 ## Quantum numbers: the address of an electron

$\require{mhchem}$

Quantum numbers are used to describe orbitals.

• Principal quantum number ($$n)$$: It describes the energy of the electron. This can only be positive integers, such as $$n=1,2,3$$ and so on.
• Orbital quantum number (Azimuthal quantum number) ($$\ell)$$: This gives the shape of the subshell electron cloud (orbital shape). $$\ell=0,1,2,.....n-1$$ (positive integers less than $$n$$).
$$\ell=0=s$$ orbital: spherical shape
$$\ell=1=p$$ orbital: dumb-bell shape.
$$\ell=2=d$$ orbital: four-leaf clovers
• Magnetic quantum number ($$m_{\ell}$$) : $$m_{\ell}=0,\pm1,\pm2,......\pm\ell$$ (integers between $$-\ell$$ and $$+\ell$$). This indicates the number of orbitals in the subshell and their orientation in space.
$$m_{\ell}=0=$$ one $$s$$ orbital,
$$m_{\ell}=0,\pm1=$$ three $$p$$ orbitals,
$$m_{\ell}=0,\pm1,\pm2=$$ five $$d$$ orbitals.
• Spin quantum number ($$m_{s}$$) : $$m_{s}=+\frac{1}{2}$$ or $$-\frac{1}{2}$$, often represented as $$\uparrow$$ or $$\downarrow$$, to indicate spin up or down. Electrons are often designated as arrows in orbital “boxes”.

For an electron $$n=2$$. What are the possible values for $$\ell$$ and $$m_{\ell}$$?
For $$\ell$$ permitted values are $$0,1,2,3\ldots\left(n-1\right)$$. If $$n=2$$, $$\ell$$ can be $$0,1$$.
$$\ell=0$$ indicates $$s$$ orbital. $$\ell=1$$ indicates $$p$$ orbital.
$$\ell$$ $$m_{\ell}$$
$$0$$ $$0$$
$$1$$ $$0,\,+1,\,-1$$
For $$\ell=0,\:m_{\ell}=0$$ only: this defines one orbital. In this case, the $$2s$$-orbital.
For $$\ell=1,\:m_{\ell}=0,+1$$ or $$-1$$ only: this defines three separate orbitals. In this case, the three $$2p$$ orbitals.