ps1_1994

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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 5.80 Small-Molecule Spectroscopy and Dynamics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MASSACHUSETTS INSTITUTE OF TECHNOLOGY Chemistry 5.76 Spring 1994 Problem Set #1 1. (a) Make the necessary conversions in order to fill in the table: Wavelength (Å) 420 Wavenumber (cm−1 ) 100 Energy (J) Energy (kJ/mole) 490 Frequency (Hz) 8.21 × 1013 (b) Name the spectral region associated with each of the last four columns of the table. 2. A 100-W tungsten filament lamp operates at 2000 K. Assuming that the filament emits like a black­ body, what is the total power emitted between 6000 Å and 6001 Å? How many photons per second are emitted in this wavelength interval? 3. (a) What is the magnitude of the electric field for the beam of a 1mW helium-neon laser with a diameter of 1 mm? (b) How many photons per second are emitted at 6328 Å? (c) If the laser linewidth is 1 kHz, what temperature would a blackbody have to be to emit the same number of photons from an equal area over the same frequency interval as the laser? 4. The lifetime of the 32 P3/2 → 32 S1/2 transition of the Na atom at 5890 Å is measured to be 16 ns. (a) What are the Einstein A and B coefficients for the transition? (b) What is the transition dipole moment in Debye? (c) What is the peak absorption cross section for the transition in Å2 , assuming that the linewidth is determined by lifetime broadening? 5. (a) For Na atoms in a flame at 2000 K and 760 Torr pressure, calculate the peak absorption cross section (at line center) for the 32 P3/2 − 32 S1/2 transition at 5890 Å. Use 30 MHz/Torr as the pressure-broadening coefficient and the data in Problem 4. (b) If the path length in the flame is 10 cm, what concentration of Na atoms will produce an absorption (I /I0 ) of 1/e at line center? (c) Is the transition primarily Doppler or pressure broadened? (d) Convert the peak absorption cross section in cm2 to the peak molar absorption coefficient � . 6. For Ar atoms at room temperature (20◦ C) and 1 Torr pressure, estimate a collision frequency for an atom from the van der Waals radius of 1.5 Å. What is the corresponding pressure-broadening coefficient in MHz/Torr? Problem Set #1 Spring, 1994 Page 2 7. Solve the following set of linear equations using matrix methods 4 x − 3y + z = 11 2 x + y − 4z = −1 x + 2y − 2z = 1. 8. (a) Find the eigenvalues and normalized eigenvectors of the matrix � � 2 4−i A= . 4 + i −14 (b) Construct the matrix X that diagonalizes A and verify that it works. 9. Given the matrices A and B as ⎛ ⎜ −1 ⎜3 ⎜ ⎜ ⎜� ⎜ ⎜ ⎜ ⎜ A=⎜ 2 ⎜ ⎜3 ⎜ ⎜√ ⎜ ⎜ ⎜2 ⎝ 3 � √ ⎟ ⎟ ⎟ 0 1 √ 3 ⎞ 2⎟ ⎟ ⎟ 3⎟ ⎟ 2 3 1⎟ ⎟ √⎟ ⎟ 3⎟ ⎟ ⎟ ⎟ ⎟ −2⎠ 3 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ B=⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 5 3 1 √ 6 1 − 3 √2 1 √ 6 3 2 1 √ 23 1⎞ − 3 √2 ⎟ ⎟ ⎟ ⎟ ⎟ 1⎟ ⎟ √ ⎟. ⎟ 2 3⎟ ⎟ ⎟ ⎟ 11 ⎠ 6 Show that A and B commute. Find their eigenvalues and eigenvectors, and obtain a unitary trans­ formation matrix U that diagonalizes both A and B. 10. Obtain eigenvalues and eigenvectors of the matrix ⎛ ⎞ ⎜1 2α 0⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 3α ⎟ H = ⎜2α 2 + α ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ 0 3α 3 + 2α to second order in the small parameter α. 11. A particle of mass m is confined to an infinite potential box with potential � ∞, � x < 0, x > L, V ( x) = � x k 1 − L , 0 ≤ x ≤ L. Calculate the ground and fourth excited-state energies of the particle in this box using first-order perturbation theory. Obtain the ground and fourth excited-state wavefunctions to first order, and sketch their appearance. How do they differ from the corresponding unperturbed wavefunctions? ...
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