Sharif University of Technology Physics Department. Modern Physics Spring 2016 Prof. Akhavan

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1 Sharif University of Technology Physics Department Modern Physics Spring 2016 Prof. Akhavan Problems Set #5. Due on: 03 th of April / 15 th of Farvardin. 1 Blackbody Radiation. (Required text book is Modern Physics, Krane, 3 rd edition.) (a) Writing the discrete Maxwell-Boltzmann distribution for Planck s cavity wall oscillators as N n = Ae E kt (where A is a constant to be determined), show that the condition A=N (1 e ε kt ) as in Eq. 3.38(Hint: Use n=0 e nx =(1 e x ) 1 ). (b) By taking the derivative with respect to x of the equation given in the hint, show that n=0 e x n e nx = (1 e x ) 2 n=0 N n =N gives (c) Use this result to drive Eq from Eq (d) Show that E av kt at large λ and E av 0 for small λ. (e) Show that the total energy density in a blackbody cavity is proportional to T 4 in accordance with the Stefan-Boltzmann law(it is wise to try to express Stephan's constant in terms of π, k, h, c.). 2 Blackbody Radiation. This problem is to drive the Wien displacement law. You are allowed to use the previous problem consequences. In addition, consider that the number of modes of oscillation per unit volume is n(λ)=8 π λ 4 (independent of the shape of the cavity). a (a) Show that the energy density distribution function can be written u=c λ 5 (e λ 1) 1, where C is a constant and a= hc kt.

2 (b) Show that the value of λ for which du / dλ = 0 satisfies the equation 5λ(1 e -a/λ ) = a. (c) This equation can be solved with a calculator by the trial-and-error method. One may try λ = α a, for various values of λ/a to a certain significant figures. Consider to seven digit answer: a λ m = Show that it implies λ m T = constant and calculate the value of the constant. 3 Blackbody Radiation. (a) Assuming the Sun to radiate like an ideal thermal source at a temperature of 6000 K, what is the intensity of the solar radiation emitted in the range nm to nm? (b) What fraction of the total solar radiation does this represent? (c) Use the numerical value of the Stefan-Boltzmann constant to find the numerical value of Planck s constant from Eq. 3.42(Required text book is Modern Physics, Krane, 3 rd edition.). 4 The Bohr Model. (a) Show that the speed of an electron in the n th Bohr orbit of hydrogen is αc n, where α is the fine structure constant, equal to e 2 4 πε 0 ħ c. (b) What would be the speed in a hydrogen-like atom with a nuclear charge of Ze? (c) Calculate the energies and speeds of electrons in circular Bohr orbits in a hydrogen-like atom using the relativistic expressions for kinetic energy and momentum. (d) The electron-positron pair that was discussed in can form a hydrogen-like system called positronium. Calculate the energies of the three lowest states and the wavelength of the Lyman lines(detection of those lines is a signature of positronium formation.). 5 The Bohr Model. Consider the electron-nucleus system. The electron of mass m and the nucleus of mass M must

3 both move so that the center of mass(cm) remains fixed in space. Carry through the details of the Bohr calculation for a nucleus of finite mass and: (a) Obtain the following equation: R M = m M m + M e 4 4 π c ħ 3, 1 λ = R M Z 2 ( 1 n f 2 1 n i 2 ) (b) Make a graph of Rydberg constant as a function of M/m. What is the maximum amount of it? Speak over the limits. (c) Calculate the speed of the proton in a grand state hydrogen atom. 6 The Bohr Model. An alternative development of the Bohr theory begins by assuming that the stationary states are those for which the circumference of the orbit is an integral number of de Broglie wavelengths. (a) Show that this condition leads to standing de Broglie waves around the orbit, and gives the angular momentum condition, Eq. 6.26, used in the Bohr theory(required text book is Modern Physics, Krane, 3 rd edition.). In the ground state of the hydrogen atom, according to Bohr's model, what are: (b) the quantum number, (c) the orbit radius, (d) the angular momentum, (e) the linear momentum, (f) the angular velocity, (g) the linear speed, (h) the force on the electron, (i) the acceleration of the electron, (j) the kinetic energy, (k) the potential energy, (l) and the total energy? How does the quantities (c) to (l) vary with the quantum number? 7 Line Spectra. (a) The shortest wavelength of the hydrogen Lyman series is nm. Find the three longest wavelengths in this series. (b) Use the Bohr theory to find the series wavelength limits of the Lyman and Paschen series of hydrogen. (c) The following wavelengths are found among the many radiations emitted by singly ionized helium: nm, nm, nm, nm. If we group the transitions in helium as we did in hydrogen by identifying the final state n 0 and initial state n, to which series does each transition belong?

