Vector model of the atom

Cones of spin angular momentum, here shown for a spin-1/2 particle

The commutator implies that for each of L, S, and J, only one component of any angular momentum vector can be measured at any instant of time; at the same the other two are indeterminate. The commutator of any two angular momentum operators (corresponding to component directions) is non-zero. Following is a summary of the relevant mathematics in constructing the vector model.

The commutation relations are (using the Einstein summation convention):

${\displaystyle {\begin{aligned}&[{\hat {L}}_{a},{\hat {L}}_{b}]=i\hbar \varepsilon _{abc}{\hat {L}}_{c}\\&[{\hat {S}}_{a},{\hat {S}}_{b}]=i\hbar \varepsilon _{abc}{\hat {S}}_{c}\\\end{aligned}}}$

where

• L = (L₁, L₂, L₃), S = (S₁, S₂, S₃) and J = (J₁, J₂, J₃) (these correspond to L = (L_x, L_y, L_z), S = (S_x, S_y, S_z) and J = (J_x, J_y, J_z) in Cartesian coordinates),
• a, b, c ∊ {1,2,3} are indices labelling the components of angular momenta
• ε_abc is the 3-index permutation tensor in 3-d.

The magnitudes of L, S and J however can be measured at the same time, since the commutation of the square of an angular momentum operator (full resultant, not components) with any one component is zero, so simultaneous measurement of ${\displaystyle {\hat {L}}_{a}}$ with ${\displaystyle {\hat {L}}^{2}}$, ${\displaystyle {\hat {S}}_{a}}$ with ${\displaystyle {\hat {S}}^{2}}$ and ${\displaystyle {\hat {J}}_{a}}$ with ${\displaystyle {\hat {J}}^{2}}$ satisfy:

${\displaystyle {\begin{aligned}&[{\hat {L}}_{a},{\hat {L}}^{2}]=0\\&[{\hat {S}}_{a},{\hat {S}}^{2}]=0\\&[{\hat {J}}_{a},{\hat {J}}^{2}]=0\\\end{aligned}}}$

The magnitudes satisfy all of the following, in terms of operators and vector components:

${\displaystyle {\begin{aligned}&{\hat {L}}^{2}={\hat {L}}_{x}^{2}+{\hat {L}}_{y}^{2}+{\hat {L}}_{z}^{2}\,\,\rightleftharpoons \,\,\mathbf {L} \cdot \mathbf {L} =L^{2}=L_{x}^{2}+L_{y}^{2}+L_{z}^{2},\\&{\hat {S}}^{2}={\hat {S}}_{x}^{2}+{\hat {S}}_{y}^{2}+{\hat {S}}_{z}^{2}\,\,\rightleftharpoons \,\,\mathbf {S} \cdot \mathbf {S} =S^{2}=S_{x}^{2}+S_{y}^{2}+S_{z}^{2},\\&{\hat {J}}^{2}={\hat {J}}_{x}^{2}+{\hat {J}}_{y}^{2}+{\hat {J}}_{z}^{2}\,\,\rightleftharpoons \,\,\mathbf {J} \cdot \mathbf {J} =J^{2}=J_{x}^{2}+J_{y}^{2}+J_{z}^{2},\\\end{aligned}}}$

and quantum numbers:

${\displaystyle {\begin{aligned}&|\mathbf {L} |=\hbar {\sqrt {\ell (\ell +1)}},\quad L_{z}=m_{\ell }\hbar ,\\&|\mathbf {S} |=\hbar {\sqrt {s(s+1)}},\quad S_{z}=m_{s}\hbar ,\\&|\mathbf {J} |=\hbar {\sqrt {j(j+1)}},\quad J_{z}=m_{j}\hbar ,\\\end{aligned}}}$

where

• ${\displaystyle \scriptstyle \ell }$, is the azimuthal quantum number,
• s, is the spin quantum number intrinsic to the type of particle,
• j, is the total angular momentum quantum number,

which respectively take the values:

${\displaystyle {\begin{aligned}&m_{\ell }\in \{-\ell ,-(\ell -1)\cdots \ell -1,\ell \},\quad \ell \in \{0,1\cdots n-1\}\\&m_{s}\in \{-s,-(s-1)\cdots s-1,s\},\\&m_{j}\in \{-j,-(j-1)\cdots j-1,j\},\\&m_{j}=m_{\ell }+m_{s},\quad j=|\ell +s|\\\end{aligned}}}$

These mathematical facts suggest the continuum of all possible angular momenta for a corresponding specified quantum number:

1. One direction is constant, the other two are variable.
2. The magnitude of the vectors must be constant (for a specified state corresponding to the quantum number), so the two indeterminate components of each of the vectors must be confined to a circle, in such a way that the measurable and un-measurable components (at an instant of time) allow the magnitudes to be constructed correctly, for all possible indeterminate components.

The geometrical result is a cone of vectors, the vector starts at the apex of the cone and its tip reaches the circumference of the cone. It is convention to use the z-component for the measurable component of angular momentum, so the axis of the cone must be the z-axis, directed from the apex to the plane defined by the circular base of the cone, perpendicular to the plane. For different quantum numbers, the cones are different. So there are a discrete number of states the angular momenta can be, ruled by the above possible values for ${\displaystyle \scriptstyle \ell }$, s, and j. Using the previous set-up of the vector as part of a cone, each state must correspond to a cone. This is for increasing ${\displaystyle \scriptstyle \ell }$, s, and j, and decreasing ${\displaystyle \scriptstyle \ell }$, s, and j> Negative quantum numbers correspond to cones reflected in the x-y plane. One of these states, for a quantum number equal to zero, clearly doesn't correspond to a cone, only a circle in the x-y plane.

The number of cones (including the degenerate planar circle) equals the multiplicity of states, ${\displaystyle \scriptstyle 2\ell +1}$.

It can be considered the extension of the Bohr model because Niels Bohr also proposed angular momentum was quantized according to:

${\displaystyle L=m\hbar }$

where m is an integer, produced correct results for the Hydrogen atom. Although the Bohr model doesn't apply to multi-electron atoms, it was the first successful quantization of angular momentum applied to the atom, preceding the vector model of the atom.