Harmonic Oscillations / Complex Numbers


 Francine Gallagher
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1 Harmonic Ocillation / Complex Number Overview and Motivation: Probably the ingle mot important problem in all of phyic i the imple harmonic ocillator. It can be tudied claically or uantum mechanically, with or without damping, and with or without a driving force. A we hall hortly ee, an array of coupled ocillator i the phyical bai of wave phenomena, the overarching ubject of thi coure. In thi lecture we will ee how the differential euation that decribe the imple harmonic ocillator naturally arie in a claicalmechanic etting. We will then look at everal (euivalent) way to write down the olution to thi differential euation. Key Mathematic: We will gain ome experience with the euation of motion of a claical harmonic ocillator, ee a phyic application of Taylorerie expanion, and review complex number. I. Harmonic Ocillation The frehmanphyic concept of an (undamped, undriven) harmonic ocillator (HO) i omething like the following picture, an object with ma m attached to an (immovable) wall with a pring with pring contant k. (There i no gravity here; only the pring provide any force on the object.) k m = 0 Note that the ocillator ha only two parameter, the ma m and pring contant k. Auming that the ma i contrained to move in the horizontal direction, it diplacement (away from euilibrium) a a function of time t can be written a () t = Bin ( ω t + φ), () where the angular freuency ω depend upon the ocillator parameter m and k via the relation ω = k m. The amplitude B and the phae φ do not depend upon m and k, but rather depend upon the initial condition (the initial diplacement ( 0) D M Riffe  /4/03
2 d and initial velocity ( 0) = & ( 0) of the object). Note that the term initial condition dt i a technical term that generally refer to the minimal pecification needed to decribe the tate of the ytem at time t = 0. You hould remember that the angular freuency i related to the (plain old) freuency ν via ω = πν and that the freuency ν and period T are related via ν = T. Now becaue the ine and coine function are really the ame function (but with jut a hift in their argument by ± π ) we can alo write E. () a () t = B co ( ω t +ψ ), () where the amplitude B i the ame, but the phae ψ = φ π (for the ame initial condition). It probably i not obviou (yet), but we can alo write E. () a ω ( t) = D co ( t) + E in( ωt), (3) where the amplitude D and E depend upon the initial condition of the ocillator. Note that the term harmonic function imply mean a ine or coine function. Note alo that all three form of the diplacement each have two parameter that depend upon the initial condition. II. Claical Origin of Harmonic Ocillation A. The Harmonic Potential The harmonic motion of the claical ocillator illutrated above come about becaue of the nature of the pring force (which i the only force and thu the net force) on the ma, which can be written a F ( ) = k. (4) Becaue the pring force i conervative, F can be derived from a potential energy function V ( ) via the general (in D) relationhip F dv ( ) ( ) =. (5) d A potential energy function for the pring that give rie to E. (4) for the pring force i D M Riffe  /4/03
3 V ( ) = k (6) (do the math!). Let' ee how the potential energy repreented by E. (6) arie in a rather general way. Let' conider an object contrained to move in one dimenion (like the ocillator above). However, in thi cae all we know i that the potential energy function ha at leat one local minimum, a illutrated in the following graph of V ( ) v. V ( ) V = 0 = 0 Let' now aume that the ma i located near the potential energy minimum on the right ide of the graph and that it energy i uch that it doe not move very far away from thi minimum. Jut becaue we can, let' alo aume that thi minimum define where = 0 and V = 0, a hown in the picture. Now here i where ome math come in. If the ma doe not move very far away from = 0 then we are only intereted in motion for mall. Let' ee what the potential energy function look like in thi cae. For mall it make ene to expand the function V ( ) in a Taylor erie 6 3 V ( ) = V ( 0) + V ( 0) + V ( 0) + V ( 0) +..., (7) where, e.g., V ( 0) i the firt derivative of V evaluated at = 0. Now the rh of E. (7) ha an infinite number of term and o i generally uite complicated, but often only one of thee term i important. Let' look at each term in order. The firt term V ( 0) i zero becaue we defined the potential energy at thi minimum to be zero. So far o good. The next term i alo zero becaue the lope of the function V ( ) i alo zero at D M Riffe 3 /4/03
