet’ quantized. • Orbital magnetic quantum number Big angular momentum Small angular momentum These orbits have same energy, a) A, but different angular momenta: b) C, r r r L=r"p c) B, Rank the angular momenta d) B, from largest to smallest: ( ) Wed. Apr 30, 2008 ! B, A C, A A, C Phy208 Lecture 27 For example: ℓ=1 gives 3 states: Wed. Apr 30, 2008 3 Electron spin Phy208 Lecture 27 4 Include spin N New electron property: Electron acts like a bar magnet with N and S pole. – m ℓ ranges from - ℓ, to ℓ in integer steps (2ℓ+1) different values – Determines z-component of L: Lz = m l h – This is also angle of L B, C e) C, A, B ! 2 Orbital mag. quantum number mℓ C B Phy208 Lecture 27 • Quantum state specified by four quantum numbers:( n, l, ml , ms ) S – Three spatial quantum numbers (3-dimensional) Magnetic moment fixed… !– One spin quantum number …but 2 possible orientations of magnet: up and down S N Described by • Spin up spin quantum number ms ms = +1/2 • Spin down ms = "1/2 z-component of spin angular momentum Sz = msh Wed. Apr 30, 2008 Phy208 Lecture 27 5 Wed. Apr 30, 2008 Phy208 Lecture 27 6 ! ! ! 1 Atomic Quantum number summary Quantum Number Question • Hydrogen atom states ( n, l, ml , ms ) How many different quantum states exist with n=2? – n: principle quantum number A. 1 B. 2 C. 4 D. 8 • Determines energy • (n=1, 2, 3…) ! – ℓ: orbital quantum number • Magnitude of orbital angular momentum L = h l(l + 1) • ℓ=0, 1, 2, … n-1 ℓ = 0 : 2s2 m l = 0 : ms = 1/2 , -1/2 ℓ=1 – mℓ: orbital magnetic quantum number r m l = +1: ms = 1/2 , -1/2 m l = 0: ms = 1/2 , -1/2 m l = -1: ms = 1/2 , -1/2 • Orientation of L ( Lz = m l h) ! • mℓ = - ℓ, - ℓ + 1, … 0, … ℓ - 1, + ℓ – ms: spin quantum r number • Orientation ! ! of S ( Sz = msh) 2 states 2 states 2 states There are a total of 8 states with n=2 • m s=-1/2, +1/2 Wed. Apr 30, 2008 2 states : 2p6 Phy208 Lecture 27 Wed. Apr 30, 2008 7 Phy208 Lecture 27 8 ! Putting electrons on atom Question • Electrons obey Pauli exclusion principle • Only one electron per quantum state (n, ℓ, mℓ, ms) How many different quantum states are in a 5g (n=5, ℓ =4) sub-shell of an atom? A. 22 B. 20 C. 18 D. 16 ℓ =4, so 2(2 ℓ +1)=18. E. 14 In detail, m = -4, -3, -2, -1, 0, 1, 2, 3, 4 unoccupied occupied n=1 states Hydrogen: 1 electron one quantum state occupied (n = 1,l = 0,ml = 0,ms = +1/2) l and ms=+1/2 or -1/2 for each. 18 available quantum states for electrons ! Wed. Apr 30, 2008 Helium: 2 electrons n=1 states two quantum states occupied (n = 1,l = 0,ml = 0,ms = +1/2) (n = 1,l = 0,ml = 0,ms = "1/2) Phy208 Lecture 27 Wed. Apr 30, 2008 9 Phy208 Lecture 27 10 ! ! Other elements: Li has 3 electrons " n=2 % $ ' $ l=0 ' $ ml = 0 ' $ 1' $ ms = + ' # 2& ! ! # n=2 & % ( % l=0 ( % ml = 0 ( % 1( % ms = " ( $ 2' " n=2 % $ ' $ l =1 ' $ ml = 0 ' $ 1' $ ms = + ' # 2& # n=2 & % ( % l =1 ( % ml = 0 ( % 1( % ms = " ( $ 2' " n=2 % $ ' $ l =1 ' $ ml = 1 ' $ 1' $ ms = + ' # 2& ! ! ! ! # n=2 & % ( % l =1 ( % ml = 1 ( % 1( % ms = " ( $ 2' # n=2 & % ( % l =1 ( % ml = "1 ( % 1( % ms = + ( $ 2' # n=2 & % ( % l =1 ( % ml = "1 ( % 1( % ms = " ( $ 2' n=2 states, 8 total, 1 occupied ! # n =1 & % ( % l=0 ( % ml = 0 ( % ( $ ms = "1/2' Wed. Apr 30, 2008 ! 