Download Quantum Chemistry II: Lecture Notes

2. Electron Spin
Contents
Electron spin angular momentum operator, eigenvalues and eigenfunctions;
hydrogen atom wavefunctions with electron spin; magnetic moment; magnetic
resonance.
Central Concepts
Spin (operator)
Difficult Points
Spin eigenfunctions which depend on no external parameters, but can be written in an
explicit way.
Notes
1. Spin Operators: Eigenvalues
The most important property for an angular momentum operator is
[Ŝi, Ŝ j]=iεijkħŜ
k
i,j,k=x,y,z.
(1.1)
(See eq.(2.6) of Lecture 1) from which almost all other properties can be
derived. For the electron, the spin is just a form of angular momentum, but it is
intrinsic meaning it cannot be expressed into the form like eqs.(2.1-3) or
eqs.(2.15,22) of Lecture 1. However, we can still find out the eigenvalues of
the spin angular momentum operators of the electron.
For convenience, we denote spin operators with S such as:
Ŝ2=Ŝ x2+Ŝ y2+Ŝ z2
(1.2)
Eq.(1) is just a compact form of the following equations:
[Ŝ x,Ŝ y]=iħŜ z, [Ŝ y, Ŝ z]=iħŜx, [Ŝz, Ŝ x]=iħŜ y, [Ŝ y, Ŝ x]=-iħŜ z, etc.
(1.1')
Similar to orbital angular momentum operators, we can easily obtain the following
equations:
[Ŝ2, Ŝ x]=0, [Ŝ2, Ŝ y]=0, [Ŝ2, Ŝ z]=0 Or compactly, [Ŝ2, Ŝ i]=0, i=x,y,z
(1.3)
Introducing ladder operators, we can find out the eigenvalues of . Ŝ2 and Ŝi (i=x,y,z)
just by replacing L with S and repeating the procedure of Section 5.4 of the textbook.
The result is: the eigenvalues of Ŝ2 are
s(s+1) ħ2, s=0,1/2,1,3/2….(1.4)
and the eigenvalues of Ŝz are
msħ, ms=-s,-s+1,…,s-1,s. (1.5)
Figure: Illustration of the angular momentum vector
component Lz.
for a tilted loop and its z
[If we change Ĺ into Ĵ, we get the same diagram for ANY angular momentum.]
Generally, if the z-component of spin angular momentum is measured to be
mJħ, the orientation of the total angular momentum relative to z-direction is
clearly
θ(Ĵ. Ĵz)= cos-1(mJ/[J(J+1)]1/2)
(1.6)
Therefore, if the z-component of spin angular momentum is measured to be
msħ, the orientation of the total angular momentum relative to z-direction is
θ(Ŝ, Ŝz)=cos-1(ms/[s(s+1)]1/2)
(1.6’)
2. Electron Spin Operators: Eigenvalues and Eigenfunctions
For electrons, experiment shows that the value of s is 1/2.[thus the procedure
in Section 5.4 can be made much simpler for electrons; see textbook pp.
300-301: Because there are only two possible spin states, we clearly have
Ŝ+α=0, Ŝ+β=ħα,, Ŝ-α=ħβ, Ŝ-β=0
(2.1)
⇛
Ŝxα=( Ŝ++Ŝ-)α/2= ħβ/2, Ŝxβ=( Ŝ++Ŝ-)β/2=ħα/2
(2.2)
and
Ŝyα=( Ŝ+-Ŝ-)α/2i= iħβ/2, Ŝyβ=( Ŝ+-Ŝ-)β/2i=-iħα/2
(2.3)
These equations are usually expressed into matrix form as
αŜxα=0, βŜxα=ħ/2, αŜxβ=ħ/2, βŜxβ=0,
(2.4a)
αŜyα=0, βŜyα=iħ/2, αŜyβ=-iħ/2, βŜyβ=0
(2.4b)
αŜzα=ħ/2, βŜzα=0, αŜzβ=0, βŜzβ=-ħ/2
(2.4c)
Can you write down the matrices Ŝx, Ŝy Ŝz? ]
Protons and neutrons also have s=1/2. Photons, the light quantum have
spin 1 (but there is no s=0 photons!!! That’s because of relativistic effect for
photons travel at the speed of light in the vacuum.)
