Simple Harmonic Oscillation (SHO)

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1 Simple Harmonic Oscillation (SHO) Homework set 10 is due today. Still have some midterms to return. Some Material Covered today is not in the book Homework Set #11 will be available later today Classical picture of a Harmonic Oscillator Physics 2170 Fall

2 Clicker question 1 Set frequency to AD When should homework #11 be turned in? We have two more homework assignments this semester. A. Friday, Nov. 22 B. Monday, Dec. 2 Physics 2170 Fall

3 Thomas Edison Friday Facts He invented the first electric light bulb. He invented the movie camera. He invented the fluoroscope. He invented the re-chargeable battery. He had more than 1300 patents in his lifetime. Nicknamed his first two sons Dot and Dash. Physics 2170 Fall

4 More facts The first s could only be sent to someone using the same local host with two computers literally side by side. In late 1971 Raymond S Tomlinson sent the first useful to a user separated from the same host by simply implementing a minor addition to the protocol that was being used and chose to use sign in the address The Apple II had a hard drive of only 5 megabytes when it was launched. A dentist is the inventor of the electric chair. Physics 2170 Fall

5 Description of Rainbow 4/08/10 Light 5

Lunar Rainbow or Moonbow A moonbow (also known as a lunar rainbow or white rainbow) is a rainbow that occurs at night. Moonbows are relatively faint, due to the smaller amount of light from the Moon.

6 Lunar Rainbow or Moonbow A moonbow (also known as a lunar rainbow or white rainbow) is a rainbow that occurs at night. Moonbows are relatively faint, due to the smaller amount of light from the Moon. As with rainbows, they are in the opposite part of the sky from the moon. This picture was taken looking south, the Southern Cross can be seen to the left of center along with some of the brighter stars in the constellation of Centaurus. (Taken from Starry Night Skies Photography Website)

7 Double Rainbow over Lake Tahoe

8 Double Rainbow Physics 2170 Fall

9 Secondary Rainbow Physics 2170 Fall

10 Double Rainbow Picture

11 Level of Importance of states Have been talking about atoms, but want to say a few words about importance of energy levels and the physics behind them. Molecules have quantum energy levels. There are Electronic states ~ levels on order ev Vibrational states ~ levels on order ~.01 1 ev Hz Rotational states ~ levels on order ~.001 ev Hz Let s tackle rotational states first --- Physics 2170 Fall

12 Rotational States Physics 2170 Fall

13 Rotational States Molecules can rotate like classical rigid bodies subject to the constraint that angular momentum is quantized in units of ħ. We estimate the energy by E = 1 2 Iω 2 ; L = Iω; E = 1 2 L 2 I We use I = Ma 0 2 and = ħ 2 /(ke 2 m e ) where M is mass of molecule and m is mass of the electron and α =ke 2 /(ħc) called fine structureconstant 1/137 Physics 2170 Fall

14 Rotational States (cont.) Look at energy changes with strongly excited at room temperature these states are With identical nuclei are 2 2. I has to be even or odd, so steps These states also have equal energy steps in emitted photon energy. Physics 2170 Fall

15 Rotational States A complex molecule will have three principal axes, and hence, three moments of inertia to use in our quantized formula. Counting degrees of freedom, which should be equal to the number of quantum numbers needed to describe the state, we have 3 coordinates to give the position of the center of mass, 3 for the rotational state, and 3N-6 for vibrational. This formula should be modified if the molecule is too simple to have three principal axes. Physics 2170 Fall

16 Absorption Spectrum of Water Physics 2170 Fall

17 SHO Background 1 dimension The simple harmonic oscillator potential is of great importance because it can be used to satisfy a broad class of problems in which a particle is oscillating about a position of stable equilibrum. At position of stable equilibrium, V(x) must have a minimum. For any V(x) which is continuous, the shape of the potential near the minimum can be approximated by a parabola. If a particle of mass m is displaced by x 0 from equilibrium, it will oscillate with frequency ν = 1 k 2π m Physics 2170 Fall

18 Harmonic Oscillator Potential Physics 2170 Fall

19 The energy-level spacing for the harmonic oscillator is constant. Expect the energy levels to diverge less rapidly than those for the square well because the higher energy states in the harmonic oscillator have effectively larger boxes than do the lower states (that is, the more energetic the oscillator, the more widely separated are its classical turning points). The harmonic oscillator has finite zero-point energy. Physics 2170 Fall

