Midterm 2 F98 solutions

# Modern Physics

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Unformatted text preview: file:///C|/Documents%20and%20Settings/Linda%20Grauer/My%20Doc...20HEP-H2/PHY251%20Sp01/Phy251%20Fa98/PHY251_M2_F98solutn.html PHYSICS 251 - Midterm II Monday, November 23, 1998 Show all work for full credit! Short answer problems, each about 3 points: 1. Describe in one sentence the decisive experimental fact which led Rutherford to the planetary atomic model and which was completely contrary to the predictions of the Thomson model. (3 points) Rutherford observed large-angle scatters from alpha particles on thin metal foils, at rates that, although small, were very many orders of magnitude above what was predicted in the Tompson model of the atom. Rutherford deduced that the atom must have a massive, but very small central nucleus (size < 5 fm). 2. Calculate the impact parameter of a Rutherford scattering of a 6.0 MeV α-particle on a zinc foil (Zn, Z =30) scattered at an angle of 90 o . (3 points) b = ( zZ ke ² / 2 K ) cot(45 o ) = ( 2×30 × 1.440 MeV fm / 2×6.0×10 6 MeV )×1 = 7.2 fm. 3. Give the quantization condition that was postulated by Bohr. (3 points) Quantization of angular momentum for stable (i.e. non-radiating orbits): L = nh , n =1,2,3,... . 4. State the "Correspondence Principle". (3 points) In the regime where classical pysics is correct and agrees with experiment, the quantum mechanics must give the same result; typically the classical regime corresponds to the limit of large quantum numbers in quantum mechanics. 5. Calculate the wavelength of the photon resulting from the first hydrogen Balmer transition ( n =2 to n =1). (3 points) ∆ E = - 1 / 2 α ² mc ² Z ² ( 1 / 2²- 1 / 1² ) = (-13.61 eV) Z ² (- 3 / 4 ) = 10.2 eV. Thus: λ = hc / ∆ E = 1240 eV nm / 10.2 eV = 121.5 nm. 6. List and name all three quantum numbers that result from the solution of the 3-dimensional Schrödinger equation for the hydrogen atom. For each of the three, list the allowed range of values it can take. (3 points) n = 1,2,3,4,... the principal quantum number, l = 0,1,2,3,..., n-1 the orbital quantum number, and m l = 0,±1,±2,±3,...,± l , the magnetic quantum number. 7. Give the time dependence of the wavefunctions of the hydrogen electrons; what factor describes the time dependence of the solutions of the static Schrödinger equation for the hydrogen atom? (3 points) For static potentials in general: e- i ϖ t . file:///C|/Documents%20and%20Settings/Linda%20Gra...251%20Sp01/Phy251%20Fa98/PHY251_M2_F98solutn.html (1 of 9) [2/5/2008 11:42:06 AM] file:///C|/Documents%20and%20Settings/Linda%20Grauer/My%20Doc...20HEP-H2/PHY251%20Sp01/Phy251%20Fa98/PHY251_M2_F98solutn.html 8. Show, using the fact that the angular parts of the wavefunction Ψ ( r ) are individually normalized, that the radial probability function P ( r ) dr = r ²| R ( r )|² dr , where Ψ ( r ) = R ( r ) Θ ( θ ) Φ ( φ ) (Hint: show that dV , a volume element at position r , can be written as dr rd θ r sin θ d φ .) (4 points) P ( r ) dV = | Ψ ( r )|² dV = | Ψ ( r )|² dr rd...
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## This homework help was uploaded on 02/05/2008 for the course PHY 251 taught by Professor Rijssenbeek during the Fall '01 term at SUNY Stony Brook.

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