The total angular momentum of an electron is therefore
[s(s+1)]1/2 ħ= [1/2(1/2+1)]1/2 ħ=√3/2 ħ
(2.5)
Applying eq.(1.5) to s=1/2 we have
Ŝzα=ħ/2 α, Ŝzβ=-ħ/2 β
(2.6)
Since Ŝ2 commutes with Ŝz, the eigenfunctions of Ŝz can also be that of Ŝ2 and
we have
Ŝ2α=3ħ2/4α, Ŝ2β=3ħ2/4β
(2.7)
Since there are only two possible spin states for electrons and they depend no
external variables as orbital angular momentum eigenfunctions. However, we
may write down the electron spin eigenfunctions as follows
α=α(ms)=δms,1/2 , β=β(ms) δms,-1/2
(2.8)
[remember Dirac’s delta-functionδms,1/2=1 if ms=1/2 or 0 otherwise]
and the normalization conditions can be readily obtained
Σms=-1/2,ms=1/2|α(ms)|2=1,
Σms=-1/2,ms=1/2|β(ms)|2=1
[Proof: e.g., Σms=-1/2,ms=1/2|α(ms)|2=|0|2+|1|2=1]
and the orthogonality condition is
(2.9)
Σms=-1/2,ms=1/2α(ms)* β(ms)=0
[proof: Σms=-1/2,
ms=1/2
(2.10
α(ms) β(ms)=0*1+1*0=0]
*
When both the space and spin variables are included, we have the following
normalization condition:
Σms=-1/2,ms=1/2∫∫∫|ψ(x,y,z,ms)|2dxdydz=1
(2.11
or equivalently
Σms=-1/2,ms=1/2∫∫∫|ψ(r,θ,φ,ms)|2 r2sinθdrdθdφ=1
(2.12
[Note sometimes, the summation over ms is omitted and the above equations become
∫∫∫|ψ(x,y,z,ms)|2dxdydz=1
and
∫∫∫|ψ(r,θ,φ,ms)|2 r2sinθdrdθdφ=1,respectively, but one should remember this
summation should always be performed whenever the normalization is carried out.]
3. Hydrogen Atom with Electron Spin
When the electron spin is considered, to a good approximation, the
Hamiltonian of hydrogen atom will be the same if there is no magnetic field
applied to it. Therefore the energy eigenequation will be the same except the
eigenfunctions are modified so that the spin wavefunctions are taken into
account:
ψ
nlm(r,θφ)→ψ nlm(r,θφ)
g(ms)
(3.1)
where g(ms) is the spin state which is generally a linear combination of α and β:
g(ms)=c1α+c2β. There are two independent choices for c1 and c2. Therefore, the
degeneracy of hydrogen atom eigenstate is changed into 2n2.
4. Magnetic Resonance
Associated with every spin S is a magnetic moment m:
m=gsqS/2m
(4.1)
where m is the mass and q the charge of the spin and gs is g-factor. For electrons, ge≈2,
q=-e, we have
m=-eS/me►|m|=-e|S|/me =-√3eħ/2me
Similarly, for protons, we have
m=geeS/2mp=√3eħ/2mp
(4.2)
(4.3)
When a spin is placed in a magnetic field, the energy is given by
E=-m．B
(4.4)
Above equation is the basis of all magnetic resonance spectroscopies such as ESR
(electron spin), NMR (nuclear spins), μSR(muon spin) etc.
Suppose the magnetic field is along z-axis, we have
E=gsqmsħB0/2m=γmsħB0
(4.5)
where γ =gsq/2m is called gyromagnetic ratio which can either be positive or
negative depending on the sign of gs and q. The resonance frequencies then are found
to be given by
ν=ΔE=γħB0(ms2-ms1)
(4.6)
For single electrons and protons, there is only one frequency (γħB0). Because the mass
of proton is about 1840 times heavier than that of electron and the g factor of proton is
5.5857/2 times larger than that of electron, the proton NMR requency is
1840*2/5.5857≈700 times lower.
Summary
The electrons have an intrinsic angular momentum called spin. The spin operator
satisfies all the properties imposed on a quantum angular momentum operator and its
spin quantum number is 1/2 and it has the total angular momentum of √3ħ/2. The
wavefunction of an atom/molecule must take the spin variable into consideration.
The spin is always related to a magnetic moment, which is the basis of ESR (electron
spin), NMR (nuclear spin), μSR (muon spin) etc. The resonance frequency is
determined by the energy difference of the spins at different orientations in spin
space(eq.(4.6))
Questions
See assignment
Problems
See assignment
References
Textbook pp.282-285, pp.299-301.
Assignment
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