The differences between spin and orbital angular momentum are:. The spin quantum numbers need not be integers -they may be half-odd-integer.. The quantum number s has a fixed value for a particular kind of fundamental particleIt is always ; for electrons (and for protons and neutrons).Wenoteinpassingthatnucleialsohavespin(importantinn.m.r.).Thenuclearspiroperator is usually designated I, with quantum numbers I and m, Some values are:1H (proton)/21I=1D (deuteron)1F-12160, 12C1=01.13C12
The differences between spin and orbital angular momentum are: • The spin quantum numbers need not be integers ——they may be half-odd-integer. • The quantum number s has a fixed value for a particular kind of fundamental particle. It is always 1 2 for electrons (and for protons and neutrons). We note in passing that nuclei also have spin (important in n.m.r.). The nuclear spin operator is usually designated I, with quantum numbers I and mI . Some values are: H (proton) 𝑰 = 1 2 D (deuteron) 𝑰 = 1 F 𝑰 = 1 2 16O, 12C 𝑰 = 0 13C 𝑰 = 1 2
Smsh1+2LCS,=h/2S=hV3/4ICYThe component of sS=hys(+1)Xover the XY-plane217
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Orbital angular momentumSpin angular momentumL(L,L,,L.),S(s,sy,s.),L =h/i(I+1),S=hys(s+1),1 =0, 1, 2,...s =1/2,L,=mh,S, =m.h,m-magnetic number.m,-magnetic spin numberm=0, ±1, ±2...m,=±1/2=++$?-$?+$,+$,[L, ]=0, X, =x, , Z,[S2S,]=0, x, =x, y, z,[,L,] =ini,,[S,s,]=ins.[L,L,]=ini,[S,s,]=ins.[i,L, ]= ihi,[s,s.]= ins.18
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Notations for spinThe m, quantum number follows the usual rules for angular momentum: it can take valuesbetwveen s and -s in integer steps. S ince s = I for an electron, m, can take only the values I or -2The state with m, = ('spin up') is denoted by the symbol α, and the state with m, = - ('spindown')by the symbol β These can bethought of as wavefunctions, butthere is no coordinate inthe conventional sense. We can also use Om. for a generic spin function; i.e., O1/2 = α and0-1/2 = β .+1|nα==nαs?α=hαs.α =421+1nβ=nβ2βs.β =-hn4The symbols 1 and I are commonly used in orbital energy level diagrams to denote electronoccupancywithspinupanddownrespectively
Notations for spin The ms quantum number follows the usual rules for angular momentum: it can take values between s and -s in integer steps. Since s = 1 2 for an electron, ms can take only the values 1 2 or - 1 2 . The state with ms = 1 2 ('spin up') is denoted by the symbol 𝛼, and the state with ms = − 1 2 ('spin down') by the symbol 𝛽 These can be thought of as wavefunctions, but there is no coordinate in the conventional sense. We can also use 𝜎𝑚𝑠 for a generic spin function; i.e., 𝜎1/2 = 𝛼 and 𝜎−1/2 = 𝛽 . The symbols ↿ and ⇂ are commonly used in orbital energy level diagrams to denote electron occupancy with spin up and down respectively. 2 2 2 2 2 2 1 1 1 3 ˆ 1 ˆ 2 2 2 4 1 1 1 3 ˆ 1 ˆ 2 2 2 4 z z s s s s
Wavefunctionsincluding spinThe spin is a newdegree of freedom, so we can construct a completewavefunctionfor thehydrogen atom by multiplying together an atomic orbital and a spin function. A general one-electronatomicwavefunctionorspin-orbitalisthen[nlm,ms ) = R. (r)Y.m, (0,0)omanditsatisfiestheeigenvalueequationsHnlm,m,)= E,nlm,m,)12|nlm,m,)=1(I+1)h2[nlm,m,)i.[nlm,m,)= m,h|nlm,m,)3 [nlm,m,)=s(s +1) h2 |nlm,m,),with s= 23.|nlm,m,)=m,hnlm,m,We don't need to list the quantum number s because it is always 2
Wavefunctions including spin The spin is a new degree of freedom, so we can construct a complete wavefunction for the hydrogen atom by multiplying together an atomic orbital and a spin function. A general oneelectron atomic wavefunction or spin-orbital is then and it satisfies the eigenvalue equations We don't need to list the quantum number s because it is always 1 2 . , , l s l s nl l m m nlm m R r Y 2 2 2 2 ˆ 1 ˆ 1 ˆ 1 ,wi 2 ˆ th l s nl l s l s l s z l s l l s l s l s z l s s l s H nlm m E nlm m nlm m l l nlm m l nlm m m nlm m nlm m s s nlm m s s nlm m m nlm m l s