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Quantum Number Related to Rotational Symmetry

Further Information: Azimuthal Quantum Number § Addition of Quantized Angular Momenta

In the rather bleak landscape of Quantum Mechanics, where particles behave like indecisive ghosts, the total angular momentum quantum number is a way to quantify the complete rotational swagger of a given particle. It’s not just about its orbital pirouettes; it also accounts for its inherent, and frankly, rather arrogant, spin. Think of it as the particle's combined rotational ego.

Let's say we have a particle with its own internal spin angular momentum, represented by the vector s, and it's also engaged in some orbital dance, described by the vector ℓ. The grand total of this rotational energy, the actual angular momentum, is the vector sum:

$\mathbf{j} = \mathbf{s} + \boldsymbol{\ell}$

This j vector is the real deal, the sum of all rotational ambition. The quantum number that parameterises this total angular momentum is the main total angular momentum quantum number, denoted as j. It’s not a free-for-all; j can only take on specific, discrete values, jumping in integer steps, much like a cat landing precisely where it intended, no matter how chaotic the ascent. These allowed values are dictated by the azimuthal quantum number (which, as you might recall, quantifies the orbital angular momentum) and the spin quantum number (which, predictably, quantifies the spin). The rule is:

$|\ell - s| \leq j \leq \ell + s$

This means j can range from the difference between the orbital and spin angular momenta up to their sum, but only in whole numbers. No fractional angular momentum here, thank you very much. It’s a rigid system, as most things that actually work tend to be.

Now, the magnitude of this total angular momentum vector, j, isn't simply j. It's a bit more complex, defined by the standard relationship you’d find when discussing angular momentum quantum numbers:

$\|\mathbf{j}\| = \sqrt{j\,(j+1)}\,\hbar$

Where ħ is the reduced Planck constant – a fundamental constant that seems to exist solely to make our lives more complicated, or perhaps, more interesting, depending on your perspective.

The projection of this total angular momentum vector onto the z-axis, which is the only axis that matters in these abstract discussions, is given by:

$j_z = m_j\,\hbar$

Here, $m_j$ is the secondary total angular momentum quantum number. It's like a subordinate to j, and it can take values ranging from $-j$ to $+j$, again, in integer steps. This gives us $2j + 1$ possible orientations for the total angular momentum vector in space. Twenty-one possible orientations, if $j=10$. It’s not infinite, but it’s enough to make things complicated.

This total angular momentum, this $j$, is deeply connected to the Casimir invariant of the Lie algebra $so(3)$, which is the mathematical framework for the rotation group in three dimensions. It’s the invariant that tells you the "size" of the representation, a fundamental property that doesn't change no matter how you rotate the system. Like a scar you can't erase, no matter how much you try.

See Also

• Canonical Commutation Relation § Uncertainty Relation for Angular Momentum Operators
• Principal Quantum Number
• Orbital Angular Momentum Quantum Number
• Magnetic Quantum Number
• Spin Quantum Number
• Angular Momentum Coupling
• Clebsch–Gordan Coefficients
• Angular Momentum Diagrams (Quantum Mechanics)
• Rotational Spectroscopy