Simply summed up what the Zeeman effect on the basis of the atomic structure. Before long, we know that this review be, the way it is calculated. LS coupling only, and assuming that the nuclear spin is 0. later on may be considered non-zero nuclear spin atoms.
Weak field Zeeman effect atoms
Electronic movement as the classic circular motion of charged ball, in accordance with the electromagnetic, magnetic moment is obtained \ (\ boldsymbol {\ MU} = Q / (2m_e) \ boldsymbol {\ ELL} \) , where \ (q = -e \) is the electronic charge, \ (M_E The base of \) is the electron mass. The magnetic moment can also be rewritten as \ (\ boldsymbol {\ MU} = Q \ hbar / (2m_e) \ boldsymbol {\ ELL} / \ hbar = - \ mu_B \ boldsymbol {\ ELL} / \ hbar \) , wherein \ (\ mu_B \) is the Bohr magneton, takes a positive value. Thus interaction energy (selection of an external magnetic field in \ (Z \) axis) \ (H '= - \ boldsymbol {B} \ CDOT \ boldsymbol {\} = eB_z MU / (2m_e) \ ell_z \) . This is the result obtained from the classic image. According to B. Canilhac "Atomic Physics (Volume)", direct analysis of static magnetic field in the Schrodinger equation single electron, two additional items may be obtained, as a \ (H '= - \ boldsymbol {B} \ CDOT \ boldsymbol {\} = eB_z MU / (2m_e) \ ell_z \) , another is proportional to the square of the magnetic field. These two items are paramagnetic and diamagnetic term. A simple calculation shows that for (1T \) \ a magnetic field, the second term is much smaller than the first, is omitted. Regardless of where to start, get additional Hamiltonian is the same. That is, in a multi-electron case
\ [H '= \ frac {
\ mu_B} {\ hbar} \ boldsymbol {B} \ cdot \ sum_i \ boldsymbol {\ ell} _i \] to give further consideration to the electron spins, the finally obtained
\ [H' = \ frac {\ mu_B} {\ hbar} \ boldsymbol {B} \ cdot \ sum_i (\ boldsymbol {\ ell} _i + g_ \ text {s} \ boldsymbol {s} _i) = \ frac {\ mu_B} {\ hbar} \ vec {B} \ cdot (
\ vec {L} + g_s \ vec {S}) \] where \ (g_ \ text {s} \) is the Lande factor of the electron spins, a value of 2. Quantum in Slater theory of Atomic Structure, the Dirac equation from the single electron in a magnetic field, the current density was calculated, respectively, to obtain the electron orbital angular momentum contributions magnetic moment (as here), and the spin magnetic moment contribution, which comprises spin magnetic moment value for the Lande factor of two. It can be said electron orbital magnetic moment of the Lande factor of 1. So far has been given additional source term in the Hamiltonian of the magnetic field, and that clear sources of electronic Lande factor.
Now look at the LS coupling case, a weak field Zeeman effect. Additional Hamilton weak magnetic field means that all smallest perturbation, so the final consideration. Necessary conditions for doing so is that the magnetic field caused by the internal energy level splitting Fine structures much smaller than the spacing between the different levels Fine structures. LS is coupled to a fine structure of the sub-space defined by \ (J ^ 2 \) and \ (J_z \) common eigenstates Zhang, they have a common \ (J \) quantum number, different \ (m_ {J} \) quantum number, it is degenerate. In the fine structure defining a subspace discussed, so, for the basis vector \ (\ {| ^ 0_J E, J, M_j \ rangle \} \) , where \ (E ^ 0_J \) is the simple fine structure energy levels energy and, in fact, this is a position supposed \ (L, S \) values, including other quantum number.
