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Chapter 05 Vibrational Motion Chapter 05 Vibrational Motion P. J. Grandinetti Chem. 4300 P. J. Grandinetti Chapter 05: Vibrational Motion Simple Harmonic Oscillator Simplest model for harmonic oscillator: mass attached to one end of spring while other end is held fixed m -x 0 +x Mass at x = 0 corresponds to equilibrium position x is displacement from equilibrium. Assume no friction and spring has no mass. P. J. Grandinetti Chapter 05: Vibrational Motion Simple Harmonic Oscillator What happens? An oscillation. Pull mass and let go. m Mass at x = 0 corresponds to equilibrium -x 0 +x position x is displacement from equilibrium. Assume no friction and spring has no Time for 1 complete cycle is T. mass. How do we get and solve equation of motion for this system? P. J. Grandinetti Chapter 05: Vibrational Motion Simple Harmonic Oscillator Hooke’s law 휅 For small displacements from equilibrium restoring force is F = * fx 휅 where f is force constant for spring. 휅 Use Newton’s 2nd law: F = ma = * fx ̈ 휅 to obtain differential equation of motion: mx.t/ + fx.t/ = 0 Propose solution x.t/ = A cos.!t + 휙/;x ̇ .t/ = A! sin.!t + 휙/; ẍ.t/ = *A!2 cos.!t + 휙/ ( ) Substitute into differential equation: 휅 * m!2 A cos.!t + 휙/ = 0 f ù !2 휅 ! ! 휅 Satisfy solution for all t by making m = f through definition: = 0 = f_m P. J. Grandinetti Chapter 05: Vibrational Motion Simple Harmonic Oscillator Solution is x.t/ = A cos.! t + 휙/ 0 ù ! ≡ ! ! 휅 where 0 natural oscillation frequency given by = 0 = f_m Velocity of mass is ̇ ! ! 휙 v.t/ = x.t/ = * 0A sin. 0t + / Make x.t/ and ẋ .t/ equations satisfy initial conditions of x.t = 0/ = A and ẋ .t = 0/ = 0 by setting 휙 = 0 and A = x.0/ to get final solution ! ! ! x.t/ = x.0/ cos 0t and v.t/ = * 0x.0/ sin. 0t/ P. J. Grandinetti Chapter 05: Vibrational Motion Simple Harmonic Oscillator – Other trial solutions ! 휙 ̈ 휅 Instead of using x.t/ = A cos. t + / as trial solution for mx.t/ + fx.t/ = 0 Í Could have used x.t/ = A cos !t + B sin !t: Í Could also have used complex variables: ★ replace x.t/ with a complex variable, z.t/, ̈ 휅 mz.t/ + fz.t/ = 0 ★ Make an initial guess of z.t/ = Aei.!t+휙/ or z.t/ = Aei!t + Be*i!t ★ Obtain real solution by taking real part of complex solution, z.t/. P. J. Grandinetti Chapter 05: Vibrational Motion Complex Variables Review P. J. Grandinetti Chapter 05: Vibrational Motion Complex Variables Review Complex variables are a mathematical tool that simplifies equations describing oscillations. Consider the 2D motion of this vector. How would you describe this mathematically? You probably would suggest: x.t/ = r cos !t and y.t/ = r sin !t P. J. Grandinetti Chapter 05: Vibrational Motion Complex Variables Review With complex notation we combine two equations into one Start with x.t/ = r cos !t and y.t/ = r sin !t First we define the square root of *1 as ù if i = *1 then i2 = *1 Second we define complex variable z as z = x + iy x is real part and y is imaginary part of z. Two circular motion equations become one circular motion equation z.t/ = r cos !t + ir sin !t P. J. Grandinetti Chapter 05: Vibrational Motion Complex Variables Review Euler’s formula In 1748 Euler showed that ei휃 = cos 휃 + i sin 휃 Euler’s formula With Euler’s formula z.t/ = r cos !t + ir sin !t becomes z.t/ = rei!t ! ! ! ! Calculate the product .cos at + i sin at/.cos bt + i sin bt/. ! ! ! ! ! ! ! ! ! ! ! ! .cos at+i sin at/.cos bt+i sin bt/ = cos at cos bt+i cos at sin bt+i sin at cos bt*sin at sin bt 2 cos 휃 cos 휙 = cos.휃 * 휙/ + cos.휃 + 휙/ 2 sin 휃 sin 휙 = cos.휃 * 휙/ * cos.휃 + 휙/ 2 sin 휃 cos 휙 = sin.휃 + 휙/ + sin.휃 * 휙/ 2 cos 휃 sin 휙 = sin.휃 + 휙/ * sin.휃 * 휙/ ! ! ! ! ! ! ! ! .cos at + i sin at/.cos bt + i sin bt/ = cos. a + b/t + i sin. a + b/t Using Euler’s formula: ! ! ! ! ! ! ! ! i at i bt i. a+ b/t ! ! ! ! .cos at + i sin at/.cos bt + i sin bt/ = e e = e = cos. a + b/t + i sin. a + b/t P. J. Grandinetti Chapter 05: Vibrational Motion Complex Variables Review Any complex number can be written in the form z = x + iy = |z|ei휃 where |z| is the magnitude of the complex number ù |z| = x2 + y2 and 휃 is the argument of the complex number tan 휃 = y_x P. J. Grandinetti Chapter 05: Vibrational Motion Complex Variables Review Complex Conjugate Complex conjugate, z<, of z is obtained by changing sign of imaginary part if z = x + iy then z< = x * iy 1 + 4i 1 * 4i if z = then z< = 4 * 5i 4 + 5i 1 + ia 1 * ia< if z = then z< = : 4 * ib 4 + ib< Related identitiesù Í |z| = zz<, since zz< = .x + iy/.x * iy/ = x2 + iyx * ixy + y2 = x2 + y2 = |z|2 Í 5 < < < < 5 .z1z2z3 / = z1z2z3 P. J. Grandinetti Chapter 05: Vibrational Motion Energy of simple harmonic oscillator P. J. Grandinetti Chapter 05: Vibrational Motion Energy of simple harmonic oscillator Total energy of simple harmonic oscillator is sum of kinetic and potential energy of mass and spring. Kinetic energy is given by 1 p2 K = mv2; or K = 2 2m Potential energy is energy stored in spring and equal to work done in extending and compressing spring, x x ¨ ¨ 휅 ¨ ¨ 1휅 2 V.x/ = * Ê F.x /dx = Ê fx dx = fx 0 0 2 Expression above is work associated with extending spring. For work in compressing spring just change integral limits to *x to 0 (get same result). P. J. Grandinetti Chapter 05: Vibrational Motion Energy of simple harmonic oscillator Potential energy is given by 1 V.x/ = 휅 x2 2 f V(x) 0 x P. J. Grandinetti Chapter 05: Vibrational Motion Energy of simple harmonic oscillator Although both K and V are time dependent during harmonic motion the total energy, E = K + V, remains constant. 1 1 p2.t/ 1 E = mv2.t/ + 휅 x2.t/ or E = + 휅 x2.t/ 2 2 f 2m 2 f Total Energy Potential Energy Energy Kinetic Energy P. J. Grandinetti Chapter 05: Vibrational Motion Energy of simple harmonic oscillator Substitute equation of motion into energy expression 1 1 1 1 E = mv2.t/ + 휅 x2.t/ = m!2x2.0/ sin2.! t/ + 휅 x2.0/ cos2 ! t 2 2 f 2 0 0 2 f 0 휅 !2 Recall that f = m 0 so 1 [ ] 1 E = m!2x2.0/ sin2.! t/ + cos2 ! t = m!2x2.0/ 2 0 0 0 2 0 Solve for initial amplitude, x.0/, in terms of energy u 1 2E x.0/ = ! 0 m and rewrite oscillation as v u 2E 2E x.t/ = x.0/ cos ! t = cos ! t and v.t/ = *! x.0/ sin.! t/ = * sin ! t 0 !2 0 0 0 m 0 0m P. J. Grandinetti Chapter 05: Vibrational Motion Simple Harmonic Oscillator - Phase Space Trajectory v u 2E 2E x.t/ = x.0/ cos ! t = cos ! t and v.t/ = *! x.0/ sin.! t/ = * sin ! t 0 !2 0 0 0 m 0 0m State of oscillator specified by point in phase space E = constant means oscillator state always lies on ellipse x2 v2 휅 + = 1 2E_ f 2E_m Trajectory moves in clockwise direction. Trajectories with different E can never cross. P. J. Grandinetti Chapter 05: Vibrational Motion Position probability distribution for harmonic oscillator P. J. Grandinetti Chapter 05: Vibrational Motion Position probability distribution for harmonic oscillator Scale x.t/ by initial amplitude x.0/, to obtain a function, y.t/, that oscillates between y = *1 and y = +1 ! y.t/ = x.t/_x.0/ = cos 0t Calculate normalized probability density, p.y/, for finding the mass at any scaled position between y = ,1. Probability of finding mass in interval dy at given y is proportional to time spent in dy interval, dy dt b p.y/ dy = b dt = b dt = b dy = dy dy dy ẏ b ≡ proportionality constant, and ẏ ≡ speed at a given y Use derivative of y.t/ ̇ ! ! y.t/ = * 0 sin 0t to obtain b b b b p.y/ = = = = ̇ ! ! ! 2 ! 1_2 ! 2 1_2 y.t/ * 0 sin 0t * 0.1 * cos 0t/ * 0.1 * y / P. J. Grandinetti Chapter 05: Vibrational Motion Position probability distribution for harmonic oscillator 10 b p.y/ = 8 ! 2 1_2 * 0.1 * y / 6 Normalizing probability distribution gives 4 m 1 p.y/ = 휋.1 * y2/1_2 2-x 0 +x 0 -1.0 -0.5 0.0 0.5 1.0 Mass spends majority of time at maximum excursions, that is, turning points where velocity is slowest and changes sign. P. J. Grandinetti Chapter 05: Vibrational Motion Diatomic molecule vibration as Harmonic oscillator P. J. Grandinetti Chapter 05: Vibrational Motion Diatomic molecule vibration as Harmonic oscillator Dashed line is harmonic oscillator potential. Solid line is Morse potential. V.r/ has minimum at re—where restoring force is zero. < > V.r/ causes repulsive force at r re and attractive force at r re. < > V.r/ increases steeply at r re but levels out to constant at r re. At r → ∞ there is no attractive force as V.r/ has a slope of zero. P. J. Grandinetti Chapter 05: Vibrational Motion Diatomic molecule vibration as Harmonic oscillator Taylor series expansion of V.r/ about equilibrium bond length, r = re, gives >0 dV.r/ 1 d2V.r / 1 d3V.r / V.r/ ù V.r / + e .r * r / + e .r * r /2 + e .r * r /3 + 5 e dr e 2@ dr2 e 3@ dr3 e V.re/ is the potential energy at equilibrium bond length.
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