HYDROGEN ATOM SELECTION RULES TRANSITION RATES

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1 DOING PHYSICS WITH MATLAB QUANTUM PHYSICS Ian Cooper Schoo of Physics, University of Sydney HYDROGEN ATOM SELECTION RULES TRANSITION RATES DOWNLOAD DIRECTORY FOR MATLAB SCRIPTS qp_rues.m Cacuates for a hydrogen atom, the transition rate and ifetime for a transition from an initia state (n1 1 m1) to the fina state (n2 2 m2). The azimutha wavefunction is given by an anaytica expression, the anguar wavefunction is found using the Matab command egendre and the radia wavefunction is soved by the Matrix Method using the function qp_fh.m. Integrations use the function simpson1d.m. The mscript aso can be used to create animated gifs for the osciation of the probabiity of compound states. The states have to be changed within the mscript by editing the statement state = [ ]; The first three numbers are n,, m for State 2 (fina State) and the second set of three numbers give the n, m, m vaues for State 1 (initia State). qp_fh.m incuded within qp_rues.m Function used to sove the radia Schrodinger Equation for the hydrogen atom [ EB(1), R1, r] = qp_fh(n(1), L(1),num, r_max); [ EB(2), R2, r] = qp_fh(n(2), L(2),num, r_max); Within this mscript you can change the maximum radia coordinate and the number of data pints for the cacuations. simpson1d.m (must have an odd number of grid points) incuded within qp_rues.m Function for doing integrations using Simpson s rue, for exampe % Normaize radia wavefunctions N(1) = simpson1d((r1.* R1),0,max(r)); N(2) = simpson1d((r2.* R2),0,max(r)); R1 = R1./ sqrt(n(1)); R2 = R2./ sqrt(n(2)); 1

2 In the input section of the script qp_rues.m you specify: The initia and fina states. The saving of an animated gif of the probabiity coud. The dispay of the animation of the probabiity coud. Can seect a contour pot or a surf pot. The maximum radia distance for the radia wavefunction r_max. This is an important variabe for accurate predicts of the ifetime of the excited state. The vaue of r_max cannot be too sma or too arge when impementing the numerica procedure to sove the radia wave equation. Figure 1 shows the pot of the radia function functions for the initia and fina states for one simuation with r_max = 30x10-9 m. The pots show that the wavefunctions approaches zero in an appropriate fashion. The cacuated ifetime is 1168 ns. Figure 2 shows the wavefunction pots for r_max=10x10-9 m. The tai is not a good representation of the wavefunction and the cacuated of the ifetime is 246 ns and is not an accurate resut. 2

3 Fig. 1. Accurate pots of the wavefunctions when r_max = 30x10-9 m. Fig. 2. Incorrect pots of the wavefunctions when r_max = 10x10-9 m. 3

4 Excited states and transition rates in a hydrogen atom If a hydrogen atom is excited to a higher energy eve, it wi at some ater time spontaneousy make transitions to successivey ower energy eves. In a transition from a higher energy state (1) to a ower energy state (2), the frequency and waveength of the emitted photon is E E 2 1 (1) f h c f The discrete waveengths emitted in a transitions that take pace constitute the ines in the emission spectrum. But, not a possibe transitions take pace. Photons are observed ony with frequencies corresponding to transitions between states whose quantum numbers satisfy the seection rues (2) 1 m 0 or 1 These seection rues are found to appy to a one-eectron atoms. Why are some transitions forbidden? We can answer this question by using a mix of cassica and quantum physics. A transmission rate R t is the probabiity per second that an atom in a certain energy eve wi make a transition to some other ower energy eve. 4

5 Typica aowed transition rates are ~10 8 s -1. This means that in about 10-8 s, the probabiity that the transition has occurred is about one. This time interva is caed the ifetime t L. (3) t L 1 R t The eigenfunctions represent stationary states, that is, they do not vary with time. However, for compound states where the wavefunction is a combination of two stationary states of energies E 1 and E 2, then, the probabiity density function contains a term that osciates in time at frequency f 12 E E 2 1 h Hence, the eectron charge distribution must aso osciate at a frequency f 12 and this is precisey the frequency of the emitted photon in the transition between the states. View animations of compound states at The atom s evoving charge distribution can be modeed as an osciating eectric dipoe. q p qr r q 5

