# 2. 빈공간 (empty space) 에서질량이 0 인 real photon이 pair production을할수없음을보이시오. photoelectron current 와 retarding potential 의관계를그래프를이용해설명하시오.

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1 현대물리학 ( 김충선교수님 ) 2 차시험 월요일 1. A photon whose energy equals the rest mass of the electron undergoes a Compton collision with an electron at rest. If the electron moves off at an angle of 60 with the original photon direction, what is the energy of the scattered photon? ( 3=1.8) 2. 빈공간 (empty space) 에서질량이 0 인 real photon이 pair production을할수없음을보이시오. 3. 아래는광전효과를보기위한실험장비입니다. 실험장비에서 metal surface는 Cesium입니다. (a) 빛의주파수를일정하게하였을때 (ν`=const; hν`=3.0 ev), 빛의세기에따른 (I, 2I, 3I) photoelectron current 와 retarding potential 의관계를그래프를이용해설명하시오. (b) 빛의세기를일정하게하였을때 (I=const), 주파수에따른 (ν 0, 2ν 0, 3ν 0 ; hν 0 =1.9 ev) photoelectron current 와 retarding potential 의관계를그래프를이용해설명하시오. (c) Metal surface가각각 Cesium, Lithium, Silver 의경우에 Maximum photoelectron energy 와 frequency 와의관계를그래프를이용해설명하시오.

2 4. 입자가길이 L 인사각형박스에가둬져있는아래와같은시스템이있다. (a) 이시스템에서입자가가질수있는 De Broglie wavelength를구하고, 왜그러한결과가생기는지설명하시오. (b) 입자가가질수있는에너지를구하시오. (c) Ground state 와 first excited state 의 wavefunction 을그리시오. 5. A particle has a de Broglie wavelength of m. The rest energy of the particle is 800 kev. (h = ev s, c = m/s) (a) Find its kinetic energy. (b) Find the group velocity of its de Broglie waves. (c) Find the phase velocity of its de Broglie waves 6. The position and momentum of a 1.00-keV electron are simultaneously determined. If its position is located to within nm, what is the percentage of uncertainty in its momentum? (ħ= ev s, m e c 2 =500 kev, 10=3)

3 MODERN PHYSICS (1), 2 nd examination (Wed) (25 points for each problem, the total will be 150 points.) Prof. C.S.Kim Student ID Name 1. X-rays of wavelength 20.0pm are scattered from a target. Estimate the electron s Compton wavelength as 2pm. <Estimate h = 4.0 x ev s> ( 1 pm = m ) A. Find the wavelength of the x-rays scattered through 60.(10pts) B. Find the possible maximum wavelength present in the scattered x-rays.(5pts) C. Find the maximum kinetic energy of the recoil electrons.(10pts) 2. A particle with a kinetic energy of 2.0MeV collides with its anti-particle at rest and the two particles are annihilated. Two photons are produced; one moves in the same direction as the incoming particle and the other moves in the opposite direction. Find the energy of the photons. <rest mass of the particle = 3.0MeV> (Anti-particles have exact same characteristics to a particle except that the charge is opposite.) (25pts) **************For #3 and #4, set the values of the following constants as ************** <h = 5x10-15 ev s, c = 3x10 8 m/s, rest energy of an electron = 500keV> 3. An electron has a de Broglie wavelength of 3.00pm. Find its kinetic energy and the phase and group velocities of its de Broglie waves. (9, 8, 8 pts each) 4. A beam of 500keV electrons is directed at a crystal and diffracted electrons are found at an angle of 120 relative to the original beam. A. Do you think relativistic calculation is needed? Show your explanation. (5pts) B. What is the minimum spacing of the atomic planes of the crystal? (20pts)

4 5. Think of a particle in a 1-dimensional box of width L. A. Write down or derive the permitted energy E n and the wavelengths, and draw how the waves look like for n = 1, 2 states.(15pts) B. Calculate the ground state energy E 1 by using the uncertainty principle.(10pts) 6. Suppose that an electron lets out a photon and changes its path. < E(e1), E(e2) >> m e 0, for convenience, set c = 1 > A. Derive the mass-square, m γ (θ) 2, of the photon as a function of θ.(15pts) B. In what situation, does the mass of the photon become 0?(10pts) (i.e. Give the values of θ and Φ.)

5 The second Exam. of Modern physics (The electron mass, m e, is ev/c 2, the proton mass, m p = 10 9 ev/c 2 and c = m/s. ) 1. A very fast moving electron(e) emits an photon(γ) which is a real photon (m γ = 0) (see Figure 1). In this process, if P c >> m e c 2, what is the magnitude of θ? Figure 1: 2. The work function of a Lithium is 2.5 ev. When the surface is illuminated by light of wavelength 300 nm, the maximum photoelectron energy is 1.5 ev. Derive Planck constant from these data. Using the value of Planck constant obtained, find the maximum wavelength of light that will cause photoelectrons to be emitted from silver. (where the work function of silver is 4.7 ev.) 3. (a) Derive the compton wavelength of an electron. (See Figure 2) (b) Find the change in wavelength of 120 pm X-ray that are scattered 90 by a target electron in compton scattering. (h = ev s, pm = m) (c) Find the angle, θ, between the directions of the recoil electron and the incident photon. 1

6 Figure 2: 4. (a) Show that the phase velocity of the de Broglie waves of a particle of mass m and de Broglie wavelength λ is given by ( mcλ ) 2 v p = c 1 + (1) h (b) Compare the phase and group velocities of an proton whose de Broglie wavelength is exactly m. Between these, what is the velocity corresponds to the motion of the proton? Why do you think so? Figure 3: 5. (a) Derive energy levels of a particle in a 1-dimensional box (See Figure 3), where the particle is free particle and the potentials of walls are infinite as well as the velocity of the particle is sufficiently small (v << c). (b) If a proton is in a 1-dimensional box with 0.5 nm across, find its permitted energies and plot its wave function of the first excited state simply. The Planck constant h = ev s. 6. The position and momentum of a 0.90-keV electron are simultaneously determined. If its position is located to within m, what is the percentage of uncertainty in its momentum? ( h = ev s) 2

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