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1 MIDTERM #1 PHYS 33 (MODERN PHYSICS II) DATE/TIME: February 16, 17 (8:3 a.m. - 9:45 a.m.) PLACE: RB 11 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. The equation sheets are on page 6 and 7. You may tear the equation sheets 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] Fine Structure of Sodium (Na) atomic number Z 11 A) Write down the ground-state electronic configurations of Na. Derive the term symbol (spectroscopic notation, see equation sheets) of the ground state. B) The two lowest excited states of Sodium involve a single electron in an unfilled 3p, subshell, and in the 3d subshell. Each of the two excited states has two states (doublets). Use the rule of addition of angular momentum to derive the term symbol (spectroscopic notation) of these excited states. Briefly explain why there are doublet states. Draw the energy-level diagram involving the ground state (part a) and the 3p and 3d excited states. Use the selection rules to draw all allowed transitions between states. NOT ON MIDTERM, W18 C) A magnetic field B 1.3T is applied to a Na atom in 3 D 3/ state. Explain why the energy split into 4 levels. Calculate the energy difference between the adjacent levels.

3 . [15 points] D and 3D Maxwell-Boltzman Distribution a) In D the speed distribution becomes f D v ( ) βmvexp β mv with v v x + v y, material. Briefly explain the mathematical meaning of f D dv, and explain why the normalization of f D requires f D dv 1. Verify by direct integration that f D dv 1. Since vexp( av )dv 1 exp av ( ) a (3 points) dvf D exp β mv exp β m m + exp β 1 b) Find the average speed in D, v of a neutron at T 3K. Use the formula in the equation sheets to calculate the average speed of the neutrons in three dimensions (3D). Data: Mass of neutron in equation sheet. HINT: x exp( ax )dx ( π ) 1/ /( 4a ) 3/. v dvvf D βmv exp β mv π 1/ πk βm BT 3/ 4 βm/ m ( ) m kg, v π J i K 1 3K kg 1/ 1/ 1974 m s 3D, v J i K 1 3K π kg 51 m s, (3 points) c) Using df D find the most probable speed of a neutron in D at T 3K, and dv compare with your answer in part b. df D dv mv βmexp β β m v exp β mv v * k T B m v * J i K 1 3K kg 1/ 1575 m s Note that this is significantly smaller than the mean speed of part b). (3 points) 1/ 3

4 3. [15 points] Bose-Einstein condensation. a) Briefly explain what are bosons and fermions. b) At very low temperature liquid helium (He4) undergoes Bose-Einstein condensation. Briefly explain what happens. NOT ON MIDTERM, W18 c) If He4 is an ideal Bose gas, the critical temperature, T C and the number density N/V obeys the relation T C h N 1. If the critical temperature is T C 3.K, calculate mk B V π (.315) the number density of He4 ( proton and neutrons). d) A container has two moles of He4 at temperature T 1.8K. Calculate the number of He4 atoms in the superfluid state? NOTE: one mole atoms. 4

5 THOUGH WHITE DWARF STAR IS NOT A TOPIC OF THIS YEAR S MIDTERM. STUDY THIS QUESTION AND NOTE THAT IT IS TH ENUMBER DENSITY OF THE ELECTRON (N/V IN m 3 ) THAT MATTERS IN FINDING THE FERMI ENERGY, ε F. 4. [15 points] Consider a white dwarf star of mass M star M sun. It is believed that its radius is similar to the earth s radius. The star is prevented from collapsing by the degenerate gas pressure due to the electron gas of the star. a) The mass of the star is mainly due to the protons and neutrons of the star. Assuming that the number of neutrons and protons are the same, calculate the number of electrons of the star. Use the result to calculate the radius of the white dwarf star, R star. M star M sun 1 3 kg, and since the mass of neutrons and protons are similar, m kg, the number of nucleons (protons plus neutrons) is N n 13 kg kg, N n Hence the number of protons should be , and for charge neutrality the number of electrons should be N (1 point), and the fermions are electrons, m fermion kg. The star s radius is R star 4GM star N 5/3 h 9 m fermion 4π ( ) 5/3 ( J s) ( )( 1 3 kg) ( kg) R star m 3 ikg 1 i s 9 4π 7 km. which is similar to the earth s radius 6371 km. ( points) m, or b) Use your answer to calculate the Fermi energy, the Fermi temperature and the degenerate gas pressure of the star. 3 The volume of the star is V ( 4 /3)πR star m 3. (1 point) ( ε F J i s) kg π m J ev. (1 point) 5

