What is the difference between shell, subshell, and orbital?

The Schroedinger equation for an electron bound to a spherically symmetric coulomb potential of a hydrogen like nuclei shows that the wave function of the electron forms standing waves called stationary states. These states are characterized by three quantum numbers.

1. Principal Quantum Number (#n#) : #n=1,2,3, \cdots \infty#
   For single electron systems the allowed energy values (energy levels) are determined purely by the principle quantum number. This quantum number can only take integer values starting with #1# with no upper bound. All the electronic states with the same principle quantum number are said to belong to the same shell.
   The #n=1# states are labeled K-shell , #n=2# states are labeled L-shell , #n=3# states are labeled M-shell , #n=4# states are labeled N-shell and so on.

2. Angular Momentum Quantum Number (#l#): #l=0,1,2, ..., n-1#.
   This quantum number determines the magnitude of the orbital angular momentum of the electron. It can take only integer values starting from 0 but has an upper bound. It can only go up to a number that is one less than the principal quantum number. While the shells are a bigger group of quantum states, this quantum number breaks them into smaller groups of quantum states called sub-shells. All quantum states with the same orbital angular momentum quantum number are said to belong to the same sub-shell. The #l=0# states are labeled s-subshell, #l=1# states are labeled p-subshell, #l=2# states are labeled d-subshell, #l=3# states are labeled f-subshell and so on. Thus the quantum states belonging to a shell with principal quantum number #n# are divided into #n# subshells.

3. Magnetic Quantum Number (#m_l#): #m_l=-l,-(l-1)\cdots,0,\cdots,+(l-1),+l#
   This quantum number determines the magnitude of the component of the orbital angular momentum vector of the electron along a reference direction (usually the direction of an applied external magnetic field). This quantum number breaks the sub-shells into further small groups called orbitals . All the electronic states with the same Magnetic Quantum Number (#m_l#) belong to the same orbital. A sub-shell characterized by a angular momentum quantum number #l# has #2l+1# orbitals. Thus the s-subshells have 1 orbital, p-subshells have 3 orbitals, d-subshells have 5 orbitals and so on.