Bohr s Correspondence Principle
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1 Bohr s Correspondence Principle In limit that n, quantum mechanics must agree with classical physics E photon = 13.6 ev 1 n f n 1 i = hf photon In this limit, n i n f, and then f photon electron s frequency of revolution in orbit. Extension of Bohr theory to other Hydrogen-like atoms
2 Bohr s Correspondence Principle In limit that n, quantum mechanics must agree with classical physics E photon = 13.6 ev 1 n f n 1 i = hf photon In this limit, n i n f, and then f photon electron s frequency of revolution in orbit. Extension of Bohr theory to other Hydrogen-like atoms He +, Li ++, Be +++, etc. (one electron orbiting nucleus of Q = +Ze)
3 Bohr s Correspondence Principle In limit that n, quantum mechanics must agree with classical physics photon i f photon h n 1 n ev E f = = In this limit, n i n f, and then f photon electron s frequency of revolution in orbit. Extension of Bohr theory to other Hydrogen-like atoms He +, Li ++, Be +++, etc. (one electron orbiting nucleus of Q = +Ze) 4 e e n n 13.6 ev Z - e Z k m n 1 E = = h
4 Bohr s Correspondence Principle In limit that n, quantum mechanics must agree with classical physics photon i f photon h n 1 n ev E f = = In this limit, n i n f, and then f photon electron s frequency of revolution in orbit. Extension of Bohr theory to other Hydrogen-like atoms He +, Li ++, Be +++, etc. (one electron orbiting nucleus of Q = +Ze) 4 e e n n 13.6 ev Z - e Z k m n 1 E = = h
5 Intrinsic linewidth λ of emitted photons
6 Intrinsic linewidth λ of emitted photons E i excited state photon E f ground state
7 Intrinsic linewidth λ of emitted photons E i excited state photon E i E f = E photon = hc λ E f ground state
8 Intrinsic linewidth λ of emitted photons E i excited state photon E i E f = E photon = hc λ E f ground state Time-energy uncertainty principle:
9 Intrinsic linewidth λ of emitted photons E i excited state photon E i E f = E photon = hc λ E f ground state Time-energy uncertainty principle: E i t i h 4π
10 Intrinsic linewidth λ of emitted photons E i excited state photon E i E f = E photon = hc λ E f ground state Time-energy uncertainty principle: E i t i h 4π The excited state of an atom is short lived (Dt i ~ 10-8 s ) before a photon is emitted.
11 Intrinsic linewidth λ of emitted photons E i excited state photon E i E f = E photon = hc λ E f ground state Time-energy uncertainty principle: E i t i h 4π The excited state of an atom is short lived (Dt i ~ 10-8 s ) before a photon is emitted. This causes an uncertainty in E i (DE i ) that induces an uncertainty in E photon, which in turn produces an uncertainty in λ.
12 Intrinsic linewidth λ of emitted photons E i excited state photon E i E f = E photon = hc λ E f ground state Time-energy uncertainty principle: E i t i h 4π The excited state of an atom is short lived (Dt i ~ 10-8 s ) before a photon is emitted. This causes an uncertainty in E i (DE i ) that induces an uncertainty in E photon, which in turn produces an uncertainty in λ. For E photon ~ ev (visible spectrum), Dl/l ~ 10-8.
13 De Broglie electron waves and the Hydrogen atom
14 De Broglie electron waves and the Hydrogen atom Meaning of: v r = n h? m e
15 De Broglie electron waves and the Hydrogen atom Meaning of: v r = n h? m e Analogy with standing waves on a vibrating string get standing waves if have integer number of l s, in this case 3l. l 3l
16 De Broglie electron waves and the Hydrogen atom Meaning of: v r = n h? m e Analogy with standing waves on a vibrating string get standing waves if have integer number of l s, in this case 3l. l 3l Instead wrap string into circle, standing wave pattern is similar.
