Page 1 of 5 PHY3101 11-8-00 Name:_______________________ 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 λ f = c 1 eV J=×−160221019. hc=1240 eV-nm f = E / h 1 MeV = 106 eV h = = ×−h / .2 10546 1034π J - s λ = h / p 1 nm = 109−m h = ×−65821 1016. eV -s k=2πλ/ ωπ=2 f 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= −FHGIKJZ Rn nl ueff2 ame00221040 53 10= = ×−πεh. m EZ e mn nne=−= −2 4202 2 22 4136h ( ).πε eV Z2b g EmLnn=π22222h µ = +FHGIKJ−1 11m me N ∆ ∆x p ≥h2 ∆ ∆E t ≥h2 $p iddx= − h $xx= $( )$( )*f x f x dx=zψ ψ x e dx n n n n nn x n−∞+z= = ⋅ − ⋅ − ⋅/! !αα011 2 2 1 bgbgL dx xxxsin sin22142z= − dx x xx xxx x 2 2326 418224sin sincosz= − −FHGIKJ− I x x dxnn= −∞zexp α 20ch I012=πα I112=α I214=απαPage 2 of 5 PHY3101 11-8-00 Name:_______________________ 1. (a) [4 points] Show that the wavefunction ψωx A em x=− 22/ h is a solution to the time-independent Schrodinger Equation for a particle of mass m in the potential energy well V x V m xaf= +02 212ω , where V0 and ω are constants. (b) [3 points] Find the energy of the particle described by this wavefunction.Page 3 of 5 PHY3101 11-8-00 Name:_______________________ 1. (c) [3 points] Normalize the previous wavefunction over the interval −∞<<∞x. 2. [3 points] An electron is confined to a 1-dimensional region of width 1010− m. What is the minimum kinetic energy of the electron, as measured in electron-volts?Page 4 of 5 PHY3101 11-8-00 Name:_______________________ 3. [4 points] Suppose that in an alternate universe, the value of h is 1026 times larger: h=×−10546108. J-s. Now suppose that an ant (mass = 0.1g = 10-4 kg) is measured to have a velocity of less than 1 mm/s = 0.001 m/s, although we don’t know exactly what it is. What is the uncertainty in the position of the ant along its direction of motion? 4. (a) [3 points] What is the wavelength of the Kα line of copper (Z = 29)? In this transition, an atomic electron from a neighboring shell fills a vacancy in the innermost shell. (b) [2 points] Calculate the energy of a photon of this wavelength. In what region of the electromagnetic spectrum does this radiation belong?Page 5 of 5 PHY3101 11-8-00 Name:_______________________ 5. [4 points] Suppose that the wavefunction for a particle is ψωx Aeikxtafaf=− for the interval −∞<<∞x, where k is the wavenumber and ω is the angular frequency. What is the “spread” in x (position) and the “spread” in p (momentum) for such a wavefunction (∆∆xp and ) ? 6. [4 points] Suppose that four wavelengths of particle fit in a 1-dimensional region of length L. What is the kinetic energy of the