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1 MIDTERM #1 PHYS 33 (MODERN PHYSICS II) DATE/TIME: February 11, 016 (8:30 a.m. - 9:45 a.m.) PLACE: RB 306 Only non-programmable calculators are allowed. Name: ID: Please read the following instructions: This midterm has 7 pages. Make sure none are missing. There are 4 questions. You must do all questions. Write your name and student ID in the provided space above. Do all questions. There is an equation sheet in the last page (page 7). You may tear the equation sheet from the midterm. Read questions carefully before attempting the solutions. 1

2 Answer all questions. Show all works. Do all works in the blank space below the question. 1. [15 points] Consider a Bohr hydrogen (neglect spin), whose energy is given by the formula E n = 13.6eV n, n = 1,, 3..., in the stationary state n = 3, l =, m l = 1. A) Write down the explicit form of wave function (in r,θ,φ ) for this hydrogen state. B) Find the most probable distance of the electron from the proton. Hint: minimize the probability distribution P nl (r) = r R nl. C) A physicist conducts an experiment on a hydrogen atom in the state n = 3, l =, m l = 1, which will determine the energy, magnitude of the orbital angular momentum, z-component of the orbital angular momentum. What values for the abovementioned quantities will he obtain? Express your answers in ev for energy, and! for angular momentum. D) Consider now an electron in the linear combination state 0.8ψ ψ 11. If the physicist measures the z-component of the orbital angular momentum, what values would he obtain, and with what probabilities?

3 . [15 points] Fine Structure of Lithium (Li) atomic number Z = 3 a) Write down the ground-state electronic configurations of Li. Derive the term symbol (spectroscopic notation, see equation sheet) of the ground state. b) The lowest excited states of lithium involve a single electron in an unfilled p subshell. This will result in two states (doublets). Use the rule of addition of angular momentum to derive the term symbol (spectroscopic notation) of these two excited states. Briefly explain why there are two states. Draw the energy-level diagram involving the ground state (part a) and the two excited states. c) As you should know the excited states of part b will split into two states of different energy. The energy difference between these two states is ΔV fs = µ B B int (see equation sheet), where! B int is the internal magnetic field. Briefly explain the physics origin of the internal magnetic field and the energy difference. Suppose! B int points in the + z direction and has magnitude B int = 0.4T, calculate the energy difference ΔV fs in ev. 3

4 3. [15 points] The ground state of Boron (B) a) Boron has an atomic number of Z = 5.Write down its ground state configuration b) Find the spectroscopic symbol (notation) of the ground state of Boron. REMARK: For this question you are expected to justify your answer with Hund s rules and the class discussion on the effects of core and valence electrons on the angular momentum (orbital, spin, and total) of an atom. Another hint is that only electrons in unfilled subshells contribute to the total angular momentum. c) Consider a Stern-Gerlach experiment on a beam of Boron in the ground state. Draw a simple diagram (just like the one on assignment ) to illustrate this experiment. The diagram should predict the results of the experiment. Does the result on Boron prove the existence of the intrinsic electron spin. Briefly justify your answer. 4

5 4. [0 points] Below is the fine-structure diagram of for all 3s and 3p states of an atom. The diagram includes the term symbol of all states (LHS). The 3 energy lines with zero applied field B ext = 0 are shown below. In the presence of an applied magnetic field of B ext = 0.6T in the +z direction the lines should split into more lines. This is the anomalous Zeeman effects. The Zeeman energy is already drawn for the 3 S 1/, which splits into two levels with energy difference ΔE S 1/ zeeman. Term Symbol of Sodium No applied Field Non-zero Field Fine Structure B ext = 0 B ext = 0.6T a) Draw the energy splitting, in the above diagram, for the 3 P 3/ and 3 P 1/. Using the allowed transition rule (equation sheet) draw all the allowed transitions between states in the above diagram for B ext = 0.6T. In total, how many transitions are there? Note that if there were no spin (normal Zeeman effect) there would be only two transitions! b) Calculate the energy difference ΔE zeeman a magnetic field of B ext = 0.6T. S 1/ between the two states of 3 S 1/ in the presence of 5