20 The wavefunctions for the harmonic oscillator are either symmetric or antisymmetric under reflection through x=0. Physics 2170 Fall

21 Harmonic Oscillator Potential The particle has a finite probability of being found beyond the classical turning points; it penetrates the barrier. This is to be expected on the basis of earlier considerations since the barrier is not infinite at the classical turning point. In the lowest-energy state the probability distribution favors the particle being in the low-potential central region of the well, while at higher energies the distribution approaches more nearly the classical result of favoring the higher potential regions. Physics 2170 Fall

22 The wave functions of the first three states are Where ω = (k/m) ½ is the classical angular frequency, and n is the quantum number Physics 2170 Fall

23 Simple Harmonic Oscillator x ψ 2 (x) = A 2 2 b e x / 2b 2 ;E = 3 2 ω dψ 2 (x) dx 1 = A 2 b x 2 e x 2 / 2b 2 b 3 d 2 ψ 2 (x) = A dx 2 2 3x b + x 3 e x 2 / 2b 2 = A 3 b b + x 2 x 2 2 b 4 b e x / 2b 2 2m E kx 2 = 3 b + x 2 2 b 4 2m 2 E = 3 b ; mk = b 4 E = 3 2 2mb = m mk = 3 2 k m = 3 2 ω Physics 2170 Fall

24 Physics 2170 Fall

25 The time independent Schrödinger equation for a quantum harmonic oscillator is d 2 ψ dx + 2mE 2 2 ωm 2 x 2 ψ = 0 Define α mω + β 2mE 2 d 2 ψ dx 2 + β α2 x 2 [ ]ψ = 0 Physics 2170 Fall 2013 β α = 2mE 2 mω = 2E ω 25

26 Change variables to Then or QHO solved d 2 ψ dx 2 + β α2 x 2 [ ]ψ = 0 dψ dx = dψ dξ dξ dx = α dψ dξ ξ α x d 2 ψ dξ + β 2 α ξ 2 ψ = 0 d 2 ψ dx = d dψ dξ 2 dξ dx dx = α d 2 ψ dξ 2 For any finite value of E, the quantity β/α will be small compared to ξ 2 as ξ goes to, so ( ) = Be ξ 2 / 2 ;ψ( ξ) = e ξ 2 / 2 H(ξ) ψ ξ β α = 2mE 2 mω = 2E ω Physics 2170 Fall

27 QHO solved.. ψ( ξ) = Be ξ 2 / 2 ;ψ( ξ) = e ξ 2 / 2 H(ξ) The H(ξ) are functions which must at ξ ->, be slowly Varying compared to e ξ 2 / 2. To evaluate these functions: We compute derivatives and factor out e ξ 2 / 2 H + ξ 2 H 2ξ dh dξ + d 2 H dξ 2 e ξ 2 / 2 + β α H ξ 2 H = 0 d 2 H dξ 2 dh 2ξ dξ + β α 1) H = 0 Hermite s Equation Physics 2170 Fall

28 QHO solved d 2 H dh 2ξ + β 1) dξ 2 H = 0 dξ α ( ) = a k ξ k H ξ dh dξ = a 0 + a 1 ξ + a 2 ξ 2 + a 3 ξ k= 0 ka kξ k 1 a 1 + 2a 2 ξ 1 + 3a 3 ξ k=1 d 2 H dξ = (k 1)ka kξ k 2 2a (3)a 3 ξ 1 + 3(4)a 4 ξ k= 2 1 2a 2 + (β /α 1)a 0 = 0 For ξ power=0 For ξ power =1 2 3a 3 + (β /α 1 2 1)a 1 = 0 For ξ power =2 For ξ power =3 4 5a 5 + (β /α 1 2 3)a 3 = 0 3 4a 4 + (β /α 1 2 2)a 2 = 0 Physics 2170 Fall

29 QHO solved So we get a k +2 a k = β 1 2k α k +1 ( )( k + 2) If we wish to terminate the series, then β = 2k +1 α This is our constraint which quantizes the energy β α = 2mE 2 mω = 2E ω or E n = (n+1/2)ħω Physics 2170 Fall

Solving Schrödinger s Wave Equation - (2)

Chapter 14 Solving Schrödinger s Wave Equation - (2) Topics Examples of quantum mechanical tunnelling: radioactive α-decay, the ammonia molecule, tunnel diodes, the scanning tunnelling microscope The quantisation