Modern Physics Exam #2



Modern Physics Exam #2 2012 Name _______________

Note that I consider a solution to a problem to be a discussion of the physical assumptions and physical principles applied not just an equation with numbers. If you can simple calculate a numerical solution for a problem, fine. But then add a few sentences of discussion so that you convey your understanding of the problem, not just of the equations. If the problem is more open-ended, then it should be obvious that you should write a clear discussion of the assumptions and physical principles that you are applying.

I expect full sentences. I expect that I should be able to read your work. Leave plenty of space so that I am not overwhelmed. Start each problem on a new piece of paper. (You may use lined or unlined paper…decide based on your normal handwriting.) Before beginning to write your solution, think about what needs to be in your solution. Make an outline of what you need to include -- I am sure that this will help you present well-organized and complete discussions of the physical principles.

Budget your time (5 questions…so just over 15 min. each…some will be shorter, others, perhaps longer.

Show what you know…even if that means redefining a question.

Mass of an electron: me = 9.11 x 10-31 kg = 0.511 MeV/c2

Mass of a proton or neutron m =1.67 x 10-27 kg

Planck constant: h = 6.63 x 10-34 Js = 4.1357 x 10-15 eVs

ħ (hbar): ħ = 1.0546 x 10-34 Js = 6.5821 x 10-16 eVs

Boltzmann constant: kB = 1.38 x 10-23 J/K

Charge of electron: e = 1.60 x 10-19 C

Eo from front cover of the text = 13.6 eV

If you want any other constants from the front covers of our text, ask and you will be given.

1. In class, we have used the Bohr’s analysis of Hydrogen to compute the energy levels, radii, and velocity of the electron’s orbit for a number of energy states.

a) Calculate the energies for the lowest four energy states of Hydrogen. (There should be four answers…please label them E1, E2, E3, E4. Make sure you show and explain your work.) Note that Eo is given on the front cover of this test so start with that. You do not need to derive it.

b) Calculate the longest wavelength for each of the Lyman, Balmer, and Pashen series. (Please label λL, λB, and λP and in your solution, clearly show how E1, E2, E3, E4 from part a are used in your calculations.)

c) State if these are infrared, visible, or ultraviolet. (They will not all have the same answer…and you should show your understanding of these terms by stating what wavelengths encompass the different ranges.)

d) What is the angular momentum of an electron in each of the four lowest energy states? (There should be four answers…please label them L1, L2, L3, L4.)

2. In class we found that solutions of the Schrödinger equation for an infinite potential well where the potential is infinite everywhere but between x=0 and x=L. This test question is going to step you through this process again.

a) Step one is to write down the time-independent Schrödinger equation with the correct potential. Since the wavefunction is zero everywhere but in the well, write down only the differential equation for in the well.

b) The general solution for the correct answer in part a) would be ( = Acos(kx) + Bsin(kx). Now let’s clean up some things in this general solution. Use boundary conditions and Normalization conditions to find values for A, B, and k. Make your work very clear, and use a sentence or two to communicate your understanding of the mathematics for each term. When you are all done you will have a wavefunction as a function of x and L.

d) To find the energies of the different modes, plug your wavefunction into the Schrödinger equation and solve for En. Show this work clearly.

e) Now assume that L = 0.100 nm. Calculate the first 5 energies of an electron stuck in this well.

f) What is the longest wavelength of light given off that would correspond to a “Pachen-like” transition of this electron in this well.

3. A person is probably not a particularly good blackbody radiator…but for this problem, let’s assume that they are a good blackbody radiator.

a) At what wavelength ( will the human body radiate the maximum radiation?

b) For that same wavelength, what is the corresponding frequency (in Hz) and energy (in eV) of the photons? (Note if you have no ideas what answer is correct for part a, it would be wise to make an educated guess and make one up so that you could calculate the frequency and energy in this part. State clearly what value you are using for the frequency.)

c) Estimate the total power radiated by a person of medium build (assume an area given by a cylinder of 175-cm height and 13-cm radius).

d) Using your answer to c), compare the energy radiated by a person in one day with the energy intake of a 2000-Cal diet.

Optional: If you do not know what you are doing on this question, you should try to show your understanding of blackbody radiation, the shape of Planck’s curve, that there is a peak at a specific wavelength, how you would figure out where that peak is located, that the energy per unit area at a specific temperature is proportional to that temperature to the fourth, …etc. But if you were on the general right track with part a through c, you probably do not need to do this.

4. Look at the attached graph carefully. The experiment was the photoelectric effect. Light of different frequencies were shown on a surface of Sodium (Na). The equation above the graph is an equation of a curve that best fits the data represented on the graph.

[pic]

Though I have not included the units in this equation, the y-axis has units of Volts and the x-axis has units of Hertz. In this equation, y is the stopping potential and x is the frequency of light used.

a) From this data, what is the value of Planck’s constant (clearly there is some experimental error since it is not exactly the same as we know it from our text)? Please remember to include your units with all of your work and answers. (For the rest of the questions in this problem, please use the value you have found here for your value of h.)

b) Based on the data given, what is the work function of the Na used in this experiment?

c) If you pointed a green pointer laser (which produces a beam of light with wavelength 532 nm at this material (Na), what would be the maximum speed of the ejected electrons?

d) If you pointed a HeNe laser (633 nm) at the material, what would be the maximum speed of the ejected electrons.

5) Draw the energy eigenfunction (wavefunction) for n=4 for a quanton whose potential energy function is shown in the diagram below. Note that the height of the fourth energy level is marked on the graph.

a) When making your drawing of the wavefunction, include x=0 and x=L markings. Also include sentences, or a bulleted list, that explain each feature you are trying to represent in your drawing.

b) On your drawing of the wavefunction, indicate the locations where you are most likely and least likely to find the quanton.

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[pic]

[pic]

E4

x=L

x=0

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