6 T F E F ev k B ev /K. 19 K (1 point) PV 3 U P U 3V andu 3 5 NE, P N F 5 V ε F m J P Pa P atm (1 point) c) What is the pressure due to the force of gravity? You do not have to do any calculation, but you must justify your answer in no more than sentences. The inward pressure due to gravity must be P atm to cancel the outward pressure due to the electron degenerate gas calculated in part b). (1.5 points) d) Degenerate gas quantum behaviors are usually associated with low temperature fermion systems. The temperature of a white dwarf is very high 1 K. Why is it valid to assume that the electrons of a white dwarf star, is a degenerate gas? A system of identical fermions becomes a degenerate gas only at very low temperature. Here low temperature is defined as being very much lower than the Fermi temperature, which in this case is T F. 1 9 K 1K. So the white dwarfs temperature of 1 K is low enough to justify a degenerate gas approximation. (1.5 points) 6

7 Useful Equations Hydrogen in n, l, ml is described by the wavefunction ψ nlml R nl ( r)y lml θ,φ a 4πε!c eV m. Conserved Quantities: E, n 1,,3... me n ; n ( l + 1) ", l,1,... 1 L l n ; L z m l ", m l,±1,±,...± l. Spin: ( ), Bohr radius S 3!, S m!; m +1 z l l (up), 1 (down) ; External Magnetic Field, B! ext :!!! 8 E µ B ext, µ magnetic moment ; Photon: E hν, λ c / ν, c 3 1 m / 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) ; 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.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, ΔJ, ±1, ΔL ±1 ( J J is not allowed); ) Zeeman Effect Δn anything, ΔS, ΔJ, ±1, ΔL ±1 ( J J is not allowed), Δm J,±1( m J m J is not allowed if ΔJ ) Maxwell-Boltzman 3D speed distribution. F MB ( v)dv 4π N ( m / ( πk B T )) 3/ v exp( mv / ( k B T ))dv ; root-mean-square speed v rms v 3k BT m ; mean speed v 8k B T πm. U f k B T f is degree of freedom; PV Nk BT. D speed distribution f D ( v) βmvexp β mv with v v x + v y, v dvvf D v Degenerate Fermion Gas: Fermi energy, E F h 3 N 8m π V ( ) Energy (average total energy), U 3 5 NE F. Pressure (P) PV 3 U, 1atm Pa. Fermi Temperature: T F E F /k B. Fermi Speedu F E F /m White Dwarfs and Neutron Star: Non-relativistic approximate theory Equilibrium Star Radius, R star 4GM star N 5/3 h 9 m fermion 4π, M star is the mass of the star, m fermion mass of the identical fermions that produces the degenerate gas pressure, G m 3 ikg 1 i s is Newton s gravitational constant. Mass of Sun M Sun 1 3 kg. Radius of sun R sun 7km. Mass of earth M earth kg. Earth s radius 6371 km. 7

8 Bose-Einstein Condensate: Critical Temperature:T c h 1 mk B π.315 ( ) N V, N N 1 T T c 3/,T T, N,T >T. C C Useful constants: Atomic mass unit (amu) 1 amu kg (1 proton or 1 neutron); 31 electron mass me kg ; proton mass m p kg; e C ; 19 electron volt 1eV J ; h J s ev s; 34 16! J s ev s; µ B e! m J / T ; k B J / K ev / K. Useful Integrals: exp( ax )dx ( π ) 1/ / a 1/ ( ) ; x exp( ax )dx xexp( ax )dx 1 a ; x 3 exp( ax )dx 1 a. ( π ) 1/ /( 4a ) 3/ ; 8

Please read the following instructions:

MIDTERM #1 PHYS 33 (MODERN PHYSICS II) DATE/TIME: February 16, 17 (8:3 a.m. - 9:45 a.m.) PLACE: RB 11 Only non-programmable calculators are allowed. Name: ID: Please read the following instructions: This