Homework Problem Set 5 Solutions ( )2

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3. continued 8 c.) Using the analytic solution to Newton's equations for the harmonic oscillator, plot the velocity as a function of time. The velocity is the time derivative of the position, € ( ) = x ˙ ( t) = d dt x( t) v t = d dt ⎡⎡ v(0) ⎣⎣ ⎢⎢ ω sin( ω t) ⎤⎤ ⎦⎦ ⎥⎥ v( t) = v( 0) cos( ω t ) . Substituting the specific parameters for this problem, the expression for the velocity becomes v( t) = ( 8400 m/s) cos( 1.58 ×10 14 s −1 t), with the velocity in m/s and the frequency in s –1 . A plot of the velocity as a function of time is shown in the figure below. € 10000 8000 6000 4000 v(t) (m/s) 2000 0 0.0 0.1 0.2 0.3 0.4 0.5 -2000 -4000 -6000 -8000 -10000 time (ps)
4. Again consider the simple one-dimensional potential used in problems 2 and 3. Assume that the force constant k is 720 kcal mol –1 Å –2 . The reduced mass is 12.0 g/mol. The initial position is 0 Å and the initial velocity is 8400 m/s. 9 a.) Using the Verlet leapfrog algorithm, carry out two time steps to determine the position and velocity of the particle at times t 1 and t 2 . Use a step size of 1 femtosecond. The Verlet leapfrog algorithm for one particle in one dimension is given by x( t n+1 ) = x( t n ) + hv x ( t n+1/ 2 ) v x ( t n+1/ 2 ) = v x ( t n−1/ 2 ) + h m F x( t n) . The force F x is obtained from the force field, € F x = − dU dx F x = − kx . Subsituting, the velocity equation becomes € ( ) = v x ( t n−1/ 2 ) − hk m x t n v x t n+1/ 2 ( ). For n = 0, the Verlet leapfrog equations are given by v x ( t 1/ 2 ) = v x ( t −1/ 2 ) − hk m x t 0 x( t 1 ) = x( t 0 ) + h v x ( t 1/ 2 ) . The time step is h = 10 −15 s . Using the initial conditions x t 0 v x ( t −1/2 ) = v x ( 0) = 8400 € m/s, the velocity equations is The position equation then becomes € ( ) v x ( t 1/ 2 ) = v x ( t −1/ 2 ) − hk m x ( t 0) = 8400 m/s − hk m ( 0) v x ( t 1/ 2 ) = 8400 m/s. x( t 1 ) = x( t 0 ) + h v x ( t 1/ 2 ) = 0 + ( 10 −15 s) ( 8400 m/s) x( t 1 ) = 8.40 × 10 −12 m. ( ) = x( 0) = 0 and €
Page 1 and 2: Chemistry 380.37 Dr. Jean M. Standa
Page 3 and 4: 1 d.) continued 3 To complete the d
Page 5 and 6: 2. continued 5 b.) What is the kine
Page 7: 3. continued 7 b.) Using the analyt
Page 11 and 12: 4 a.) continued 11 Averaging the se
Page 13 and 14: 5a, b.) continued 13 The velocity d
Page 15 and 16: 15 6. The equipartition theorem sta

Homework Problem Set 5 Solutions ( )2

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