4 8 Line Spectra. Three isotopes of hydrogen occur in nature; ordinary hydrogen, deuterium, and tritium. Their nuclei consist of, respectively, 1 proton, 1 proton and 1 neutron(deuteron), and 1 proton and 2 neutrons(triton)(required text book is Modern Physics, Krane, 3 rd edition.). (a) Use following Equation to determine Rydberg constants for deuterium and tritium. R = μ k2 e 4 4 π c ħ 3 = m k2 e 4 4 π c ħ 3 ( m /M ) = R ( m/ M ) (b) Determine the wavelength difference between the Balmer a lines of deuterium and tritium. (c) Determine the wavelength difference between the Balmer a lines of hydrogen and tritium. (d) Show that a small change in the reduced mass of the electron produces a small change in a spectral line given by Δλ λ = Δμ μ. Use this to calculate the difference Δλ in the Balmer red line λ=656.3 nm between hydrogen and deuterium, which has a nucleus with twice the mass of hydrogen. 9 Franck-Hertz Experiment. Consider the Franck-Hertz experiment with Hg vapor in the tube and the voltage between the cathode and the grid equal to 4.0 V, that is, not enough to for the electrons to excite the Hg atom s first excited state. Therefore, the electron Hg atom collisions are elastic. (a) If the kinetic energy of the electron is E k, show that the maximum kinetic energy that a recoiling Hg atom can have is approximately 4mE k /M, where M is the Hg atom mass. (b) What is the approximate maximum kinetic energy that can be lost by an electron with E k = 2.5 ev? (c) The first excited state of sodium decays to the ground state by emitting a photon of wavelength 590 nm. If sodium vapor is used for the Franck-Hertz experiment, at what voltage will the first current drop be recorded? 10 Franck-Hertz Experiment. Electrons in the Franck-Hertz tube can also have elastic collisions with the Hg atoms. If such a collision is a head-on, what fraction of its initial kinetic energy will an electron lose, assuming the Hg atom to be at rest? If the collision is not head-on, will the fractional loss be greater or less than this?

5 11 The Bohr Model. (a) The current i due to a charge q moving in a circle with frequency f rev is q f rev. Find the current due to the electron in the first Bohr orbit. (b) The magnetic moment of a current loop is ia, where A is the area of the loop. Find the magnetic moment of the electron in the first Bohr orbit in units A-m2. This magnetic moment is called a Bohr magneton. (c) Derive a relation connecting the frequency of the electromagnetic radiation, emitted in a transition between two states of a Bohr atom, and the orbital frequencies of the electron in these states. Study this relation in the limit of large quantum numbers, and comment on its correspondence with the predictions of classical physics. 12 The Bohr Model. (a) Assume the angular momentum of the earth of mass 6.0 E24 kg due to its motion around the sun at radius 1.5 E11 to be quantized according to Bohr's relation L = nh/2π. What is the value of the quantum number n? Could such quantization be detected? (b) If electric charge did not exist and electrons were bound to protons by the gravitational force to form hydrogen, derive the corresponding expressions for a 0 and E n,and compute the energy and frequency of the H α line(which is the second line of the Balmer series predicted by Bohr s model. The H α line is emitted in the transition from n i =4 to n f =2.) and the limit of the Balmer series. Compare these with the corresponding quantities for real hydrogen. 13 Questions. (a) For the Bohr hydrogen atom orbits, the potential energy is negative and greater in magnitude than the kinetic energy. What does this imply? (b) If only lines in the absorption spectrum of the hydrogen need to be calculated, how would you modify Eq to obtain them? (c) On emitting a photon, the hydrogen atom recoils to conserve momentum. Explain the fact that the energy of the emitted photon is less than the energy difference between the energy levels involved in the emission process. (d) In the figure below(on the next page), you can see the result Franck-Hertz experiment using mercury vapor. Why in the structure of the Franck-Hertz current versus curve, not sharp?

6 (e) What examples of degeneracy in classical physics, other than planetary motion, can you think of? (f) According to classical mechanism, an electron moving in an atom should be able to do so with any angular momentum whatever. According to Bohr's theory of the hydrogen atom, however, the angular momentum is quantized to L = nh/2π. Can the correspondence principle reconcile these two statements? 14 General Problem. When an atom emits a photon in a transition from a state of energy E 1 to a state of energy E 2, the photon energy is not precisely equal to E 1 E 2. Conservation of momentum requires that the atom must recoil, and so some energy must go into recoil kinetic energy K recoil. Show that K recoil (E 1 E 2 ) 2 / 2Mc 2 where M is the mass of the atom. Evaluate this recoil energy for the n = 2 to n = 1 transition of hydrogen. Compare the wavelength of the light when the recoil is taken into account to the wavelength without accounting for recoil. 15 General Problem. In a muonic atom, the electron is replaced by a negatively charged particle called the muon. The muon mass is 207 times the electron mass. (a) What is the radius of the first Bohr orbit of a muonic lead(z = 82) atom? (b) Calculate the magnitude of the lowest energy state for this atom. (c) Ignoring the correction for finite nuclear mass, what is the shortest wavelength of the Lyman series in a muonic hydrogen atom? In what region of the electromagnetic spectrum does this belong? (d) How large is the correction for the finite nuclear mass in this case(hint: See the discussion at the beginning of Section 6.8 of Modern Physics, Krane, 3 rd edition.)?

is the minimum stopping potential for which the current between the plates reduces to zero.

is the minimum stopping potential for which the current between the plates reduces to zero. Module 1 :Quantum Mechanics Chapter 2 : Introduction to Quantum ideas Introduction to Quantum ideas We will now consider some experiments and their implications, which introduce us to quantum ideas. The