PHYS 3313 Section 001 Lecture #13 Wednesday, Feb. 27, 2019 Dr. Jaehoon Yu Wed. Feb. 27, 2019 Bohr Radius Bohrs Hydrogen Model The Correspondence Principle Importance of Bohrs Hydrogen Model Success and Failure of Bohrs Model Characteristic X-ray Spectra PHYS 3313-001, Spring 2019 Dr. Jaehoon Yu 1 Homework #3 Announcements End of chapter problems on CH4: 5, 14, 17, 21, 23 and 45 Due: Monday, March 18 Reminder: Midterm Exam In class next Wednesday, March. 6 Covers from CH1.1 through what we learn March 4 plus the math refresher in the
appendices Mid-term exam constitutes 20% of the total Please do NOT miss the exam! You will get an F if you miss it. BYOF: You may bring a one 8.5x11.5 sheet (front and back) of handwritten formulae and values of constants for the exam No derivations, word definitions or setups or solutions of any problems! No Lorentz velocity addition formula! No Maxwells equations! No additional formulae or values of constants will be provided! Special Colloquium this Friday 4pm, Friday, Mar. 1, SH101 Mr. Steve Battel, Battel Engineering, member of National Academy of Engineering Wed. Feb. 27, 2019 PHYS 3313-001, Spring 2019 Dr. Jaehoon Yu 2 The Planetary Model is Doomed From the classical E&M theory, an accelerated electric charge radiates energy (electromagnetic radiation) which means total energy must decrease. Radius r must decrease!! Electron crashes into the nucleus!? Physics had reached a turning point in 1900 with Plancks hypothesis of the quantum behavior of radiation. Wed. Feb. 27, 2019 PHYS 3313-001, Spring 2019 Dr. Jaehoon Yu
4 The Bohr Model of the Hydrogen Atom The assumptions Stationary states or orbits must exist in atoms, i.e., orbiting electrons do not radiate energy in these orbits. These orbits or stationary states are of a fixed definite energy E. (Niels Bohr was awarded Nobel in 1922 for this!) The emission or absorption of electromagnetic radiation can occur only in conjunction with a transition between two stationary states. The frequency, f, of this radiation is proportional to the difference in the energies of the two stationary states: E = E1 E2 = hf where h is Plancks Constant Bohr thought this has to do with the fundamental length of order ~10-10m Classical laws of physics govern the dynamic equilibrium of the stationary state but they do not apply to the transitions between stationary states. The mean kinetic energy of the electron-nucleus system is quantized as K = nhforb/2, where forb is the frequency of rotation. This is equivalent to the angular momentum of a stationary state to be an integral multiple of h 2p Wed. Feb. 27, 2019 PHYS 3313-001, Spring 2019 Dr. Jaehoon Yu 5 How did Bohr Arrived at the angular momentum quantization? The mean kinetic energy of the electron-nucleus system is quantized as K = nhforb/2, where forb is the frequency of rotation in the given orbit. This is equivalent to the angular momentum of a stationary state to be an
integral multiple of h/2p. nhf 1 2 Kinetic energy can be written K = = mv 2 2 Angular momentum is defined as L = r p =mvr The relationship between linear and angular quantifies v=rw ; w =2p f 1 1 1 nhf Thus, we can rewrite K = mvrw = Lw = 2p Lf = 2 2 2 2 h h ,where = 2p L =nh L =n =n 2p 2p Wed. Feb. 27, 2019
PHYS 3313-001, Spring 2019 Dr. Jaehoon Yu 6 Bohrs Quantized Radius of Hydrogen The angular momentum is L = r p =mvr =n So the speed of an orbiting e can be written ve = n mer From the Newtons law for a circular motion 2 e 1 e2 meve ve = Fe = = 2 r 4pe 0 mer 4pe 0 r So from the above two equations, we can get n e 4pe 0 n2 2 ve = = r= 2 mer