Since the Wigner-Eckart theorem, each component irreducible tensor of rank two operators with the angular momentum vector group \ (| \ alpha, J, m_J \ rangle \) is equal to the ratio of the matrix elements, i.e., (assuming \ (K \) rank)
\ [\ FRAC {\ langle \ alpha'J'm_J '| ^ {T (K)} _ Q | \ Alpha Jm_J \ rangle} {\ langle \ alpha'J'm_J' | ^ {the U- (k)} _ q | \
alpha Jm_J \ rangle} = C \] where the constant \ (C \) and (q, m_J, m'_J \) \ independent. Select Here \ (U ^ {(k) } _ q \) is a rank of angular momentum operators \ (J_q \) , there are
\ [\ langle \ alpha'J'm_J '| \ boldsymbol {A} | \ alpha Jm_J \ rangle = C \ langle
\ alpha'J'm_J '| \ boldsymbol {J} | \ alpha Jm_J \ rangle \] wherein \ (\ hat {\ boldsymbol { A}} \) is a vector operator. Because the quantum number or other \ (J \) orthogonal quantum numbers, the formula \ (\ alpha ', J' \) apostrophes can be omitted without loss of generality. Compute
\[ \begin{aligned}&\langle\alpha Jm_J|\boldsymbol{A}\cdot\boldsymbol{J}|\alpha Jm_J\rangle\\ &=\sum_{\alpha' j'm'}\langle\alpha Jm_J|\boldsymbol{A}|\alpha' J'm_J'\rangle\cdot\langle\alpha 'J'm_J'|\boldsymbol{J}|\alpha Jm_J\rangle\\ &=C\sum_{\alpha 'j'm'}\langle\alpha Jm_J|\boldsymbol{J}|\alpha' J'm_J'\rangle\cdot\langle\alpha'J'm_J'|\boldsymbol{J}|\alpha Jm_J\rangle\\ &= C \ hbar ^ 2J (J + 1) \ end {aligned} \] {J}} \ rangle} {\ [\ hat {\ boldsymbol {because now only discuss the issue of a fine structure subspace, so the equation can be written directly{A} \ cdot \ boldsymbol {\ [\ langle \ alpha Jm_J |
so
Lande formula or called projection theorem, it is important physical meaning: vector \ (\ boldsymbol {A} \ ) meaningful only part of which is projected on the angular momentum.
From the above analysis Wigner-Eckart theorem, a fine structure found in the subspace, the additional applied magnetic field Hamiltonian \ (H '= ({\ mu_B} / {\ hbar}) {B} _z \ sum_i ({\ ell } _ {iz} + g_ \ text {s} {s} _ {iz}) = ({\ mu_B} / {\ hbar}) {B} _z (L_z + g_sS_z) \) calculating matrix elements writable is
\ [\ langle E ^ 0_JJm_J | (L_z + g_sS_z) | E ^ 0_JJm_J '\ rangle = g_J \ langle E ^ 0_JJm_J | \ hat {J} _z | E ^ 0_JJm_J' \ rangle = \ hbar g_Jm_J \ delta_ {m_J , m_J '} \\ \ langle E
^ 0_JJm_J | H' | E ^ 0_JJm_J '\ rangle = \ delta_ {m_J, m_J'} \ mu_Bg_Jm_JB_z \] wherein \ (G_J \) is the operator \ ((L_z + g_sS_z) \) and operator \ (J_z \) the difference between the factor \ (C \) , also known as Lande factor, it is actually a measure of the total magnetic moment and the ratio of the total angular momentum. There formula \ (\ Delta \) symbol, and therefore a fine structure in the sub-space, if the select \ (J ^ 2, J_z \ ) together are an additional eigenstates automatically Hamiltonian Diagonalization, each \ (M_j \) quantum number has its own energy correction, degeneracy is removed.
The remaining is the Lande factor of $ G_J \ (calculated because Lande factor is \) C $, so
\ [g_J = \ langle (\ hat {\ boldsymbol {L}} + g_s \ hat {\ vec {S}} ) \ cdot \ hat {\ vec
{J}} \ rangle / \ langle \ hat {\ vec {J}} ^ 2 \ rangle \] look molecule, \ ((\ Hat {\ boldsymbol} + {L} G_S \ hat {\ vec {S} }) \ cdot \ hat {\ vec {J}} = \ hat {\ vec {J}} \ cdot [\ hat {\ vec {J}} + (g_s-1) \ {Hat \ VEC} {S}] \) , on the other hand take \ (\ hat {\ vec { L}} = \ hat {\ vec {J}} - \ hat {\ vec {S}} \) on both sides there square \ (\ hat {\ vec { J}} \ cdot \ hat {\ vec {S}} = (\ hat {\ vec {J}} ^ 2+ \ hat {\ vec {S}} ^ 2- \ Hat {\ VEC {L}} ^ 2) / 2 \) , so
\ [(\ hat {\ boldsymbol {L}} + g_s \ hat {\ vec {S}}) \ cdot \ hat {\ vec { J}} = \ hat {\ vec {J}} ^ 2+ \ frac {g_s-1} {2} (\ hat {\ vec {J}} ^ 2+ \ hat {\ vec {S}} ^ 2 - \ hat {\ vec {L
}} ^ 2) \] in the fine structure subspace, the square of angular momentum are conserved quantity, so immediately
\ [g_J = \ frac {g_s + 1} {2} + \ frac {g_s-1} {2} \ frac {S (S + 1) -L (L + 1)} {J (J + 1)} \]
Thus the calculation is complete, the introduction of an external magnetic field in LS coupling, reduced symmetry, each of the fine structure of the original degenerate subspace split in different \ (M_j \) quantum number of different energy correction.