1s1 He 1s2 Li 1s22s1 Be 1s22s2 B 1s22s22p1 Ne one spin up, one spin down Phy208 Lecture 27 Configuration H 1s shell filled (n=1 shell filled noble gas) ! n=1 states, 2 total, 2 occupied " n =1 % $ ' $ l=0 ' $ ml = 0 ' $ ' # ms = +1/2& Electron Configurations Atom 11 Wed. Apr 30, 2008 etc 2s shell filled 1s22s22p6 2p shell filled Phy208 Lecture 27 (n=2 shell filled noble gas) 12 ! 2 The periodic table Atoms with more than one electron • Atoms in same column have ‘similar’ chemical properties. • Quantum mechanical explanation: similar ‘outer’ electron configurations. Be 2s2 Mg 3s2 Ca 4s2 Sc 3d1 Y 3d2 8 more transition metals Wed. Apr 30, 2008 B 2p1 Al 3p1 Ga 4p1 Phy208 Lecture 27 C 2p2 Si 3p2 Ge 4p2 N 2p3 P 3p3 As 4p3 O 2p4 S 3p4 Se 4p4 F 2p5 Cl 3p5 Br 4p5 He 1s2 Ne 2p6 Ar 3p6 Kr 4p6 13 Hydrogen wavefunctions p: d: f: g: 14 – Ne core = 1s 2 2s2 2p6 (closed shell) – 1 electron outside closed shell Na = [Ne]3s 1 • Outside (11th) electron easily excited to other states. Phy208 Lecture 27 15 Wed. Apr 30, 2008 • How does electron in excited state decide to make a transition? Phy208 Lecture 27 Na 16 Another possibility: Stimulated emission How do atomic transitions occur? • Atom in excited state. • Photon of energy hf=ΔE ‘stimulates’ electron to drop. Additional photon is emitted, Same frequency, in-phase with stimulating photon • One possibility: spontaneous emission • Electron ‘spontaneously’ drops from excited state One photon in, two photons out: – Photon is emitted ‘lifetime’ characterizes average time for emitting photon. Wed. Apr 30, 2008 Phy208 Lecture 27 • 11 electrons ℓ=1 ℓ=2 ℓ=3 ℓ=4 Wed. Apr 30, 2008 Wed. Apr 30, 2008 Na Optical spectrum • Radial probability • Angular not shown • For given n, probability peaks at ~ same place • Idea of “atomic shell” • Notation: – s: ℓ=0 – – – – States fill in order of energy: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d 589 nm, 3p -> 3s H 1s1 Li 2s1 Na 3s1 K 4s1 • Electrons interact with nucleus (like hydrogen) • Also with other electrons • Causes energy to depend on ℓ in addition to n Phy208 Lecture 27 17 hf=ΔE light has been amplified ΔE Before After If excited state is ‘metastable’ (long lifetime for spontaneous emission) stimulated emission dominates Wed. Apr 30, 2008 Phy208 Lecture 27 18 3 Ruby Laser LASER : Light Amplification by Stimulated Emission of Radiation Atoms ‘prepared’ in metastable excited states …waiting for stimulated emission Called ‘population inversion’ (atoms normally in ground state) Excited states stimulated to emit photon from a spontaneous emission. • Ruby crystal has the atoms which will emit photons Two photons out, these stimulate other atoms to emit. • Flashtube provides energy to put atoms in excited state. • Spontaneous emission creates photon of correct frequency, amplified by stimulated emission of excited atoms. Wed. Apr 30, 2008 Phy208 Lecture 27 19 Relaxation to metastable state (no photon emission) 20 • Hydrogen atom: single electron orbiting around single positively-charged proton • Hydrogen atom can be in different quantum states, corresponding classically to different orbits. 2 eV 1 eV Phy208 Lecture 27 Good description of atom Ruby laser operation 3 eV Wed. Apr 30, 2008 • Can also have more than one electron orbiting around the nucleus. Metastable state • # of electrons determines the chemical properties, and hence the element. PUMP Transition by stimulated emission of photon Ground state Wed. Apr 30, 2008 Phy208 Lecture 27 21 Wed. Apr 30, 2008 Phy208 Lecture 27 22 4