4 the minimum. The next term i not zero and neither, in general, are any of the other. However, if i mall enough, thee other term are negligible compared to the third (uadratic) term. Thu, if the object' motion i ufficiently cloe to the minimum in potential energy, then we have V ( ) V ( 0). (8) So thi i pretty cool. Even though we have no idea what the potential energy function i like, except that it ha a minimum omewhere, we ee that if the object i moving ufficiently cloe to that minimum, then the potential energy i the ame a for a ma attached to a pring where the effective pring contant k i imply the curvature V evaluated at the potential energy minimum ( = 0 ), V ( 0). Euation (8) i known a the harmonic approximation to the potential V ( ) (near the minimum). B. Harmonic Ocillator Euation of Motion OK, o we ee that the potential energy near a minimum i euivalent to the potential energy of an harmonic ocillator. If we happen to know how an harmonic ocillator behave, then we know how our ma will behave near the minimum. But let' aume for the moment that we know nothing about the pecific of a harmonic ocillator. Where do we go from here to determine the motion of the ma near the minimum? Well, a in mot claical mechanic problem we ue Newton' econd law, which i generally written (for D motion) a Fnet a =, (9) m where a i the acceleration of the object and F net i the net force (i.e., um of all the force) on the object. In the cae at hand, in which the object' acceleration i d dt = & and the net force come only from the potential energy (near the minimum) E. (9) become In phyic we are often intereted in comparing term in expreion uch a E. (7), and we often ue the comparator >> and << to compare thee term. Thee two comparator actually mean < 0 and > 0, repectively. For example, a >> b indicate that a > 0b. Exception can occur. If the potential minimum i o flat that V ( 0 ) = 0 then the third term will be zero and it will be ome higherorder term() that determine() the motion near the potentialenergy minimum. The object will ocillate, but it will not ocillate harmonically. D M Riffe 4 /4/03
5 V ( 0) & =, (0) m or, identifying V ( 0) a the pring contant k we can write E. (0) a k & + = 0. () m Euation () i known a the euation of motion for an harmonic ocillator. Generally, the euation of motion for an object i the pecific application of Newton' econd law to that object. Alo uite generally, the claical euation of motion i a differential euation uch a E. (). A we hall hortly ee, E. () along with the initial condition ( 0) and & ( 0) completely pecify the motion of the object near the potential energy minimum. Note that two initial condition are needed becaue E. () i a econdorder euation. Let' take few econd to claify thi differential euation. It i econd order becaue the highet derivative i econd order. It i ordinary becaue the derivative are only with repect to one variable ( t ). It i homogeneou becaue or it derivative appear in every term, and it i linear becaue and it derivative appear eparately and linearly in each term (where they appear). An major coneuence of the homogeneity and linearity i that linear combination of olution to E. () are alo olution. Thi fact will be utilized extenively throughout thi coure. C. HO Initial Value Problem The olution to E. () can be written mot generally a either E. (), E. (), or E. (3) (where ω = m ). Let' conider E. (3) k ω ( t) = D co ( t) + E in( ωt), (3) and ee that indeed the contant D and E are determined by the initial condition. By applying the initial condition we are olving the initial value problem (IVP) for the HO. Setting t = 0 in E. (3) we have ( 0 ) = D () Similarly, taking the time derivative of E. (3) ω & ( t) = Din( ωt) + ωe co( ωt) (3) D M Riffe 5 /4/03
6 and again etting t = 0 give u &( 0 ) = E. (4) ω Hence, the general olution to the (undamped, undriven) harmonic ocillator problem can be written a () ( ) ( ) ( ) t ω & 0 = 0 co t + in( ωt). (5) ω To ummarize, for a given et of initial condition ( 0) and & ( 0) olution to E. (), the harmonic ocillator euation of motion., E. (5) i the III. Complex Number In our dicuion o far, all uantitie are real number (with poibly ome unit, uch a = 3 cm). However, when dealing with harmonic ocillator and wave phenomena, it i often ueful to make ue of complex number, o let' briefly review ome fact regarding complex number. The key definition aociated with complex number i the uare root of, known a i. That i, i =. From thi all ele follow. Any complex number z can alway be repreented in the form z = x + iy, (6) where x and y are both real number. Common notation for the real and imaginary part of z are x = Re( z) and y = Im( z). It i alo often convenient to repreent a complex number a a point in the complex plane, in which the x coordinate i the real part of z and the y coordinate i the imaginary part of z, a illutrated by the picture on the following page. A thi can be inferred from thi picture, we can alo ue polar coordinate r and θ to repreent a complex number a = r co( θ ) + ir in( θ ) = r[ co( θ ) i in( θ )]. (7) z + Uing the infamou Euler relation (which you hould never forget!) D M Riffe 6 /4/03
7 Imaginary Axi ( y ) z = x + i y y r θ x Real Axi ( x ) z = x i y = co( θ ) i in( θ ) (8) θ e i + we ee that a complex number can alo be written a iθ z = re. (9) The lat important definition aociated with complex number i the complex conjugate of z defined a z * = x iy. (0) A i apparent in the diagram above, thi amount to a reflection about the real ( x ) axi. Note the following relationhip: z + z z z i * * = x = Re = y = Im ( z), () ( z), and D M Riffe 7 /4/03