4pe 0 mer mee Wed. Feb. 27, 2019 PHYS 3313-001, Spring 2019 Dr. Jaehoon Yu 7 Bohr Radius The radius of the hydrogen atom for the nth stationary state is 4pe 0 2 n2 2 rn = = a0 n 2 mee Where the Bohr radius for a given stationary state is: 2 2 1.055 10 J s) 4pe 0 ( - 10 = = 0.53 10 m
a0 = 2 2 9 2 2 - 31 - 19 mee ( 8.99 10 N m C ) ( 9.11 10 kg) 1.6 ( 10 C ) - 34 The smallest diameter of the hydrogen atom is d = 2r1 = 2a0 10 - 10 m1A OMG!! The fundamental length!! n = 1 gives its lowest energy state (called the ground state) Wed. Feb. 27, 2019 PHYS 3313-001, Spring 2019 Dr. Jaehoon Yu 8 Ex. 4.6 Justification for nonrelativistic treatment of orbital e Are we justified for the non-relativistic treatment
of the orbital electrons? When do we apply relativistic treatment? When v/c>0.1 Orbital speed: Thus ( 1.6 10 ) ( 9 10 ) 2.2 10 ( m s) < 0.01c ( 9.110 ) ( 0.5 10 ) - 16 ve = e ve = 4pe 0 mer Wed. Feb. 27, 2019 9 6 - 31 - 10 PHYS 3313-001, Spring 2019 Dr. Jaehoon Yu 9 Uncertainties
Statistical Uncertainty: A naturally occurring uncertainty due to the number of measurements Usually estimated by taking the square root of the number of measurements or samples, N Systematic Uncertainty: Uncertainty due to unintended biases or unknown sources Biases made by personal measurement habits Some sources that could impact the measurements In any measurement, the uncertainties provide the significance to the measurement Wed. Feb. 27, 2019 PHYS 3313-001, Spring 2019 Dr. Jaehoon Yu 10 The Hydrogen Atom Rydberg Equation Recalling the total E of an e in an atom, the n th stationary states, En E0 e2 e2 En === 2 n2 8pe 0 rn 8pe 0 a0 n ( 2
E0 = 1.6 10- 19 ) 2 e =13.6 ( eV ) = 8pe 0 a0 8p 8.85 10- 12 0.5310- 10 ( )( ) where E1 is the lowest energy or ground state energy Emission of light occurs when the atom is in an excited state and decays to a lower energy state (nu n). hf =Eu - El where f is the frequency of a photon. 1 1 1 f Eu - El E0 1 1 = = = 2 - 2 =R 2 - 2 l c hc hc nl nu
nl nu R is the Rydberg constant. R =E hc 0 Wed. Feb. 27, 2019 PHYS 3313-001, Spring 2019 Dr. Jaehoon Yu RH =1.096776 107 m- 1 11 Transitions in the Hydrogen Atom Lyman series (n=1): The atom will remain in the excited state for a short time before emitting a photon and returning to a lower stationary state. All hydrogen atoms exist in n = 1 (invisible). Balmer series (n=2): When sunlight passes through the atmosphere, hydrogen atoms in water vapor absorb the wavelengths (visible). Wed. Feb. 27, 2019 PHYS 3313-001, Spring 2019 Dr. Jaehoon Yu 12 Fine Structure Constant The electrons speed on an orbit in the Bohr model: n ve =
= mern 2 n 1 e = 4pe 0 n2 2 n 4pe 0 me mee2 On the ground state, v1 = 2.2 106 m/s ~ less than 1% of the speed of light The ratio of v1 to c is the fine structure constant, . 2 e v1 = = = ma0 c 4pe 0 c c ( 8.99 10 N m C ) 1.6 ( 10 C ) 1 137 ( 1.055 10 J s) ( 3 10 m s) 9 2 2 - 34
Wed. Feb. 27, 2019 PHYS 3313-001, Spring 2019 Dr. Jaehoon Yu - 19 2 8 13 Ex. 4.7: Determine the longest and shortest wavelengths observed in Paschen Series (n=3) We use the equation 1 1 1 = R 2 - 2 nl nu l 1 l Max 1 1 =1.0974 10 2 - 2 = 5.335 105 m- 1 3 4 1 ( )
7 l Max =1875( nm) 1 1 6 -1 =1.219 10 m =1.0974 10 2 - 2 l Min 3 7 l Min Wed. Feb. 27, 2019 PHYS 3313-001, Spring 2019 Dr. Jaehoon Yu ( ) =820 ( nm) 14
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