Homework Assignments XXVI - XXXIX

PHYSICS 262

Spring, 2003 Daniel Finley

Homework Assignments XXVI - XXXIX, prior to the third Exam

There is a homework assignment due (almost) every class period!
If on paper, they should be turned in at the alphabetically-labeled box at the front of the classroom, before class begins!
If on WebAssign, they are due at 1 AM of the night of the day mentioned, which is of course actually very early in the morning of the next day.

No. XXVI: due viernes, 4 de abril

  1. There are no paper problems due this Friday.
  2. There are 2 questions and 2 problems due on WebAssign tonight.
Complete solutions to HW 26 may be accessed after they are due by clicking right here.

No. XXVII: due lunes, 7 de abril

  1. There are no paper problems due today.
  2. There are 1 (very simple) question and 3 problems, on WebAssign, due this Monday night.
Complete solutions to HW 27 may be accessed after they are due by clicking right here.

No. XXVIII: due miercoles, 9 de abril,

  1. There are no paper problems due today.
  2. There are 4 reasonably quick problems, on WebAssign, due tonight.
Complete solutions to HW 28 may be accessed after they are due by clicking right here.

No. XXIX: due viernes, 11 de abril,

  1. There is a re-exam today, in class, on time and space relationships in special relativity.
  2. There are two WebAssign problems due Saturday night (instead of the usual Friday night).
Complete solutions to HW 29 may be accessed after they are due by clicking right here.

No. XXX: due lunes, 14 de abril,

  1. There is a paper problem due today, at classtime.
    Consider the radiation from a blackbody maintained at 1000 K.
    1. What is the intensity emitted by the blackbody in the yellow part of the spectrum, which we take to be (approximately) between 580 and 590 nm. [Assume this region of the spectrum is sufficiently narrow that we may approximate the integral by simply using the value of the average wavelength over the region multiplied by the width of that region.]
    2. Do the same thing for the region in the far infrared, between 5800 and 5810 nm.
  2. There are 3 problems and 1 question due tonight on WebAssign; they all concern Heisenberg's uncertainty principle.
Complete solutions to HW 30 may be accessed after they are due by clicking right here.
Note that the WebAssign answers were wrong for the first two parts of the last problem!

No. XXXI: due miercoles, 16 de abril,

  1. There are no paper problems due today.
  2. On WebAssign there are 1 question and 3 problems from Chapter 40, all having to do with the spectral lines of the hydrogen atom.
Complete solutions to HW 31 may be accessed after they are due by clicking right here.

No. XXXII: due viernes, 18 de abril,

  1. There are 2, related, problems due on paper, at classtime Friday.
    1. Problem 39-66. What follows is not a change in content of this problem, but at least some motivation for why one should be doing it. Schrödinger's equation requires the use of complex numbers, i.e., numbers of the form a + i b, where a and b are ordinary (real) numbers, and i is the basic idea behind complex numbers, namely
      i2 = -1 .
      If is an arbitrary complex number, i.e., there are some real numbers a and b such that = a + i b, then the complex conjugate of is * = a - i b. The quantities a and b are called the real part and the imaginary part, respectively, of .
      It is of course true that nothing in the physical world in which we live is directly described by a complex number; however, in the study of Schrödinger's equation, we use the product of the solution with its complex conjugate to describe/determine probability densities. Therefore, first show that if is any complex number, then (*) () is a non-negative real number, and therefore could be used to describe the result of some physical process. The quantity (*) () is often written in the form |(*)|², and is referred to as the square of the absolute value of the complex number, .
      This justifies part (a) of the problem. Please do both parts (a) and part (b).
    2. In the text, in the paragraph prior to the one that contains Eq.(39-19), the real and imaginary parts of the exponential, eix, are given, namely
      eix = cos(x) + i sin(x) .
      1. Use this relationship to show that the absolute value of eix is +1;
      2. Use this relationship to determine the relationship between cos(3x) and various powers of cos(x) and sin(x). [Hint: take the third power of both sides of the equation.]
      3. Using this relationship, find the real and imaginary parts of the quantity given on the right hand side of Eq.(39-17); assume that the constants A and B given there are real numbers.
  2. There are two WebAssign problems due tonight, still from Ch. 39.
Complete solutions to HW 32 may be accessed after they are due by clicking right here.

No. XXXIII: due lunes, 21 de abril,

  1. There are 2 closely related paper problems, due at classtime today:
    Chapter 39, Problems 69P, 71P.
  2. On WebAssign there are 3 straightforward problems concerning probability waves, due tonight.
Complete solutions to HW 33 may be accessed after they are due by clicking right here.

No. XXXIV: due miercoles, 23 de abril,

  1. Please do the one paper problem, 39-73P, for today at classtime. Note that it builds on the paper problem from last time.
    Note also that I have no idea what it means when it says, in part (b), that you should "demonstrate that it describes the square of the amplitude of a standing matter wave." Therefore, just plotting it will be fine for that part.
  2. There are 2 problems and 3 very simple questions due on WebAssign this night.
    Please use the questions as a helpful, learning experience.
Complete solutions to HW 34 may be accessed after they are due by clicking right here.

