dx Solution to 1-D infinite square well of length L: PHYS2170 Fall 2007: Mid-Term #2 PRACTICE

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1 PHYS2170 Fall 2007: Mid-Term #2 PRACTICE IMPORTANT INFORMATION that you may need: Speed of light in empty space (c) 3.0 x 10 8 m/s Planck s constant (h) x J-sec h = h / 2! Coulomb s constant (k) 8.99 x 10 9 N-m 2 /C 2 Charge of an electron (e) 1.6x10-19 C Ground state energy of electron in Hydrogen -13.6eV Mass of electron (kg) 9.11 x kg Mass of proton or Mass of neutron (kg) 1.67 x kg hc = 1240 ev-nm ke 2 = ev-nm 1 electron Volt = x Joules 1 nm = 1x10-9 m Representative wavelengths: Red (680 nm); Orange (610 nm); Yellow (580 nm); Green (540 nm); Blue (470 nm); Violet (410 nm) Work functions of common metals: Sodium=2.28eV; Cadmium=4.07eV; Aluminum=4.08eV; Copper=4.7eV; Lead= 4.14eV; Silver= 4.73eV; Carbon= 4.81eV; Nickel= 5.01eV; Atomic configurations: neutral Hydrogen (H): 1 proton, 1 electron; neutral Helium (He): 2 protons, 2 neutrons, 2 electrons Some Useful Equations: 1D Schrodinger Equation: 2 2 h! "( x, t)!"( x, t) # + V ( x, t) "( x, t) = ih 2 2m! x! t 2 " h 2m 2 d! ( x) + V ( x)! ( x) = 2 dx Solution to 1-D infinite square well of length L: "( x, t) = 2 n# x sin( ) e L L! iet / h E! ( x) PRACTICE

2 1. True(a)/False(b) Photons are just packets of energy. They do not involve oscillating electric and magnetic fields. 2. A cell phone tower transmits a signal over a broad area by generating a 100 Watts of power in a E/M wave with a wavelength of 0.25 m. At what frequency are the electrons in the transmitter oscillating? (pick the closest answer) a. 400 Hz b. 120 Hz c. 3 x 10 8 Hz d. 1.2 x 10 9 Hz e. 4 x Hz 3. In the photoelectric effect, if you shine in light of 300 nm, most energetic electrons come out with some kinetic energy, KE 300. A voltage diff of 1.8 V is required to stop these electrons. What is the work function Φ for this plate? (e.g. the minimum amount of energy needed to kick e out of metal?) a. 1.2 ev b ev c. 2.9 ev d. 6.4 ev e ev 4. Which graph would represent current vs. battery voltage curves for light of the same frequency and two different intensities shined on a plate in the photoelectric effect? A B C 0 Batt. V 0 Batt. D V 0 Batt. V E 0 Batt. V 0 Batt. V

3 5. Light is shining on a sample metal plate in a photoelectric effect experiment, but no electrons are being ejected. Which of the following changes to the experiment could make the sample start ejecting electrons? A. Increasing the intensity of the light beam B. Decreasing the frequency of the light C. Increasing the voltage of the battery a moderate amount D. Replacing the target with a material that has a smaller work function E. More than one of the above changes would work 6. You observe the following spectrum from a discharge lamp. Which energy level configuration is consistent with this spectrum? nm 496 nm 620nm A B C D E 0eV 0eV 0eV 0eV 0eV -2eV -2.5eV -3eV -2eV -4.5eV -4eV -4eV -5.5eV -4.5eV -6eV -6.5eV -6.5eV -7.5eV -8.5eV -8eV 7-9. The first three energy levels of the fictitious element X are shown in the figure. 7.. What is the ionization energy of element X? a) 6.5 ev b) 3 ev, c) 2 ev d) 4.5 ev e) 1 ev 8.What energy would NOT be observed in the absorption spectrum of element X? a) 3.5 ev b) 4.5 ev c) 1 ev

4 9. What is the color of the photon with the least energy that is absorbed? a) Ultraviolet b) Blue c) Green d) Red e) InfraRed, 10. An atom (inside a discharge lamp) is hit by a free electron. The atom subsequently gives off a photon. a) True b) False The color (energy) of the photon does not depend upon the energy of the free electron that hit the atom. 11. How many possible colors can an atom with 6 electronic energy levels (Ground state through level 5) emit? a) 6 b) 10 c) 12 d) 15 e) In a three level laser system, what change would increase the likelihood that an atom in the first excited state (the first one above the ground level) would undergo stimulated emission: a. decreasing the lifetime of that excited state b. increasing the lifetime of that excited state c. decreasing the pump lamp intensity (which pumps from ground to the second excited state) d. increasing the wavelength of light coming out of the lamp (assume you start at a wavelength where stimulated emission does occur) e. decreasing the wavelength of light coming out of the lamp (assume you start at a wavelength where stimulated emission does occur) 13. In order to get a laser to work: a. More than half the atoms need to be in the ground state b. You need half the atoms in the ground state, half in the excited state c. More than half the atoms need to be in the excited state d. The number of excited state atoms is irrelevant 14. A ruby laser emits a 100MW, 20ns long pulse of light with a wavelength of 690 nm. How many atoms undergo stimulated emission to generate this pulse? a) 0.5 b) 345 c) d) e)

15-16. Suppose you have an electron and a photon both moving through space each with a kinetic energy of 9 ev. 15. What is the debroglie wavelength in nm of the photon? a) 0.

