Chaper Q1 Inroducion o Quanum Mechanics End of 19 h Cenury only a few loose ends o wrap up. Led o Relaiviy which you learned abou las quarer Led o Quanum Mechanics (1920 s-30 s and beyond) Behavior of aomic and subaomic world Newon s Laws don hold Foundaion of mos Physics Research ha occurs hese days. Basic Ideas behind Quanum Mechanics Small Paricles behave like waves No localized Inerference effecs ( more soon) Observaions aler he sysem. o Basic quaniies x, p o In QM if you ry o deermine x you will effec p and visa versa. Implies bound sysems have quanized saes (Aomic Specra) Waves (EM) behave like paricles. We need o undersand Classical wave firs. 3/28/2004 H133 Spring 2004 1
Classical Waves Basics Demo Waves are a disurbance ha ravel in a medium. e.g. waer waves Noe: Paricles in he medium are no raveling along wih he wave. We will look a 1-D wave o begin. Two ypes of waves Tension Wave (someime called ransverse) e.g. wave on a sring. Equilibrium Posiion Compress ional Wave (someimes call longiudinal) e.g. sound wave Tube 3/28/2004 H133 Spring 2004 2
Mahemaical Represenaion Funcion of posiion and ime e.g. f f ( x, ) ( x, ) = Ae ( x b ) 2σ 2 2 = 20s = 10s = 0s x = 0 Also common o hink abou funcions wih sines and cosines more laer. x 3/28/2004 H133 Spring 2004 3
Superposiion Principle If wo waves described by f 1 (x,) and f 2 (x,) are moving in a medium, he combined wave is described by he algebraic sum of he wo waves f o (x,) = f 1 (x,)+f 2 (x,) This is a powerful saemen even hough i seems quie simple. I holds for many (bu no all) waves. For wha we sudy we will assume i holds. 3 > 2 2 > 3 1 > 0 0 = 0 3 > 2 2 > 3 1 > 0 0 = 0 3/28/2004 H133 Spring 2004 4
Q1B.1 8 Example 4 4 10 20 30 40 50 60 70 = 0 8 4 4 10 20 30 40 50 60 70 = 2s 8 4 4 10 20 30 40 50 60 70 = 3s 8 4 4 10 20 30 40 50 60 70 = 4s 8 4 4 10 20 30 40 50 60 70 = 6s 3/28/2004 H133 Spring 2004 5
Reflecion Fixed End One of he more ineresing feaures of waves is wha occurs when hey encouner a boundary. Examples: A rope ied off a an end Waves in a pool Densiy changes of a sring/rope. The issue of wha happens o waves a a boundary will also be imporan when we look a waves of paricles in QM. When waves encouner a boundary hey are parly or enirely refleced from he boundary. Le s sar by looking a wo exreme cases for boh ension and pressure waves: Compleely fixed end Compleely free end. Sring (Tension) ied down Sound in Tube (Pressure/Densiy) Pressure/Densiy fixed o environmen open end Free End Fricionless Ring Closed end Ampliude no consrained Pressure/Densiy free o be any value 3/28/2004 H133 Spring 2004 6
Reflecions Wha happens when he waves srike hese boundaries? Fixed Ends : Wave reflecs bu is invered Open End : Wave reflecs bu is no invered. See ex for argumen why his happens Inermediae Case (Mos Cases) (1) Ligh o heavy (2) Heavy o Ligh Wha is he analogy for sound waves? 3/28/2004 H133 Spring 2004 7
Sine Waves When discussing fixed (or free) boundaries where we have reflecions we can se up an ineresing phenomena called sanding waves. A sine wave raveling in he x direcion has he form f ( x, ) = Asin( k x ω + φ) Where A is he ampliude, k is he wave number and ω is he angular frequency. 2π 2π 1 k = ω = = 2π f T = λ T f By following a peak in he wave we can measure he velociy of his raveling wave. The peak is defined by having he phase = π/2. π = kx peak ω 2 Solving for x peak we find x v v peak peak peak phase π ω = + 2k k dxpeak ω ω = = 0+ = d k k 2π ω T λ = = = = λ f k 2π T λ Phase Consan : Make 0 by redefining x=0 or =0 3/28/2004 H133 Spring 2004 8
Example A sine wave raveling in he +x direcion has an ampliude of 15.0 cm, a wavelengh of 40.0 cm, and frequency of 8.00 Hz. The verical displacemen a =0, x=0 is 15.0 cm. a) k =?, T =?, ω =?, v =? b) Wha is he phase consan? 3/28/2004 H133 Spring 2004 9
Sanding Waves Wha abou a wave raveling in he x direcion? f ( x, ) = Asin( k x+ ω ) How does all his connec o sanding waves? Imagine a wave raveling in he negaive x direcion oward a fixed poin. When he wave is refleced an invered wave raveling he +x direcion. According o he superposiion principle we add he wo funcions f ( x, ) = Asin( k x+ω )? Asin( k x ω ) Wha should he sign? be? A + or a? Think abou a he fixed boundary (x=0), we hink he wave should be invered: f ( x = 0, ) = Asin( ω ) = Asin( ω ) ( Asin( ω )) = 0 Since he firs erm represens he incoming + wave and he second erm represens he ougoing invered wave he sign? mus be posiive. 3/28/2004 H133 Spring 2004 10?? Asin( ω )
Sanding Waves Now le s change he form of he superposiion of he wo waves so ha i is in a more revealing form: f ( x, ) = Asin( kx + ω) + = A { sin kx cosω + cos kxsinω + sin kx cosω cos kxsinω} = 2Asin kx cosω Asin( kx ω) Noe he srucure of his relaionship he posiion and he ime dependence has been facorized. f ( x, ) = A( x) B( ) A fixed x=x o we have f()=a(x o )cos(ω), so we have somehing ha is oscillaing wih a frequency ω and a maximum ampliude A(x o ). A fixed = o we have f(x)=sin(kx)(b( o )), so we have a wave wih wavenumber k and max. ampliude B( o ). 3/28/2004 H133 Spring 2004 11
Sanding Waves If we consider a sring of lengh L fixed a boh ends (x=0, x=l): sin(k0)= 0 sin(kl)= 0 kl=nπ n=1,2,3, So only cerain wave numbers (freq.) will allow he exisence of a sanding wave. k=n(π/l) or L=n(λ/2) n=1,2,3, This implies we mus fi an exac number of half wavelenghs beween he fixed ends. We can also express his in erms of frequency f = ω/(2π) = (ω/k)(k/2π) = nν/(2l) This concep ha only cerain frequencies of waves work, i imporan. We say his is quanized somehing ha is imporan when we ge o quanum mechanics. 3/28/2004 H133 Spring 2004 12
Sanding Waves We jus considered he case when he sring is fixed a wo ends. We can consider wo oher cases. (A) Fixed one end x=0 and oally free a x=l sin(k0)=0 sin(kl) = 1 kl=n(π/2) n=1,3,5, L=n(λ/4) and f = nν/(4l) We mus fi an odd number of ¼ wavelenghs. Noice ha for he case of 1 free end he fundamenal frequency is ½ he fundamenal frequency if boh ends were fixed. 3/28/2004 H133 Spring 2004 13
Sanding Waves (B) We could also include he case where boh ends are free. L=λ/2 L=λ L= (3/2)λ L = n (λ/2) n=1,2,3, This has he same condiions as when boh ends are fixed in place. Please Read Secions 1.6 and 1.7 you will be responsible for hem. 3/28/2004 H133 Spring 2004 14