IMAGE DESCRIPTION:
QUANTUM NUMBERS ARE A SET OF NUMBERS USED TO DESCRIBE THE STATE OF AN ELECTRON IN AN ATOM, SPECIFYING ITS ENERGY, POSITION, AND SPIN. THE FOUR QUANTUM NUMBERS ARE THE PRINCIPAL QUANTUM NUMBER (\(N\)), THE ORBITAL ANGULAR MOMENTUM QUANTUM NUMBER (\(L\)), THE MAGNETIC QUANTUM NUMBER (\(M_{L}\)), AND THE SPIN QUANTUM NUMBER (\(M_{S}\)). TOGETHER, THESE NUMBERS DEFINE THE UNIQUE ORBITAL OF AN ELECTRON, AS DESCRIBED BY THE PAULI EXCLUSION PRINCIPLE, WHICH STATES THAT NO TWO ELECTRONS IN AN ATOM CAN HAVE THE SAME SET OF ALL FOUR QUANTUM NUMBERS. THE FOUR QUANTUM NUMBERS:
PRINCIPAL QUANTUM NUMBER (\(N\)): DESCRIBES THE ELECTRON'S MAIN ENERGY LEVEL AND THE SIZE OF THE ATOMIC ORBITAL. CAN BE ANY POSITIVE INTEGER: \(1,2,3,...\). HIGHER VALUES OF \(N\) INDICATE HIGHER ENERGY LEVELS AND LARGER ORBITALS.
ORBITAL ANGULAR MOMENTUM QUANTUM NUMBER (\(L\)): DETERMINES THE SHAPE OF THE SUBSHELL (OR ORBITAL). THE ALLOWED VALUES ARE INTEGERS FROM \(0\) TO \(N-1\). \(L=0\) CORRESPONDS TO AN S-ORBITAL (SPHERICAL), \(L=1\) TO A P-ORBITAL (DUMBBELL-SHAPED), \(L=2\) TO A D-ORBITAL, AND SO ON. MAGNETIC QUANTUM NUMBER (\(M_{L}\)): SPECIFIES THE ORIENTATION OF THE ORBITAL IN SPACE WITHIN A SUBSHELL. THE ALLOWED VALUES ARE INTEGERS FROM \(-L\) TO \(+L\), INCLUDING \(0\). FOR EXAMPLE, IF \(L=1\) (A P-ORBITAL), \(M_{L}\) CAN BE \(-1,0,+1\), REPRESENTING THE THREE DIFFERENT P-ORBITALS (\(P_{X},P_{Y},P_{Z}\)).
SPIN QUANTUM NUMBER (\(M_{S}\)): DESCRIBES THE INTRINSIC ANGULAR MOMENTUM OF AN ELECTRON, OFTEN VISUALIZED AS ITS "SPIN". CAN HAVE ONE OF TWO VALUES: \(+1/2\) ("SPIN UP") OR \(-1/2\) ("SPIN DOWN"). THIS IS THE ONLY QUANTUM NUMBER THAT DOESN'T DESCRIBE THE ORBITAL'S SHAPE OR ORIENTATION, BUT RATHER AN INTRINSIC PROPERTY OF THE ELECTRON ITSELF.