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Runge-Lenz Method for the Energy Level of Hydrogen Atom Masatsugu Sei Suzuki Department of Physics, SUNY at Binghamton (Date: January 06, 2015)

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Runge-Lenz Method for the Energy Level of Hydrogen Atom Masatsugu Sei Suzuki Department of Physics, SUNY at Binghamton (Date: January 06, 2015) Runge-Lenz method for the energy level of hydrogen atom Masatsugu Sei Suzuki Department of Physics, SUNY at Binghamton (Date: January 06, 2015) Runge-Lentz (or Laplace-Runge-Lentz) vector In classical mechanics, the Runge–Lenz vector (or simply the RL vector) is a vector used chiefly to describe the shape and orientation of the orbit of one astronomical body around another, such as a planet revolving around a star. For two bodies interacting by Newtonian gravity, the LRL vector is a constant of motion, meaning that it is the same no matter where it is calculated on the orbit; equivalently, the LRL vector is said to be conserved. More generally, the RL vector is conserved in all problems in which two bodies interact by a central force that varies as the inverse square of the distance between them; such problems are called Kepler problems. The hydrogen atom is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law of electrostatics, another inverse square central force. The RL vector was essential in the first quantum mechanical derivation of the spectrum of the hydrogen atom, before the development of the Schrödinger equation. However, this approach is rarely used today. http://en.wikipedia.org/wiki/Laplace%E2%80%93Runge%E2%80%93Lenz_vector Wolfgang Pauli in 1926 used the matrix mechanics of Heisenberg to give the first derivation of the energy levels of hydrogen and their degeneracies. Pauli's derivation is based on the Runge- Lenz vector multiplied by the particle mass. [W. Pauli, Z. Physik 36, 336 (1926).] 1. Kepler's law of planetary motion The Kepler's laws (I, II, and III) describe the motion of planets around the Sun, (I) The orbit of a planet is an ellipse with the Sun at one of the two foci. (II) A line segment joining a planet and the Sun sweeps out equal areas during equal interval of time. (III) The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit. The Sun is at the one focus of the ellipse (the planet orbit). The ellipse orbit is described by x2 y 2 1. a2 b2 where a is the semi-major axis, b is the semi-minor axis, and e is the eccentricity (0<e<1). 1 K0 aba p0 ae ae aaF1 ae,0 O F2 ae,0 r1 r2 (i) The eccentricity e From the definition of ellipsoid, we have F1K0 F2K0 2a When F1K0 F2K0 , we have F1K0 F2K0 a. We apply the Pythagorean theorem to the triangle F1OK0 , a2 b2 a2e2 , Then we have the expression for the eccentricity b2 1 a2 Note that 2 b a 1 e2 . (ii) The perihelion and aphelion The focus is at (ae,0) and (-ae,0). For simplicity, we assume that Sun is located at focus (- ae,0). The perihelion (r1) the point nearest the Sun r1 a(1 e) , The aphelion ( r2) the point farthest the Sun r2 a(1 e) (iii) Area of the ellipsoi The area of the ellipse orbit is given by A ab a2 1 e2 . (iv) The mathematical formula the ellipsoid: Here we show that mathematically, an ellipse can be represented by the formula p r 0 , 1 ecos where p0 is the semi-latus rectum, and e is the essentricity of the ellipse, r is the distance from the Sun to the planet, and is the angle to the planet's current position K0 from its closest point P (perihelion), as seen from the Sun at F1. We use p0 instead of p since p is typically a linear momentum in physics. We also use instead of , for convenience. 3 K0 rr1 P a,0 O F1 ae,0 F2 ae,0 In the triangle F1F2K0 (see the above figure), we have r r1 2a , (1) from the definition of the ellipse. Using the cosine law, we have 2 2 2 2 2 2 2 r1 r 4a e 4aer cos( ) r 4a e 4aer cos . (2) where is the angle between the vector F1P and F1K0 . From Eqs.