Zeeman effect in strong fields
Consider a strong external magnetic field, the additional time that is less than the residual electrostatic potential Hamiltonian, but greater than the additional Hamiltonian LS coupling. Thus in the spectral term consider additional space external magnetic field Hamiltonian, consider additional Hamiltonian LS coupling. Brief spectral terms of space and degree \ ((2S +. 1) (2L +. 1) \) , a non-selected basis vector coupling vectors \ (| E_0LSm_Lm_S \ rangle \) , benefits are \ (\ hat {L} _z \) and \ (\ hat {S} _z \) operator automatically diagonalized, additional outer field Hamiltonian exactly \ (({\ mu_B} / {\ hbar}) {B} _z (L_z + g_sS_z) \) .
Since the automatic diagonalization, you can directly write a revised \ (\ mu_BB_z (M_L + 2m_S) \) , should be noted that the same may have several levels of a correction, that is not fully lifted Jane and.
Next consider additional LS coupling Hamiltonian \ (\ sum_i \ xi_i \ VEC {\ ELL} _i \ CDOT \ {S} _i VEC \) , considering that the problem only in the space of spectral term, using Wigner-Eckart theorem, the additional Hamiltonian can be rewritten as \ (A '\ L VEC {} \ CDOT \ VEC {S} \) , where \ (A' \) is a constant. Because the magnetic field perturbation has an outer front portion such that the level of degeneracy is released, the non-degenerate state, the energy correction can be calculated directly as
\ [A '\ langle LSm_Lm_S | \ vec {L} \ cdot \ vec {S} | LSm_Lm_S \ rangle = a '\
langle LSm_Lm_S | L_zS_z | LSm_Lm_S \ rangle = Am_Lm_S \] degenerate states considered separately. First \ (L_zS_z \) necessarily and automatically diagonalized, followed by computing \ (L_xS_x \) and \ (L_yS_y \) matrix elements. In \ (L_xS_x \) Case
\ [L_xS_x = (L _ ++
L _-) (S _ ++ S _-) = L_ + S _ ++ L_-S _- + L_ + S _- + L_-S_ + \] If wherein the \ (L_ + S _ + \ ) matrix element between a non-zero two degenerate states, then there must be \ (= M_L M_L '. 1 + \) , \ (= M_S M_S'. 1 + \) holds. On the other hand, because the two states degenerate, so there must be\ (\ mu_BB_z (M_L 2m_S +) = \ mu_BB_z (M_L '+ 2m_S') \) , these two contradictory formula. Thus \ (L_ + S _ + \ ) any two degenerate matrix elements between states is zero, the other three the same way, so in the case of degenerate energy correction is still \ (Am_Lm_S \) .
To sum up, Splittings energy in the strong field interval \ (\ mu_BB_z (M_L + 2m_S) + Am_Lm_S \) .
Intermediate case
If the LS coupling term introduced an additional external magnetic field Hamiltonian in size, the need to process a perturbation, i.e., calculate the matrix elements in the spectral space entries. Calculation generally selected coupling representation yl vectors when the matrix element is calculated, in the set of basis vectors, \ (A '\ VEC {L} \ CDOT \ VEC {S} \) is diagonal, the matrix element is
\ [\ frac { A '} {2} \ left
[J (J + 1) -L (L + 1) -S (S + 1) \ right] \] However \ (({\ mu_B} / {\ hbar}) {B } _z (L_z + g_sS_z) \ ) is not diagonal, in order to calculate the matrix elements between two basis vectors is a generally used CG basis vector coefficients to expand a non-coupled linear superposition of basis vectors appearance, and then calculated . After completion of the calculation matrix diagonalization, you can get all the energy correction.
Note that even if the two applied perturbation, the system is still in the magnetic field direction along the outer cylindrical symmetry, that is to say \ (J_z \) and a full and easy Hamiltonian is conserved quantity, the \ (M_j \ ) is a good quantum number. Thus different perturbation Hamiltonian \ (M_j \) matrix elements between states is zero, if the basis vector by \ (M_j \) are arranged, is based on the perturbation Hamiltonian \ (M_j \) block diagonal , in which case it is only necessary to each block diagonalization.
Appendix: Calculation intermediate coupling MATLAB code. For reference only, if wrong, please let me know.