8 zz = * = x + y r. () Note alo the abolute value of a complex number, which i eual to r, i given by z r = * = zz. (3) IV. Complex Repreentation of Harmonic Ocillator Solution Becaue the HO euation of motion [E. ()] i linear and homogeneou, linear combination of olution are alo olution. Thee linear combination can be complex combination. For example, becaue co ( ω t) and in ( ω t) are both olution to E. () (we are not worrying about any particular initial condition at the moment), another olution to E. () i the complex linear combination t ( t) ( t) i ( t) e i ω = α co ω + α in ω = α, (4) where α i ome complex number. So what i the point here? Well, a we hall ee a we go along, it i often convenient to work with complex repreentation of olution to the harmonic ocillator euation of motion (or to the wave euation that we will be dealing with later). So what doe it mean to have a complex diplacement? Nothing, really a diplacement cannot be complex, it i indeed a real uantity. So if we are dealing with a complex olution, what do we do to get a phyical (real) anwer? There are at leat three approache: () The firt approach i to, up front, make the olution manifetly real. For example, let' ay you want to work with the general complex olution t i t () e i ω ω t = α + βe, (5) where α and β are complex number. You can impoe the condition β = α *, which reult in () t being real. () Another approach i to imply work with the complex olution until you need to doing omething uch a impoe the initial condition. Then, for example, if we are working with the form in E. (5), the initial condition might be omething like Re [ ( 0) ] = A, Im [ ( 0) ] = 0, Re [ &( 0) ] = B, and Im [ & ( 0) ] = 0. Thee four condition would then determine the four unknown, the real and imaginary part of α and β (and again would reult in () t being real). D M Riffe 8 /4/03
9 (3) A third approach i to imply ue the real part of a olution when omething like the initial condition are needed, and then impoe the initial condition. (Alternatively, becaue the imaginary part of the olution i alo a real number, you could ue the imaginary part of the olution if you were o inclined.) Note that thee different approache work becaue the harmonic ocillator euation of motion i real and linear. We will not worry about the detail of thi at the moment, but if you are intereted you can read more about thi on pp. 90 of Dr. Torre text, Foundation of Wave Phenomena (FWP). Exercie *. Show that E. () and () are olution to E. (), the euation of motion for the harmonic ocillator. *. Auming the form of E. () for the olution to E. (), find the value of B and φ in term of the initial condition ( 0) and & ( 0). *.3 Auming the form of E. () for the olution to E. (), find the value of B and ψ in term of the initial condition ( 0) and & ( 0). [Thi exercie along with Exercie. how that E. () and () are euivalent.] **.4 Quadratic approximation to a particular potential energy. Conider the potential energy function V ( ) = co( κ ), where κ i ome poitive contant. (a) Carefully (!) plot (uing a computer math program uch a Mathcad, Maple, etc.) thi function for π < κ < π (For implicity in making thi and the following graph you may et κ to ) Make ure that your axe are carefully labeled on thi and all other graph. (b) Find the force F ( ) aociated with thi potentialenergy function (for arbitrary poitive κ ). (c) Taylor erie expand the potentialenergy function about the point = 0 ; keep all term up to the 4 term. (d) What i the effective pring contant k aociated with thi potentialenergy function? (e) What condition on i neceary o that the 4 term (in the Taylor erie expanion) i much maller (<< ) that the (harmonic) term? (See footnote on p. 4.) (f) Replot the original function V ( ) = co( κ ) and the harmonic approximation to thi function from π < κ < π. Baed on thi graph, for what value of κ do you expect the harmonic approximation to be valid? How doe thi compare to your anwer in (e)? D M Riffe 9 /4/03
10 θ **.5 The Euler relation. The Euler relation e i = co( θ ) + i in( θ ) can be hown to be true by comparing, term by term, the Taylor erie of both ide of the relation. Calculate the firt five Taylor erie term (about θ = 0, o thee will be the θ 0, θ, θ 3 4 5, θ, θ, and θ term) of each ide of the euation and how that they are euivalent. *.6 Write the following complex expreion in the form x + iy (a) 5e i ( π 4) (b) ( 5 4i)( 3 + i) (c) 6 7i iπ i( π ) (d) e 6e *.7 Write the expreion in Exercie.6 in the form Ae iθ, where A and θ are both real. iωt iωt *.8 Find the real part of () t = αe + α e. Here α i complex. Write your olution in term of Re ( α ), Im ( α ), in ( ω t), and co ( ω t). *.9 Complex olution and initial condition t i t (a) Show that () e i ω ω t = α + βe i a olution to E. (), the harmonic ocillator euation of motion. (b) Auming that α and β are complex, find α and β in term of the initial condition ( 0) and & ( 0). (Do thi uing one of the three method dicued on p. 8.) D M Riffe 0 /4/03
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