Q. Obtain the Hamiltonian for a one electron atom in the presence of an external magnetic field.

Transcription

1 Syed Ashad Hussain Lectue Deatment of Physics Tiua Univesity Q. Obtain the Hamiltonian fo a one electon atom in the esence of an extenal magnetic field. To have an idea about the one electon atom laced in a magnetic field we need to know the inteaction Hamiltonian in esence of the magnetic field. We know the Hamiltonian in the absence of extenal magnetic field is; Η 0 = T + V = T + V() Now if we take into account the sin obit inteaction tem then the Hamiltonian will be; uu H0 = + V ( ) + ξ ( )( L. S)...(). m dv ( ) Ze h Whee ξ ( ) =. = 3 mc d mc Then the eigen states ae nls,,, jm, whee m = - to +, (j + ) nos. When we have the sin obit tem in the Hamiltonian, the Eigen states have a degeneacy in the ojection of j. Thee ae (j + ) fold degeneacy. All this (j+ ) states have the same enegy eigen function and the total angula momentum will be conseved. Now let this atom with a single electon at the outside coe, is laced in a constant unifom magnetic field, the enegy level unde the influence of extenal magnetic field ae slit u into comonents giving chaacteistics Zeeman atten. Now the inteaction enegy which oduces these dislacements consists of the two ats, that aising fom the obital motion of the electon and that aising fom the sin of the electon. The effect of the extenal magnetic field on the obital motion of the electon is obtained by using the vecto otential u A whee, u u u B = A The Hamiltonian o K.E fo a aticle of chage e in a magnetic field of otential u A is u e u obtained by witing + A in lace of u in the esective exessions c u e u u K.E. T = + A Whee A = vecto otential m c u eu u eu = + A + A m c c e u u u u e = + ( A+ A) + A...(3). m mc mc Atom in extenal magnetic field i

2 Syed Ashad Hussain Lectue Deatment of Physics Tiua Univesity u u Now does not commute with A in geneal and uu u u u A = i A = ih ( ) ψ h(. ) ψ u u u = ih( ψ.. A+ A. ψ) If wechoosethecoulomb gauge. A = 0, then we have,(. A) ψ = A( ih ψ ). A= A. Hence fom equation () we have; e T = + ( A. ) + e A m mc mc Now we conside the magnetic moment associated with the electon sin; eh µ s = gsβs; whee β = mc gs = (s+ ) = + = Until u to this oint we have not taken into account the intinsic magnetic moment of the electon, which also inteacts with the extenal field u B. Thus the inteaction of the intinsic sin magnetic moment with the magnetic field is given by, uu u H = µ s. B uu = gsβ S. B u u = β B.S Thus we have the comlete Hamiltonian as, e u uu e uu u H = + ( A. ) + A +V ( ) + ξ( ) ( LS. ) + βb. S...() m mc mc Again, fo a unifom magnetic field we can wite, u u A= ( B ) e uuu e u u e u u e u u ( A. ) = ( B ). = B( ) = BL. mc mc mc mc e u u uu = BL. = β BL. mc Using the above we have fom (); Atom in extenal magnetic field ii

3 Syed Ashad Hussain Lectue Deatment of Physics Tiua Univesity e uu u H = + V( ) + βbl. + A + ξ ( ) ( LS. ) + βb.s m mc u u u e = H0 + β B( L+ S) + A mc uu u u u e H = + ( ) + ξ( )( LS) + βb( L+ S) + A m mc V....(3) Equation (3) gives the total Hamiltonian of a single electon in an atom in the esence of extenal magnetic field. N. B. In case of weak magnetic field (Zeeman effect) e A mc tem is neglected because the extenal field is small comaed to the field oduced by the electon and nucleus. Theefoe equation (3) can be educed to ; uu u u u H = + V( ) + ξ( )( LS. ) + βb( L+ S) m u u u H = H0 + β B( L+ S) uu whee H0 = + V( ) + ξ ( )( L. S ) m Q. Descibe the effect of weak magnetic field on the sectal lines emitted by a one electon atom. The exession fo the total Hamiltonian of a single electon in an atom in the esence of extenal magnetic field is given by; u u u H = H0 + β B L+ S...(4) ( ) We conside the magnetic field be weak enough so that β B is small comaed to the fine stuctue slitting. In this case, tem othe than H 0 (in equation 4) may be egaded as the etubation. So the etubed Hamiltonian is; u u u H = β B L+ S...(5) ( ) In esence of weak magnetic field ( H < H ), L-S couling occus and out of S one S emains. The enegy shift uto the fist ode is given by; E = ψβb.( L+ S) ψ LSM = ψβb.( + S) ψ LSM = β B.[ z + Sz ] LSM [ if B isin z diection...(6) so Atom in extenal magnetic field iii

4 Syed Ashad Hussain Lectue Deatment of Physics Tiua Univesity Fom symmety the aveage value of S is a vecto aallel to as this the only vecto which is conseved. Thus, S = constant = C S = C. = C S S. S C = = = S. S =.... (7) S. Sz = M... (8) Now = L+ S S = L + S. S = L S. = [ + S L ] S. = [ ( + ) + S( S+ ) LL ( + )] Theefoe fom equation (6) E= β B.[ z + Sz ] LSM + S L = β B.[ z LSM +. z LSM ( + ) + S( S+ ) L( L+ ) = βbm + βbm [ ] ( + ) ( + ) + S( S+ ) L( L+ ) = β BM [ + ] ( + ) = β BM g ( + ) + S( S+ ) L( L+ ) whee g = + ( + ) = lande' g ' facto. Since the diagonal elements of the etubation oeato ae the only non vanishing one, the enegy of the atom in the st etubation aoximation u is given by, E = E mhβ Bg nljm nj Whee m = 0, + to - Atom in extenal magnetic field iv