--------------------------------------%
% CG.m CG coefficients calculated
% --- -----------------------------------
function ret = CG(j1,j2,m1,m2,j,m)
if m1+m2~=m
ret=0;
else
c=0;
for z=max([0,j2-j-m1,j1+m2-j]):min([j1+j2-j,j1-m1,j2+m2])
c=c+(-1)^z/(gamma(z+1)*gamma(j1+j2-j-z+1)*gamma(j1-m1-z+1)*gamma(j2+m2-z+1)*gamma(j-j2+m1+z+1)*gamma(j-j1-m2+z+1));
end
ret=c*sqrt((2*j+1)*gamma(j1+j2-j+1)*gamma(j+j2-j1+1)*gamma(j+j1-j2+1)/gamma(j+j1+j2+1+1))*sqrt(gamma(j+m+1)*gamma(j-m+1)*gamma(j1+m1+1)*gamma(j1-m1+1)*gamma(j2+m2+1)*gamma(j2-m2+1));
end
end% ------------------------------------------------- -------
% H_B.m additional matrix elements calculated magnetic field Hamiltonian
% ----------------------------- ---------------------------
function ret = H_B(j1,j2,j,m,j_,m_)
% 只在谱项空间内计算，即LS给定。因此只有j1,j2，无j1',j1'.
% 返回值的单位为u*B，u为玻尔磁子(取正值)，B为外磁场。如返回1.5，则矩阵元为1.5uB.
% 实际上m_j是好量子数，不同m_j的态之间矩阵元为零。因此实际上这里m和m_只需保留一个即可。
right_vec=zeros((2*j1+1),(2*j2+1));
num_i=1;
for m_r=-j1:j1
num_j=1;
for m_c=-j2:j2
right_vec(num_i,num_j)=CG(j1,j2,m_r,m_c,j_,m_)*(m_r+2*m_c);
num_j=num_j+1;
end
num_i=num_i+1;
end
left_vec=zeros((2*j1+1),(2*j2+1));
num_i=1;
for m_r=-j1:j1
num_j=1;
for m_c=-j2:j2
left_vec(num_i,num_j)=CG(j1,j2,m_r,m_c,j,m);
num_j=num_j+1;
end
num_i=num_i+1;
end
ret=sum(sum(left_vec.*right_vec));
end% ------------------------------------------------- -------
% Zeeman.m calculated energy eigenvalue
% --------------------------------- -----------------------
function energy = Zeeman(L,S,B)
% B单位为T，Boh磁子为u
A=200e-25; %这里随便设一个数字
u=9.27400949e-24;% Bohr
J_max=L+S;
J_min=abs(L-S);
J_tab=J_min:J_max;
block_diag=cell(1,round(2*J_max+1));
num=0;
for m_j=-J_max:J_max
num=num+1;
if abs(m_j)>=J_min
block_diag{num}=zeros(round(J_max-abs(m_j)+1));
for J=abs(m_j):J_max
block_diag{num}(J-abs(m_j)+1,J-abs(m_j)+1)=0.5*A*(J*(J+1)-L*(L+1)-S*(S+1));
end
for J1=abs(m_j):J_max
for J2=abs(m_j):J_max
block_diag{num}(J1-abs(m_j)+1,J2-abs(m_j)+1)=block_diag{num}(J1-abs(m_j)+1,J2-abs(m_j)+1)+B*u*H_B(L,S,J1,m_j,J2,m_j);
end
end
else
block_diag{num}=zeros(length(J_tab));
for J=J_tab
block_diag{num}(J-min(J_tab)+1,J-min(J_tab)+1)=0.5*A*(J*(J+1)-L*(L+1)-S*(S+1));
end
for J1=J_tab
for J2=J_tab
block_diag{num}(J1-min(J_tab)+1,J2-min(J_tab)+1)=block_diag{num}(J1-min(J_tab)+1,J2-min(J_tab)+1)+B*u*H_B(L,S,J1,m_j,J2,m_j);
end
end
end
end
energy=[];
for i=1:length(block_diag)
[~,val]=eig(block_diag{i});
energy=[energy,(diag(val))'];
end
end% ------------------------------------------------- -------
% run.m run code
% ------------------------------------ --------------------
clear all;
B=[0:500:100000]*1e-4;
L=1;
S=1/2;
for i=1:length(B)
energy(i,:)=Zeeman(L,S,B(i));
end
plot(B,energy);
xlabel('Magnetic Field (T)');
ylabel('Energy Level');
title('Zeeman Splitting of L=1, S=1/2');operation result:
