13. Adiabatic Invariants and Action-Angle Variables Michael Fowler

Transcription

1 3 Adiabatic Invaiants and Action-Angle Vaiables Michael Fowle Adiabatic Invaiants Imagine a paticle in one dimension oscillating back and foth in some potential he potential doesn t have to be hamonic, but it must be such as to tap the paticle, which is executing peiodic motion with peiod Now suppose we gadually change the potential, but keeping the paticle tapped hat is, the potential depends on some paameteλ, which we change gadually, meaning ove a time much geate than the time of oscillation: dλ / λ A cude demonstation is a simple pendulum with a sting of vaiable length, fo example (see figue) one hanging fom a fixed suppot, but the sting passing though a small loop that can be moved vetically to change the effective length If λ wee fixed, the system would have constant enegy E and peiod As λ is gadually changed fom outside, thee will be enegy exchange in geneal, we ll wite the Hamiltonian H( qpλ, ; ), the enegy of the system will be E ( λ ) (Of couse, E also depends on the initial enegy befoe the vaiation began) Remembe now that fom Hamilton s equations dh / = H / t, so duing the vaiation de H H dλ = = t λ It s clea fom the diagam that the enegy fed into the system as the ing moves slowly down vaies thoughout the cycle -- fo example, when the pendulum is close to vetically down, its enegy will be almost unaffected by moving the ing Moving slowly down means λ vaies vey little in one cycle of the system, we can aveage ove a cycle: de H dλ =, λ whee H = λ H λ Now Hamilton s equation dq / H / p time fo going ound one complete cycle is = means that we can eplace with / ) dq, so the

2 = = dq ) / (his won t integate to zeo, because on the etun leg both dq and q = / p) heefoe, eplacing in / λ ) as well, / λ ) dq de dλ H / p = dq / ) will be negative) Now, we assume λ, E ae vaying slowly enough that they ae close to constant ove one cycle, meaning that at a given point q on the cicuit, the momentum can be witten p p( qeλ ;, ) =, egading E, λ as constant and independent paametes (We can always adjust E at fixedλ by giving the pendulum a little push!) λ = E with espect to λ, keeping E constant (appopiate infinitesimal pushes equied!), we get, at point q on the cicuit, If we now patially diffeentiate H( qp,, ) H / λ H / λ + / p)( p/ λ) =, o p( q, λ, E) / λ =, H / p which is the integand in the numeato of ou expession fo de, so ( p / ) ( / ) de dλ λ dq E =, p E dq In the denominato, we ve eplaced / / ) by ( p ) Reaanging, his can be witten λ / E λ p de p dλ + dq = E λ λ E di =, whee I = p ( q, λ, E) dq π

3 3 I is an adiabatic invaiant: hat means it stays constant when the paametes of the system change gadually, even though the system s enegy changes Impotant! he patial deivative with espect to enegy I / E detemines the peiod of the motion: I p dq dq π = dq = = =, o E / I = ω E E H p q λ ( / ) (Note: hee is anothe connection with quantum mechanics If the system is connected to the outside wold, fo example if the obiting paticle is chaged, as it usually is, and can theefoe emit adiation, since in quantum mechanics successive action numbes I diffe by integes, and the quantum of action is, the enegy adiated pe quantum dop in action is ω his is of couse in the classical limit of high quantum numbes) Notice that I is the aea of phase space enclosed by the integal, dpdq I = pdq π = π Fo the SHO, it s easy to check fom the aea of the ellipse that I = E / ω : λ ake ( / )( ω ) H = m p + m q he phase space elliptical obit has semi-axes with lengths πab = πe / ω me, / E mω, so the aea enclosed is he bottom line is that as we gadually change the sping stength (o, fo that matte, the mass) of an oscillato (not necessaily hamonic), the enegy changes popotionally with the fequency Adiabatic Invaiance and Quantum Mechanics his finding, the invaiance of E / ω fo slow vaiation of the potential stength in a simple hamonic oscillato, connects diectly with quantum mechanics, as was fist pointed out be Einstein in 9 Suppose the (quantum mechanical) oscillato is in the enegy eigenstate with E = ( n+ ) ω hen the spatial wave function has n zeos If the potential is changed slowly enough (meaning little change ove one cycle of oscillation) the oscillato will not jump to anothe eigenstate (o, moe pecisely, the pobability will go to zeo with the speed of change) he wave function will gadually stetch (o compess) but the numbe of zeoes will not change heefoe the enegy will stay at ( n + ) ω, and tack with ω Of couse, the classical system is a little diffeent: the quantum system is locked in to a paticula state if the petubation has vanishingly small fequency components coesponding to the enegy diffeences ω to available states he classical system, on the othe hand, can move to states abitaily close in enegy Landau gives a nontivial analysis of the classical system, concluding that the change in the adiabatic invaiant is of ode e ωτ fo an extenal change acting ove a time τ