5 Syed Ashad Hussain Lectue Deatment of Physics Tiua Univesity The ( + ) fold degeneacy is thus lifted in the esence of the magnetic field. The shift of the level is symmetic with esect to the unetubed enegy level E nj. The u distance between the neighboing substances, is E = mh β Bg, which is ootional to the magnetic field stength and to the Lande s facto which deends on the quantum numbe j, l and s. The slitting of the enegy levels detemined by the above equation is called the anomalous Zeeman Effect. Fo a sineless aticle s = 0 and the Lande s facto g =. In that case, the distance between neighboing sublevels doesn t deends on all the chaacte of the state and equal to u E = mh β Bg. Such a slitting of enegy level is called the nomal Zeeman Effect. Nomal Zeeman Effect is secial case of anomalous Zeeman Effect fo which s = 0. Thus fo weak field, each enegy level is slitted symmetically into (j + ) equally saced states the slitting being ootional to the magnetic field and is indeendent of the total quantum numbe n of the atom. Q3. Show that the sectal lines of a hydogen atom in state slits in a stong magnetic field. The inteaction etubed Hamiltonian of an atomwith an extenal magnetic field is given by; u u u H = β B L+ S ( ) Ina weak magnetic field the quantity β u B is small enough comaed to the fine stuctue slitting and we get the Zeeman slitting. When the magnetic field is stong enough such that the quantity β u B is lage o comaable to the fine stuctue slitting. The enegy level slitting diffe fom the Zeeman atten and the henomenon is called Paschen-Back effect. Hee the sin-obit couling is boken and eigen functions ae labeled by L, M L, S and M S which ae good quantum numbes. Consideing the case when the magnetic inteaction is lage comaed to slitting due to Coulomb eulsion, we can wite the Hamiltonian as; H = H0 + H = H0 + β B( L+ S) = H0 + β B( LZ + SZ ) whee B is the magnitude of the field along Z-diection. The fist ode enegy shift due to magnetic field inteaction is given by; Atom in extenal magnetic field v

6 Syed Ashad Hussain Lectue Deatment of Physics Tiua Univesity E= H LM LSM S = β B LZ + SZ LMLSMS = β BM ( L + MS) The multilet slitting due to S-O inteaction is sueosed on the slitting due to extenal magnetic field. The enegy shift due to S-O is given by; ES O = ξ ()(.) LS LM SM R 4 yα Z = ςnlmlms whee, ςnl = 3 nll ( + )( L+ ) Hence the total shift of enegy levels is; E = β B( ML + MS) + ςnlmlms and E = E0 + E E = E0 + βb( ML + MS) + ςnlmlms In wave numbe; β B ς T = T0 + ( M ) nl L + MS + MLMS ch ch Selection ule M L = 0, ± M L = 0 π comonent MS = 0 =± σ comonent To calculate the stong field atten, eg. fo Na-D line, let us constuct the following table: L S No field Stong field (Paschen Back effect) Tem L value M L M S M L + M S am L M S P 3 a 0 0 P a a a S Atom in extenal magnetic field vi

7 Syed Ashad Hussain Lectue Deatment of Physics Tiua Univesity Atom in extenal magnetic field vii

8 Syed Ashad Hussain Lectue Deatment of Physics Tiua Univesity Q.4. Calculate the weak field atten (Zeeman atten) fo Na-D line. Tem L S =L+S P 3 P S 0 3 g No. of g = + Zeeman levels (j+) ( + ) + S( S = ) L( L+ ) ( + ) m (+) values m =, -,..- +, 3 3,,,,, m.g,,, 3 3, 3 3, The selection ules fo tansitions ae mj =± σ -comonent. = 0 π comonent. and L = ± Atom in extenal magnetic field viii

9 Syed Ashad Hussain Lectue Deatment of Physics Tiua Univesity Samle questions: Q. Show qualitatively the fine stuctue slitting of 3S, 3P, S and P levels of hydogen atom and show the allowed tansition. Q. Show the slitting of 3 P,3P and 3S when laced in an weak magnetic field. Q3. A H-atom is in the P state. It is laced in a stong magnetic field B. Neglecting the sin-obit inteaction find the slitting of P level and the enegy sacing. Q4. Show the enegy level diagam fo the sectal lines of n= n= tansition of H- atom. Wite the selection ules. Q5. Fo the electonic tansition D P (a) Daw the enegy level diagam and show the Zeeman slitting of the enegy levels D, P in the esence of magnetic field. (b) Show all the allowed tansition (c) How many distinct lines ae obseved in the Zeeman sectum. Q6. Daw the fine stuctue slitting and show the tansition fo n=3 to n= lines. Q7. Comute the Zeeman atten fo ( a) F 3 D tansition ( b) D 3 P tansition () c D 5 P3 tansition Q8. Obtain the total Hamiltonian fo a one electon atom in the esence of extenal magnetic field. Q9. An alkali atom contains seveal electons but the alkali secta ae geneally undestood in tems of one electon secta-why? Q0. Why ae the enegy levels of alkali atoms ae diffeent fom that of H-atom? Q. How V() exected to behave fo 0 and α in case of an alkali atom? Atom in extenal magnetic field ix