6 The eectric dipoe moment p for the osciating charge distribution is the product of the eectron charge e and the expectation vaue of its dispacement r vaue from the nuceus. A charge distribution that is constant in time wi not emit eectromagnetic radiation, whie a charge distribution with an osciating eectric dipoe moment emits radiation at the frequency of the osciator. The osciating eectric dipoe is the most efficient radiator of eectromagnetic radiation. We can use a cassica formua for the rate of emission of energy by an osciating eectric dipoe (4) R t f 3 hc 0 3 p 2 where p is the ampitude of its osciating eectric dipoe moment and f is the frequency of the osciation. R t is the atomic transition rate, it gives the probabiity per second that a photon is emitted and thus it is equa to the probabiity per second that the atom has undergone the transition. Reative to the Origin at the nuceus, the atomic eectric dipoe moment of a one eectron atom is (5) p e r where r is the position vector from the nuceus at the Origin. 6

7 We can cacuate the expectation vaue of p to obtain an expression for the ampitude p of the osciating eectric dipoe moment of the atom in a compound state as shown in the foowing arguments: (6) exp i E t / exp i E t / (7) 1 2 where 1 and 2 are both normaized wavefunctions. The time independent parts of the wavefunctions are of the form (8) r,, R( r) g( r) / r g r r R( r) where gris ( ) the soution of the radia Schrodinger Equation (9) 2 2 d g() r U ( ) ( ) ( ) 2 eff r g r E g r 2m dr The soutions ( ) for the anguar equation are poynomias in cos known as m the associated Legendre poynomias P (cos ) where = 0, 1, 2, and m = 0, 1, 2, 3,. m 7

8 The normaized soution to equation (6) can be written as m (10) N P cos m where N m is an appropriate normaization constant such that m (11) 0 N P cos sin d 1 m The normaized soution of the azimutha equation is (12) ( ) exp 1 2 im The time independent part of the normaized wavefunctions that can be expressed as m (13) r g r r N P i m,, ( ) / cos exp / 2 m m where gris ( ) aso normaized and both g(r) and P cos are rea quantities. Since the probabiity of finding the eectron is 1, then 2 integrating over a voume eement dv r dr d d 8

9 The eectric dipoe moment can be expressed as 2 (14) p e r dv e r r sin d d dr * * This is a vector quantity and has components (p x, p y, p z ) where (15) x r sin cos y r sin sin z r cos (16) 2 m2 m p e r g ( r) g ( r) dr P cos P cos sin d exp i m exp i m cos d x m 2 m2 1 y p e r g ( r) g ( r) dr P cos P cos sin d exp i m exp i m sin d 2 m2 m p e r g ( r) g ( r) dr P cos P cos cos sin d exp i m exp i m cos d z Let I Rx I Ry I Rz r g2( r) g1( r) dr 0 m I I P cos P cos sin d m Tx Ty m2 m1 (17) I P cos P cos cos sin d Tz Px I exp i m m cos d 2 Py I exp i m m sin d 2 Pz I exp i m m d 9

10 Therefore px e I Rx ITx I Px (18) p y e I Ry ITy I Py pz e I Rz ITz I Pz (19) p px py pz (20) p p p p x y z The radia wavefunction g(r) can be found numericay by soving the radia equation (equation 9) by the Matrix Method and the associated Legendre m functions P (cos ) can be evauated using the Matab command egendre. The normaization constants can be found by numerica integration using Simpson s rue. The integras in the set of equations (17) are computed numericay to find the numerica vaue of the ampitude of the osciating eectric dipoe moment p. Then from equation (3) and equation (4) the transition rate R t and t L can be cacuated respectivey. A summary of the cacuations is given in Tabe 1. The mscript qp_rues.m is used for the eectric dipoe cacuations. 10