Bonus Problem Number 4: worth 1 week's worth of homework points (as usual),
due on Wednesday, 30 April, at classtime.

Determine a method to determine the allowed energy levels for a finite, 1-dimensional well that constrains an electron, which has arbitrary width L and depth U0. We want to know both the number of allowed (quantized) energy levels, below the non-quantized ones that have energy greater than U0, and also their numerical values (above zero).
To do this, first constrain your description of the well so that it is symmetric about x=0, as was done in class; i.e., put the origin for the x-axis at the center of the well, so that its edges are at ±(L/2). Then note that the allowed wave functions may be split into two sorts: those that are even (trigonometric) functions of x, such as the cosine function that was discussed in class, and those that are odd functions of x, such as a sine function. To verify that you understood the derivation given in class that began with a cosine form for the wave function, follow carefully through the derivation that was given in class, that ended in a transcendental equation for the energies. Then, also consider a sine form for the wave function and repeat a similar derivation to obtain an appropriate transcendental equation for that case.
Then for an electron in a well of width 100 pm, use numerical methods to determine the allowed energies for depths of 10 eV, 30 eV, 150 eV, 300 eV, and, finally 450 eV. Make sure that you have determined all those levels that are allowed.

Lastly, explain why there are sometimes no solutions to allowed energies for the odd functions, based on sine functions, while there is always at least one solution for allowed energies for the even functions, based on cosine functions.
This fact is sometimes referred to, in simple English, as there must always be an allowed ground state, sometimes referred to as that state that has the zero-point energy.

A complete solution to this problem may be accessed after it is due by clicking right here.

No. XXXV: due viernes, 25 de abril,

  1. Two paper problems:
    1. Problem 40-21P,
    2. Problem 40-22P.
  2. There are 4 WebAssign problems due tonight, from Chapters 39 and 40.
Complete solutions to HW 35 may be accessed after they are due by clicking right here.

No. XXXVI: due lunes, 28 de abril,

  1. For one paper problem, please do, for classtime,
    Problem 58P from Chapter 40.
    For part (a), just some English words concerning the relation of your plot of |200(r)|² and the dot plot in the text will be quite sufficient.
    On the other hand, for that plot, do please use some sort of graphing program.
    For part (c), do remember to include the "differential volume element" in your determination of the probability density.
  2. There are 5 problems from WebAssign due tonight.
    They are all from Chapter 41, and concern the labelling of quantum states, and our measurements of spin, via the Stern-Gerlach experiment.
Complete solutions to HW 36 may be accessed after they are due by clicking right here.

No. XXXVII: due miercoles, 30 de abril,

  1. There are no paper problems due today; however, you might want to begin work on the one due Friday.
  2. There are 1 question and 3 problems due on WebAssign:
    they come from Chapter 43, and concern the decays of radioactive nuclei.
Complete solutions to HW 37 may be accessed after they are due by clicking right here.

No. XXXVIII: due viernes, 2 de mayo,

  1. There is a single paper problem that in itself is not very long, and is due on Friday at classtime.
    However, hoping that you did read the description for a "modified Stern-Gerlach apparatus" as given in Chapter 5 of (Vol. 3 of) the Feynman lectures, I have nonetheless also given a somewhat lengthy summary description here of the details of the action of any apparatus of the general nature of the one used by Stern and Gerlach, and explained in some detail how the orientation of the gradient of the magnetic field in such a device affects the outcome, comparing them to the action of an optical polaroid. The formulae I give are considerably simpler in appearance than the similar ones in Chapter 6.
    At the link above, where the problem is described, near the end there are 3 questions to be answered, labelled (i), (ii), and (iii). These constitute the actual problem.
  2. There are 4 problems involving radioactive decays, due on WebAssign, for this night.
Complete solutions to HW 38 may be accessed after they are due by clicking right here.

No. XXXIX: due lunes, 5 de mayo,

  1. There are no paper problems due today.
  2. On WebAssign, there are 1 question and 3 problems, considering nuclear fission and fusion, due tonight.
Complete solutions to HW 39 may be accessed after they are due by clicking right here.

THIRD EXAM will be on Friday, 9 May.

Coverage will be all the material we have discussed on Quantum Physics, basically Chapters 39-41 and those portions of 43 and 44 that we discussed in class. This of course excludes all skipped material, BUT also includes all the added material that was put on the class webpage, including those parts of Chs. 1-2 and 5-6 from the Feynman lectures that were discussed in class.
A listing of equations for the exam, in Acrobat format, may be accessed here. You may bring these sheets with you to the exam itself, and write on the backs if you so desire.
Solutions may be accessed here.

Back to Phys. 262 Home Page.
Last updated/modified: 14 April, 2003