10.2: Angular Momentum in Hydrogen Atom
( \newcommand{\kernel}{\mathrm{null}\,}\)
In a hydrogen atom, the wavefunction of an electron in a simultaneous eigenstate of L2 and Lz has an angular dependence specified by the spherical harmonic Yl,m(θ,ϕ). (See Section [sharm].) If the electron is also in an eigenstate of S2 and Sz then the quantum numbers s and ms take the values 1/2 and ±1/2, respectively, and the internal state of the electron is specified by the spinors χ±. (See Section [spauli].) Hence, the simultaneous eigenstates of L2, S2, Lz, and Sz can be written in the separable form ψ(1)l,1/2;m,±1/2=Yl,mχ±.
Because the eigenstates ψ(1)l,1/2;m,±1/2 are (presumably) orthonormal, and form a complete set, we can express the eigenstates ψ(2)l,1/2;j,mj as linear combinations of them. For instance,
ψ(2)l,1/2;j,m+1/2=αψ(1)l,1/2;m,1/2+βψ(1)l,1/2;m+1,−1/2,
Now, it follows from Equation ([e11.26]) that
J2ψ(2)l,1/2;j,m+1/2=j(j+1)ℏ2ψ(2)l,1/2;j,m+1/2,
It follows from Equations ([e11.32]) and ([e11.34])–([e11.39]) that J2Yl,mχ+=[l(l+1)+3/4+m]ℏ2Yl,mχ+=+[l(l+1)−m(m+1)]1/2ℏ2Yl,m+1χ−,
(x−m)α−[l(l+1)−m(m+1)]1/2β=0,−[l(l+1)−m(m+1)]1/2α+(x+m+1)β=0,
ψ(2)l+1/2,m+1/2=(l+m+12l+1)1/2ψ(1)m,1/2+(l−m2l+1)1/2ψ(1)m+1,−1/2,
ψ(2)l−1/2,m+1/2=(l−m2l+1)1/2ψ(1)m,1/2−(l+m+12l+1)1/2ψ(1)m+1,−1/2.
ψ(1)m,1/2=(l+m+12l+1)1/2ψ(2)l+1/2,m+1/2+(l−m2l+1)1/2ψ(2)l−1/2,m+1/2,ψ(1)m+1,−1/2=(l−m2l+1)1/2ψ(2)l+1/2,m+1/2−(l+m+12l+1)1/2ψ(2)l−1/2,m+1/2.
m,1/2 | m+1,−1/2 | m,ms | |
[0.5ex] l+1/2,m+1/2 | √(l+m+1)/(2l+1) | √(l−m)/(2l+1) | |
[0.5ex] l−1/2,m+1/2 | √(l−m)/(2l+1) | −√(l+m+1)/(2l+1) | |
[0.5ex] j,mj |
As an example, let us consider the l=1 states of a hydrogen atom. The eigenstates of L2, S2, Lz, and Sz, are denoted ψ(1)m,ms. Because m can take the values −1,0,1, whereas ms can take the values ±1/2, there are clearly six such states: that is, ψ(1)1,±1/2, ψ(1)0,±1/2, and ψ(1)−1,±1/2. The eigenstates of L2, S2, J2, and Jz, are denoted ψ(2)j,mj. Because l=1 and s=1/2 can be combined together to form either j=3/2 or j=1/2 (see previously), there are also six such states: that is, ψ(2)3/2,±3/2, ψ(2)3/2,±1/2, and ψ(2)1/2,±1/2. According to Table [t2], the various different eigenstates are interrelated as follows:
ψ(2)3/2,±3/2=ψ(1)±1,±1/2,ψ(2)3/2,1/2=√23ψ(1)0,1/2+√13ψ(1)1,−1/2,ψ(2)1/2,1/2=√13ψ(1)0,1/2−√23ψ(1)1,−1/2,ψ(2)1/2,−1/2=√23ψ(1)−1,1/2−√13ψ(1)0,−1/2,ψ(2)3/2,−1/2=√13ψ(1)−1,1/2+√23ψ(1)0,−1/2,
ψ(1)±1,±1/2=ψ(2)3/2,±3/2,ψ(1)1,−1/2=√13ψ(2)3/2,1/2−√23ψ(2)1/2,1/2,ψ(1)0,1/2=√23ψ(2)3/2,1/2+√13ψ(2)1/2,1/2,ψ(1)0,−1/2=√23ψ(2)3/2,−1/2−√13ψ(2)1/2,−1/2,ψ(1)−1,1/2=√13ψ(2)3/2,−1/2+√23ψ(2)1/2,−1/2,
−1,−1/2 | −1,1/2 | 0,−1/2 | 0,1/2 | 1,−1/2 | 1,1/2 | m,ms | |
[0.5ex] 3/2,−3/2 | 1 | ||||||
[0.5ex] 3/2,−1/2 | √1/3 | √2/3 | |||||
[0.5ex] 1/2,−1/2 | √2/3 | −√1/3 | |||||
[0.5ex] 3/2,1/2 | √2/3 | √1/3 | |||||
[0.5ex] 1/2,1/2 | √1/3 | −√2/3 | |||||
[0.5ex] 3/2,3/2 | 1 | ||||||
j,mj |
The information contained in Equations ([ecgs])–([ecge]) is neatly summed up in Table [t3]. Note that each row and column of this table has unit norm, and also that the different rows and different columns are mutually orthogonal. Of course, this is because the ψ(1) and ψ(2) eigenstates are orthonormal.
Contributors and Attributions
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)