4 4 Action-Angle Vaiables Fo a closed one-dimensional system undegoing finite motion (essentially a bound state), the equations of motion can be efomulated using the action vaiable I = pdq π in place of the enegy E I is a function of enegy alone in a closed one-dimensional system, and vice vesa We e visualizing hee a paticle moving back and foth in a one-dimensional well with potential zeo at the oigin, and the potential neve deceasing on going out fom the oigin to infinity Obviously, if a potential has two low points, local bound states can aise in diffeent places, and the I, E elationship is complicated, with diffeent banches, possibly coming togethe at high enegies Impotant! Notice the integal sign in the expession fo the action vaiable I is signifying an integal aound a closed path, a cicuit Don t confuse this integal with the abbeviated action integal, which q has the same integand, but is an integal pdq along a contou fom a fixed stating point, say the oigin, to the endpoint q, not going aound a closed path (Apologies fo using the same lette fo the diffeential and the endpoint, just following Landau) In the spiit of the discussion of constants of motion above, we make a canonical tansfomation to I as the new momentum, using as geneating function the abbeviated action S ( qi, ) he oiginal momentum ( ) ( ) ( ) p= S / q = S qi, / q E I he new coodinate conjugate to the momentum I will be ( ) w= S qi, / I his is called an angle vaiable, I is the action vaiable, they ae canonical o find Hamilton s equations in the tansfomed vaiables, since thee is no time-dependence in the tansfomation, and the system is closed, the enegy emains constant Also, the enegy is a function of I (meaning not of w ) Hence ( ) ( ) ( ) I = E I / w=, w = E I / I = de I / di, so the angle is a linea function of time: w = ( ) de / di t + constant One futhe point about the action vaiable and the action: since we define the action as (, ) S q I = pdq q

5 5 it follows that if we tack the change in this integal as time goes on and the system moves ound and ound the cicuit in phase space, an additional tem S = π I will be added to the action fo each time ound, so the action is multi-valued *Keple Obit Action-Angle Vaiables We have not yet coveed Keple obits, so skip this section fo now: it's hee to efe back to late It's fom Landau, p 67 Fo motion confined to a plane, we can take the cental potential analysis with θ = π /, p θ = and p = mv, the angula momentum, so the Hamiltonian is he Hamilton-Jacobi equation is theefoe p H = p + V + ( ) m S V ( ) S + + = E m m So, following the pevious analysis of sepaation of vaiables fo motion in a cental potential, hee ( ) ( ) ( ) ( ) S, = S + S = S + p he action vaiable fo the angula motion is just the angula momentum itself, π π I = pd = L And the adial action vaiable, with potential V ( ) = k /, is max k L m π E min I = m E + d = L + k (Details on doing the integal ae given in the Appendix, Mathematica can do it too) So the enegy is mk E = ( I + I ) he motion is degeneate: the two fundamental fequencies coincide, I / E = I / E his has majo consequences in quantum mechanics: the actions ae all quantized in units of Planck's constant, fo the hydogen atom, fom the fomula above, the enegy depends only on the sum of the quantum

6 6 numbes: above the gound state, enegy levels ae degeneate, which is why the enegy spectum has the deceptively simple fom so successfully explained by the Boh model he obital paametes, semi-latus ectum and eccenticity, fom E k /a ae = and L kma ( e ) =, I, e + I = = mk I I Recall the semi-majo axis is given by E = k / a, and fom the above expession b I m = = a I + I n, in the hydogen atom quantum numbe notation Appendix: Doing the Integal fo he Radial Action I he integal can be put in the fom β C I = x x π α dx x ( α)( β ), which can be integated by taking a contou encicling the cut fom α to β he integal will have a contibution fom the pole at the oigin equal to C is αβ, and anothe fom the cicle at infinity, which I ( α + β) C α β C = idz = π z z Equating coefficients (multiplying the tem inside the squae oot by ( ) ) C = m E, C α + β = mk, C αβ = L So the contibution fom the oigin gives the L, the cicle at infinity mk / m E = k m /E

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