11 Tabe 1. Data for the transition from a higher energy eve to a ower eve. The initia and fina states are given by the set of quantum numbers (n m). The mscript qp_rues.m is used to cacuate the binding energies of the two states (EB = - E), the integras given by equation (17), the ifetime tl of the higher energy state and the waveength of the photon emitted in the transition. Ony the integra IP components for the azimutha coordinate are shown in the tabe. Pubished vaues are given for the ifetime so that the mode resuts can be compared to accepted vaues. Reference * : fina state n2 2 m2 initia state n1 1 m1 ifetime tl (ns) web * ifetime tl (ns) simuation IPx IPy IPz EB2 (ev) EB1 (ev) (nm) Forbidden transitions 1 1s (1 0 0) 2s (2 0 0) ~ ~ s (1 0 0) 3s (3 0 0) ~ ~ s (1 0 0) 3d (3 2 0) ~ ~ s (1 0 0) 3d (3 2 1) ~ ~ s (1 0 0) 3d (3 2 2) ~ ~ p (3 1 1) 4f (4 3 0) ~ ~ Forbidden transitions m 0 or 1 3d (3 2 0) 4d (4 3 2) ~ ~ Aowed transitions = 1 and m = 0 or 1 1s (1 0 0) 2p (2 1 0) s (1 0 0) 2p (2 1 1) s (1 0 0) 3p (3 1 0) s (1 0 0) 3p (3 1 1) s (1 0 0) 4p (4 1 0) s (2 0 0) 3p (3 1 0) s (2 0 0) 3p (3 1 1) s (2 0 0) 4p (4 1 0) s (2 0 0) 4p (4 1 1) p (2 1 0) 3s (3 0 0) p (2 1 1) 3s (3 0 0) p (2 1 0) 3d (3 2 0) p (2 1 0) 3d (3 2 1) p (2 1 1) 3d (3 2 0) p (2 1 1) 3d (3 2 1) p (2 1 0) 4s (4 0 0) p (2 1 1) 4s (4 0 0)

12 From Tabe 1, it is quite cear that not a transitions give rise to spectra emissions where spectra ines are not observed or very weak. For the case of dipoe radiation, the transition rates are zero when the dipoe matrix eements are zero. This gives rise to a set of seection rues which are conditions on the quantum numbers of the eigenfunctions of the initia state (state 1) and the fina state (state 2) energy eves. For the dipoe radiation, aowed transitions ( p 0 ) are given by the seection rues Aowed transitions 1 and m 0 or m 1 Transitions rates of atom are typicay R t ~ 10 8 s -1. If these conditions are not satisfied, p = 0 and the transition is caed forbidden. Seection rues arise because of the symmetry properties of the osciating charge distribution of the atom. The compound state formed by the superposition of two stationary states must have an osciating charge distribution which osciates as the same frequency of the emitted photon. Aso, for the system of the atom and the emitted photon, anguar momentum must be conserved. The anguar momentum of the emitted photon in units of is one, hence the change in the anguar momentum quantum number of the atom shoud be 1. Seection rues do not absoutey prohibit transitions that vioate them, but ony make such transitions very unikey. The transition may occur through some other mechanism such as magnetic dipoe moment radiation or eectric quadrupoe moment radiation. 12

13 From the inks beow, you can view schematic animations of compound states showing the variation in eectron probabiity for the osciations between two stationary states. The animations were created with the mscript qp_rues.m. Two frames of one of the animations are shown beow Transitions Type of animated pot (1 0 0) (511) pcoor surf (3 2 0) (4 1 1) pcoor surf (2 0 0) (3 1 0) pcoor surf 13

14 A summary of the numerica quantities is shown in a Figure Window Reference Acknowedgements Thanks to Duncan for comments and corrections to the script qp_rues.m Duncan Carsmith Professor of Physics University of Wisconsin-Madison duncan@hep.wisc.edu 14