An electron wave function between 0 and L is described by the following function: (x,t)=sqrt(2/l)sin(2π x/l)e -i t between 0 and L (x,t)=0 for x<0 and x>l At t = 0: What does this wave look like at

5 Suppose you have an electron and a photon both moving through space each with a kinetic energy of 9 ev. 15. What is the debroglie wavelength in nm of the photon? a) 0.41 nm b) 138 nm c) 276 nm d) 410 nm e) photons do not have debroglie wavelengths 16. What is the debroglie wavelength of the electron? a) 0.41 nm b) 0.58 nm, c) 138 nm d) 276 nm e) 726 nm 17. An electron wave function between 0 and L is described by the following function: (x,t)=sqrt(2/l)sin(2π x/l)e -i t between 0 and L (x,t)=0 for x<0 and x>l At t = 0: What does this wave look like at t=0? 18. For a different wave function, plotted on the right, how do the probabilities of finding the electron very close (within a very small distance dx) to x=g, H, I, J, and K compare? (G=Probability of finding the electron near point G): a. G=H=I=J=K b. H>J=G=K>I c. I>H>G=J=K d. J>H>I=G=K e. H>I>J>G=K

6 Consider the following two cases: Case 1: At time t=0, an electron is described by a plane wave, ψ(x)=ae ikx, where we have drawn the real part below (the wave keeps going forever off the edge of the paper): Case 2: At time t=0, an electron is described by a wave packet as drawn below: 19. Are the following statements true or false? I. There is no uncertainty in the momentum of the electron in Case 1. II. The uncertainty in the position of the electron in Case 1 is less than the uncertainty in the position of the electron in Case 2. a. I=true, II=true b. I=true, II=false c. I=false, II=true d. I=false, II=false 20. Which of the following statements best explains what is happening with the uncertainty in momentum for Case 2 and why: a. The uncertainty in momentum for case 2 is the same as for case 1, because the wavelength is the same. b. The uncertainty in momentum for case 2 is less than for case 1, because the wave is less spread out. c. The uncertainty in momentum for case 2 is greater than for case 1, because to create a localized wave packet requires the superposition of sine waves with a range of wavelengths and a range of momentum. d. The uncertainty in momentum for case 2 is less than for case 1, because the wave is more spread out and uncertainty in momentum and position trade off. e. The uncertainty in momentum for case 2 is greater than for case 1, because the wave packet is made up of a number of electrons with a range of momentum A particle is described by the wave function shown below where ψ(x) =0 for x<-l and x>l!(x) -L a/3 L X -a

7 21. What does a need to be for this wave function to be properly normalized? A. 3 2L B. 9 10L C. 4L 3 D. 11L 6 E. 3 4L 22. The probability of finding the particle between L and 0 is the probability of finding the particle between 0 and L. a. 9 times b. 3 times c. 1/3 times d. 1/9 times e. (some other value)

8 Long Answers 1. Suppose you were to perform the photoelectric effect experiment using light with a wavelength of 400nm and a target made of cadmium. You find that when the voltage measured across the electrodes V is equal to zero volts, the ammeter reads zero current. Would the ammeter read zero current or a non-zero current if you were to: a. Double the intensity of the light source on the cadmium target? Explain your reasoning. b. Increase the voltage V of the battery from 0 volts to +5.0 volts (using the cadmium target)? Explain your reasoning. c. what can you do to change this?

9 2. DeBroglie Model a. What new idea did debroglie introduce that went beyond the Bohr model? b. Name one thing about the hydrogen atom that the Bohr model was unable to explain but that the debroglie model could explain. c. How was the debroglie model able to explain to thing you named in part b? d. Name at least one limitation of the debroglie Model

10 3. A low energy electron sitting in a ccd pixel has a wave function of approximately the following functional form: =Ce (x+d)/l at x<-d =C between x=-d and x=d =Ce -(x-d)/l at x>d Draw the wave function and calculate what C needs to be in terms of L and d in order for the wave function to be properly normalized (where d and L are positive real constants)

PH300 Spring Homework 06

PH300 Spring Homework 06 PH300 Spring 2011 Homework 06 Total Points: 30 1. (1 Point) Each week you should review both your answers and the solutions for the previous week's homework to make sure that you understand all the questions