17 De Broglie electron waves and the Hydrogen atom Meaning of: v r = n h? m e Analogy with standing waves on a vibrating string get standing waves if have integer number of l s, in this case 3l. l 3l Instead wrap string into circle, standing wave pattern is similar. De Broglie standing waves in an electron orbit
18 De Broglie electron waves and the Hydrogen atom Meaning of: v r = n h? m e Analogy with standing waves on a vibrating string get standing waves if have integer number of l s, in this case 3l. l 3l Instead wrap string into circle, standing wave pattern is similar. De Broglie standing waves in an electron orbit πr = nλ
19 De Broglie electron waves and the Hydrogen atom Meaning of: v r = n h? m e Analogy with standing waves on a vibrating string get standing waves if have integer number of l s, in this case 3l. l 3l Instead wrap string into circle, standing wave pattern is similar. De Broglie standing waves in an electron orbit πr = nλ n = 1,, 3
20 De Broglie electron waves and the Hydrogen atom Meaning of: v r = n h? m e Analogy with standing waves on a vibrating string get standing waves if have integer number of l s, in this case 3l. l 3l Instead wrap string into circle, standing wave pattern is similar. De Broglie standing waves in an electron orbit λ = h m e v πr = nλ n = 1,, 3
21 De Broglie electron waves and the Hydrogen atom Meaning of: v r = n h? m e Analogy with standing waves on a vibrating string get standing waves if have integer number of l s, in this case 3l. l 3l Instead wrap string into circle, standing wave pattern is similar. De Broglie standing waves in an electron orbit λ = h m e v πr = nλ n = 1,, 3
22 De Broglie electron waves and the Hydrogen atom Meaning of: v r = n h? m e Analogy with standing waves on a vibrating string get standing waves if have integer number of l s, in this case 3l. l 3l Instead wrap string into circle, standing wave pattern is similar. De Broglie standing waves in an electron orbit λ = h m e v πr = nλ n = 1,, 3
23 De Broglie electron waves and the Hydrogen atom Meaning of: v r = n h? m e Analogy with standing waves on a vibrating string get standing waves if have integer number of l s, in this case 3l. l 3l Instead wrap string into circle, standing wave pattern is similar. De Broglie standing waves in an electron orbit λ = h m e v πr m e v r = = nλ n = 1,, 3 nh
24 Quantum Mechanics and the Hydrogen Atom
25 Quantum Mechanics and the Hydrogen Atom Schrodinger wave equation was solved for Hydrogen atom
26 Quantum Mechanics and the Hydrogen Atom Schrodinger wave equation was solved for Hydrogen atom A revision of Bohr theory: n = 1 state actually has zero angular momentum!
27 Quantum Mechanics and the Hydrogen Atom Schrodinger wave equation was solved for Hydrogen atom A revision of Bohr theory: n = 1 state actually has zero angular momentum! How is this possible? Won t electron fall into proton?
28 Quantum Mechanics and the Hydrogen Atom Schrodinger wave equation was solved for Hydrogen atom A revision of Bohr theory: n = 1 state actually has zero angular momentum! How is this possible? Won t electron fall into proton? Invoke Heisenberg Uncertainty Principle
29 Quantum Mechanics and the Hydrogen Atom Schrodinger wave equation was solved for Hydrogen atom A revision of Bohr theory: n = 1 state actually has zero angular momentum! How is this possible? Won t electron fall into proton? Invoke Heisenberg Uncertainty Principle As electron is localized near proton, the uncertainty of linear momentum will increase, causing its kinetic energy to rise.