6 3b) continued c) In the absence of an applied field, B ext = 0, the energy difference between the 3 S 1/ and 3 P 3/ is ΔE 0 = 3.0eV. In the presence of an applied field of B ext = 0.6T, calculate the wavelength, λ, of the photon emitted for a transition from an initial 3 P 3/, m J = 3 / to a final state 3 S 1/, m J = 1 /. NOTE: 1) E photon = hc λ with hc = 139.8eVinm ; ) in the equation sheet the Zeeman energy is V z = gµ B B ext m J, where the Lande factor g are different for the two states. 6

7 Useful Equations Hydrogen in R 1,0 = a 0 3/ e r/a 0, R,0 = 4 R 3, = 3/ 81 30a 0 n, l, ml is described by the wavefunction ψ nlml = R nl ( r)y lml θ,φ 1 r 3/ a a 0 0 e r/a 0 1, R,1 = 3/ 6a 0 r a 0 e r /3a 0 Y 1,±1 = 1 5 ( π 3cos θ 1), Y,±1 = 1 3 π sinθe±iφ, Y 1,0 = 1 15 cosθ sinθe±iφ π r a 0 e r/a 0, ( ) 3 π cosθ, Y 0,0 = 1 π, Y,0 = 1 4 Bohr radius a 0 = 4πε!c 0 = eV m. Conserved Quantities: E =, n = 1,,3... me n ; n ( l + 1) ", l = 0,1,... 1 L = l n ; L z = m l ", m l = 0,±1,±,...± l. Spin: S = 3!, S = m!; m = +1 z l l (up), 1 (down) ; Radial probability: P (r)dr = r R nl nl dr! Averages: f ( r) = f ( r,θ,φ) f = π π r dr sinθdθ dφ f ( r, θ, φ) ψ nlm ; f = f r l f = r f ( r)r π nl dr ; f = f ( θ ) f = f ( θ )sinθ Y 0 lml dθ. 0 Magnetic moment of an electron:! µ L = e! m L (orbital);! µ S = e! m S (spin). External Magnetic Field, B!!!! : E = µ, µ magnetic moment ; Spherical Coordinate: x = ext B ext 8 rsinθcosφ, y = rsinθsinφ, z = rcosθ; Photon: E = hν, λ = c / ν, c = 3 10 m / s, h = ev s = J s.fine-structure Energy ΔV fs = µ B B int Zeeman Effect V z = gµ B B ext m J, Lande Factor g = 1 + J J + 1 ( ) + S( S + 1) L( L + 1) J ( J + 1) ( ) Bohr magneton µ B = e! m = ev / T Spectroscopic (Term) Symbol n S +1 L J Energy order of electron subshell 1s, s, p, 3s, 3p, 4s, 3d, 4p, 5s. Hund s rule 1) The total spin angular momentum is maximized without violating the Pauli s exclusion principle; ) Insofar as rule 1 is not violated, L should also be maximized; 3) For an gatom with subshell less than half full, J should be minimized. More useful constant Mass of electron kg ;!!!! = J s = ev s. Addition of Angular Momentum: J = L + S, J = L+ S,L+ S 1,... L S, M J = ± J,± ( J 1),± ( J ),.. Selection rules: 1) atom with fine structure Δn anything, ΔS = 0, ΔJ = 0, ±1, ΔL = ±1 ( J = 0 J = 0 is not allowed); ) Zeeman Effect Δn anything, ΔS = 0, ΔJ = 0, ±1, ΔL = ±1 ( J = 0 J = 0 is not allowed), Δm J = 0,±1( m J = 0 m J = 0 is not allowed if ΔJ = 0 ) 7