(1) and (2), we get (2a r)2 r 2 4a2e2 4aer cos or 4a2 4ar r 2 r 2 4a2e2 4aer cos or 2 r(1 ecos ) a(1 e ) p0 4 or p r 0 , 1 ecos with 2 p0 a(1 e ) , where (r, ) are heliocentric polar coordinates for the planet, p0 is the semi latus rectum, and e is the eccentricity, which is less than one. For = 0 the planet is at the perihelion at minimum distance: p a(1 e2 ) r 0 a(1 e) . (perihelion) 1 1 e 1 e For , 2 r p0 . (semi latus rectum) For , the planet is at the aphelion at maximum distance: p r 0 a(1 e) . (aphelion) 2 1 e Note that 1 1 1 e 1 e 2 . r1 r2 p0 p0 p0 The semi-major axis a is the arithmetic mean between r1 and r2, r r a 1 2 . 2 The semi-minor axis b is the geometric mean between r1 and r2, 5 p0 b r1r2 1 e2 ((Note)) The meaning of semi latus rectum The chord through a focus parallel to the conic section directrix of a conic section is called the latus rectum, and half this length is called the semilatus rectum (Coxeter 1969). "Semilatus rectum" is a compound of the Latin semi-, meaning half, latus, meaning 'side,' and rectum, meaning 'straight.' 2. Kepler problem in classical mechanics (hydrogen atom) The classical Hamiltonian for the Kepler problem is 1 H p2 , 2m r where m is a reduced mass and is a positive quantity. For the case of the hydrogen-like atom, we can identify Ze 2 . This system is invariant under the rotation. So the angular momentum is conserved. The classical orbit of the particle is elliptical. The Runge-Lentz vector is defined as 1 M r p L , r m or A mM mer p L , where L (= r p ) is the angular momentum and p is the linear momentum. 6 7 Fig. A (=mM) (denoted by the arrows with purple) on the ellipse orbits. The vector A points in the direction of the perihelion. The magnitude is constant. The angular momentum L is always perpendicular to the orbit. The perihelion, point of the orbit the nearest to the focal point F). The aphelion, the point of orbit from the focal point. It is obvious from the definition that L M 0 , (M lies in the plane of motion). p M p r , r and 8 1 r M r ( r p L) r m 1 r r ( p L) m . 1 r L2 m 1 r l 2 m or r A r mM mr l 2 , since r ( p L) L (r p) L2 , A mM . Since the angular momentum (along the z axis) is conserved, we can calculate L l r1 p1 , where p1 is the linear momentum at the perihelion. M is a conserved quantity since d d 1 1 d M r ( p L) dt dt r m dt p r(e p) 1 [ r ] [F L p (r F)] m r r 2 m p r(e p) [ r ] [e (r p) p (r e )] m r r 2 mr 2 r r p r(e p) [ r ] [(e p)r rp ( p e )r ( p r)e ] m r r 2 mr 2 r r r p r(e p) p ( p e )r [ r ] [ r ] m r r 2 m r r 2 0 where d 1 1 p r(e p) r [ r ] , dt r m r r 2 9 d p F e dt r 2 r and d τ L r F . (torque). dt ((Mathematica)) Proof of dM/dt = 0. Clear "Global`" ;r t_ : x t , y t , z t ; R t_ : r t .r t ; p t_ : m D r t , t ;L t_ : Cross r t ,pt ; Ft_ : r t ; R t 3 eq1 r t 1 D ,t Cross F t ,L t R t m 1 Cross p t , Cross r t ,F t m FullSimplify 0, 0, 0 ______________________________________________________________________________ We note that p L p (r p) p2r ( p r) p Then the vector M (A mM ) always points in the direction of the perihelion from the focal point. At the perihelion, p r 0 We have 10 A mM 2 mer p r1er l 2 (m )er r1 l 2 (m )(ex ) r1 A0 (ex ) where er ex and l 2 A0 m . r1 In general case, we have the orbital equation as 2 r A A0r cos mr l , where is the angle between r and the perihelion direction. From this equation we get 2 r(A0 cos m) l , or A l 2 r( 0 cos 1) m m l 2 p r m 0 , A 1 0 cos 1 ecos m Then we have p0 is given by l 2 p . (semi latus rectum) 0 m and 11 A A0 me . l 2 Since A0 m , we have r1 l 2 m(1 e) r1 3. Derivation of the Runge-Lentz vector The equation of motion of a particle of mass m in the attractive potential is dp F e . dt r 2 r We take the cross product of both sides with the angular momentum L. dp L F L r . dt r3 Since L is constant in time, the left-hand side can be written as dp d L (L p) .
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