30 Quantum Mechanics and the Hydrogen Atom Schrodinger wave equation was solved for Hydrogen atom A revision of Bohr theory: n = 1 state actually has zero angular momentum! How is this possible? Won t electron fall into proton? Invoke Heisenberg Uncertainty Principle As electron is localized near proton, the uncertainty of linear momentum will increase, causing its kinetic energy to rise. r p r h 4 π
31 Quantum Mechanics and the Hydrogen Atom Schrodinger wave equation was solved for Hydrogen atom A revision of Bohr theory: n = 1 state actually has zero angular momentum! How is this possible? Won t electron fall into proton? Invoke Heisenberg Uncertainty Principle As electron is localized near proton, the uncertainty of linear momentum will increase, causing its kinetic energy to rise. r p r h 4 Thus electron never falls into proton. Instead it forms a spherical probability cloud around nucleus. π
32 Quantum Mechanics and the Hydrogen Atom Schrodinger wave equation was solved for Hydrogen atom A revision of Bohr theory: n = 1 state actually has zero angular momentum! How is this possible? Won t electron fall into proton? Invoke Heisenberg Uncertainty Principle As electron is localized near proton, the uncertainty of linear momentum will increase, causing its kinetic energy to rise. r p r h 4 Thus electron never falls into proton. Instead it forms a spherical probability cloud around nucleus. π probability cloud n = 1 nucleus
33 Quantum Mechanics and the Hydrogen Atom (cont.) Quantum numbers
34 Quantum Mechanics and the Hydrogen Atom (cont.) Quantum numbers
35 Quantum Mechanics and the Hydrogen Atom (cont.) Quantum numbers Need to include Spin Magnetic Quantum Number: m s = ½
36 Quantum Mechanics and the Hydrogen Atom (cont.) Quantum numbers Need to include Spin Magnetic Quantum Number: m s = ½ m s = + ½
37 Quantum Mechanics and the Hydrogen Atom (cont.) Quantum numbers Need to include Spin Magnetic Quantum Number: m s = ½ m s = + ½ m s = - ½
38 Pauli Exclusion Principle (195) and the Periodic Table Wolfgang Pauli ( )
39 Pauli Exclusion Principle (195) and the Periodic Table Wolfgang Pauli ( ) No two electrons in an atom can ever be in the same quantum state; that is, no two electrons in the same atom can have exactly the same value for the set of quantum numbers: n, l, m l, m s.
40 Pauli Exclusion Principle (195) and the Periodic Table Wolfgang Pauli ( ) No two electrons in an atom can ever be in the same quantum state; that is, no two electrons in the same atom can have exactly the same value for the set of quantum numbers: n, l, m l, m s.
41
42
43
44 +
45
46 +
47
48 +
49 Characteristic X-rays Mo: Z = 4
50 Characteristic X-rays Mo: Z = 4 State 1 n = 1 K shell n = L shell
51 Characteristic X-rays Mo: Z = 4 State 1 High energy electron n = 1 K shell n = L shell n = 1 K shell n = L shell
52 Characteristic X-rays Mo: Z = 4 State 1 n = 1 K shell n = L shell n = 1 K shell n = L shell
53 Characteristic X-rays Mo: Z = 4 State 1 State n = 1 K shell n = L shell n = 1 K shell n = L shell
54 Characteristic X-rays Mo: Z = 4 State 1 State n = 1 K shell n = L shell n = 1 K shell n = L shell n = 1 K shell n = L shell
55 Characteristic X-rays Mo: Z = 4 State 1 State State 3 n = 1 K shell n = 1 K shell n = 1 K shell n = L shell n = L shell n = L shell E ph
56 Characteristic X-rays Mo: Z = 4 State 1 State State 3 n = 1 K shell n = 1 K shell n = 1 K shell n = L shell n = L shell n = L shell E ph In K shell for State 1, each electron partially shields the other. Thus effective nuclear charge Z eff = 4 1 = 41. In State, there is only one electron between L-shell electrons and nucleus, thus Z eff = 4 1 = 41.
57 E K = 13.6 ev Z eff n = 13.6 ev Z eff
58 E E K L Z = 13.6 ev n Z = 13.6 ev n eff eff = 13.6 ev Z = 13.6 ev eff eff Z 4
59 E E K L Z = 13.6 ev n Z = 13.6 ev n eff eff = 13.6 ev Z = 13.6 ev eff eff Z 4 Eph = EL EK = 17.1 kev
60 E E K λ L K Z = 13.6 ev n Z = 13.6 ev n α = h c E ph eff eff = 13.6 ev Z = 13.6 ev ev m = 17.1 kev eff eff Z 4 Eph = EL EK = 17.1 kev
61 E E K λ L K Z = 13.6 ev n Z = 13.6 ev n α = h c E ph eff eff = 13.6 ev Z = 13.6 ev ev m = 17.1 kev eff eff Z 4 Eph = EL EK = 17.1 kev λ K = 7 pm α
62 E E K λ L K Z = 13.6 ev n Z = 13.6 ev n α = h c E ph eff eff = 13.6 ev Z = 13.6 ev ev m = 17.1 kev eff eff Z 4 Eph = EL EK = 17.1 kev λ K = 7 pm α
Physics 102: Lecture 24. Bohr vs. Correct Model of Atom. Physics 102: Lecture 24, Slide 1
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