Are certain combinations of quantum numbers (basis vectors) forbidden?

In summary, certain combinations of quantum numbers, which describe the states of quantum systems, can indeed be forbidden due to the underlying physical principles such as the Pauli exclusion principle and the rules governing angular momentum. These restrictions arise from the symmetries and statistical properties of particles, leading to specific allowed and disallowed combinations that define the behavior and arrangement of electrons in atoms and other quantum entities.
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The electron's wavefunction is usually expressed in the standard basis {n, l, m_l, s, m_s}, but how to express it in the basis {n, l, m_l, s, m_j} ? (Note that m_s is replaced with m_j.) Or is it that certain combinations of quantum numbers are forbidden?
I've seen the hydrogen electron's wavefunction expressed in the basis or , but so far, never in . My question is, are certain combinations of quantum numbers, eg, , forbidden?

If is not forbidden, how do we get it from the standard basis ?

I know how to get from using Clebsch-Gordan coefficients:
Screenshot 2024-07-12 at 5.29.38 AM.png


where .
is the total angular momentum.

But other than that, I do not know how to express the wavefunction in other bases.
 
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is equivalent to due to the relation . This relation does render some states invalid, though, eg., for a spin-1/2 particle we have the requirement . I think this just makes a less-desirable way to write out the state, though it is equivalent.

In general, with the addition of angular momentum, if you know all three angular momentum quantum numbers (i.e. l, s, and j), you may know at most one magnetic quantum number. If you know all three magnetic quantum numbers, you may know at most two angular momentum quantum numbers. You can know less information, but not more (with the exception of the "top" or "bottom" states, where and ).

Also, in the case of adding orbital angular momentum and spin, we always know the s quantum number, because it is an intrinsic property of the particle, so that also effectively limits what combination of quantum numbers you can have.
 
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FAQ: Are certain combinations of quantum numbers (basis vectors) forbidden?

1. What are quantum numbers and why are they important?

Quantum numbers are sets of numerical values that describe the unique quantum state of a particle, such as an electron in an atom. They include principal, azimuthal, magnetic, and spin quantum numbers. These numbers are important because they determine the energy levels, shapes, and orientations of atomic orbitals, which in turn influence the chemical properties of elements.

2. What combinations of quantum numbers are considered forbidden?

Forbidden combinations of quantum numbers occur when the values assigned do not comply with the rules of quantum mechanics. For instance, the principal quantum number (n) must be a positive integer, the azimuthal quantum number (l) must be in the range from 0 to n-1, the magnetic quantum number (m_l) must range from -l to +l, and the spin quantum number (m_s) can only be +1/2 or -1/2. Any combination that violates these conditions is considered forbidden.

3. How do forbidden combinations affect atomic structure?

Forbidden combinations of quantum numbers can lead to non-physical states that cannot exist in a quantum system. For example, if an electron were assigned a quantum state with an invalid set of quantum numbers, it would not be able to occupy that state, thus affecting the overall structure and stability of the atom. This can have implications for electron configurations and the behavior of atoms in chemical reactions.

4. Can forbidden quantum states still be observed in certain conditions?

While forbidden quantum states cannot exist under normal conditions, they may be temporarily observed in specific situations, such as in high-energy environments or through processes that allow for transitions between states. However, these observations are typically short-lived and do not represent stable configurations of matter.

5. How are forbidden combinations of quantum numbers determined in practice?

Forbidden combinations of quantum numbers are determined through the application of quantum mechanical principles and the rules governing atomic structure. Physicists and chemists analyze the allowed values for each quantum number and apply the Pauli exclusion principle, which states that no two electrons in an atom can have the same set of quantum numbers. This helps identify which combinations are valid and which are not.

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