Page 1 of 5 PHY3101 3-24-99 Name:_______________________ SSN: ________________________ Midterm 2 Closed book exam. One 8.5x11" study sheet is allowed. A calculator is allowed also. Exam is worth 30 points, 15% of your total grade. Forms of the Schrodinger equation: −∂∂+ =∂∂− + =hhh2 222 222 2m xV x t it mddxV x Eψψψ ψψ ψ( , ) ( ) Constants and equations: h=×−662611034. J-s c=×30108.m/s e=×−160221019. C h=×−413571015. eV-s λν = c 1 eV J=×−160221019. hc=1240 eV-nm Eγ = h ν 1 MeV = 106 eV h = = ×−h / .2 10546 1034π J - s λ = h / p 1 nm = 109−m h = ×−65821 1016. eV -s 1 Watt = 1 J/s 1 MHz = 106s-1 me= ×−911 1031. kg mp= ×−1673 1027. kg mµ= ×−1881 1028. kg me=0511. MeV / c2 mp=9383. MeV / c2 mµ=1057. MeV / c2 e2094144 10πε= ×−. eV- m FEvB=+×q() WqV= REhc∞= = ×07109737 10. m-1 REhcH= = ×07109678 10. m-1 1 1 12 2λ= −FHGIKJRn nl u ame00221040 53 10= = ×−πεh. m EZ e mn nne=−= −2 4202 2 22 4136h ( ).πε eV Z2b g µBeem= = ×−h257884 105. eV / T ∆∆xp ≥h/2 ∆∆Et ≥h/2 f x x f x x dx( ) ( ) ( ) ( )*=zψ ψPage 2 of 5 PHY3101 3-24-99 Name:_______________________ SSN: ________________________ 1. Consider the semi-infinite potential well shown below. The kinetic energy E of a particle of mass m is smaller than the barrier height V0, so it is trapped in the well. V xxx LV x L( ) =∞ <≤ <≥RS|T| 00 00 (a) [2 points] What is the wavefunction solution to the time-independent Schrodinger Equation for region 1 ( x < 0 ) ? (b) [4 points] What is the general form of the wavefunction solution for region 2 (0≤<xL) taking into account the boundary condition at x = 0 ? Express your answer as a function of ()kx, and determine the constant k. V(x) x 0 E V0 1 2 L 3Page 3 of 5 PHY3101 3-24-99 Name:_______________________ SSN: ________________________ (c) [4 points] What is the general form of the wavefunction solution for region 3 (xL≥) taking into account the boundary condition at x=∞? Express your answer as a function of ()α x, and determine the constant α . (d) [4 points] Determine two relations between the solution of region 2 and the solution of region 3 by applying boundary conditions at xL=.Page 4 of 5 PHY3101 3-24-99 Name:_______________________ SSN: ________________________ (e) [4 points] Sketch the wavefunction solution of the lowest energy state (ground state). 2. [4 points] Estimate the minimum bound on the momentum spread of a particle of mass m trapped in a one-dimensional box of size L. Determine the typical kinetic energy of the particle.Page 5 of 5 PHY3101 3-24-99 Name:_______________________ SSN: ________________________ 3. [4 points] The ground state wavefunction for a particle of mass m trapped in an infinite potential well between x=0 and xL= is ψπ12( ) sinxLxL=FHGIKJ . Determine the kinetic energy of the particle. Compare your answer with the previous problem. 4. [4 points] A measurement of the z-component of the angular momentum for an atom of hydrogen yields the value −4h . What are the minimum possible values for the angular momentum quantum number l and the principal quantum number n ? What is the degeneracy (total number of states with equal energy) for this value of n, neglecting