Quantum Numbers and Schrodinger’s Wave Equation

Schrodinger wrote an equation that described both the particle and wave nature of the electron. This is a complex equation that uses wave functions to relate energy values of electrons to their location within the atom. A more qualitative analysis can at least describe

Wave function (ψ) describes:

  • energy of e- with a given
  • probability of finding e- in a volume of space

In addition to being mathematically complex, another downfall of Schrodinger’s equation is that it can be solved exactly for only the hydrogen atom—since it contains 1 lone electron.  For all other elements, the ψ must be approximated and best that can be done to find ψ2, or the area of highest probability of finding an electron.

Schrodinger’s wave equation takes into account 4 quantum numbers which are further discussed below:

  • Principal Quantum # (n):
    • Represents the shell #;
    • Possible values are n= 1, 2, 3, 4…7
    • Refers to the 7 rows/periods on the periodic table
  • Angular Momentum Quantum # (l):
    • Represents the subshell and shape of the orbital
    • Has numerical values of 0, 1, 2, and 3 which also refer to letters as below
    • Possible values:
    • 0 1 2 3
      s p d f
    • Therefore, you can refer to a subshell by its numerical or letter value (i.e., l=2 or d)
    • Refers to one of four sections of the periodic table as illustrated in the color-coded table below:

spdftable

NOTE: “s” and “p” sections have a quantum number value equal to the row number or “n,”  while the “d” section has a value of “n-1” and the f section as a value of “n-2.”

This means that the principle quantum number for an electron on the 6th row would be 6s, 6p, (6-1) or 5d, and (6-2) or 4f.

  • Magnetic Quantum # (m):
    • Refers to orbital #
    • Possible values are ml = 0, ±1, ±2, ±3
    • No reference to periodic table but orbitals are drawn as diagrams (a box represents each orbital) and electrons are placed inside them.

spdf1

  • Magnetic Spin Quantum # (m):
    • Refers to electron spin
    • Electrons are drawn as arrows
    • Since like charges of electrons in a pair repel, electrons spin away from one another or have opposite spin directions, noted as a + ½ or – ½  here.
    • Each arrow is an electron, point in opposite directions

spdf2

  • Each value (1 electron) refers to one block or square on the periodic. A good analogy is like rolling dice and moving spots on a board game.

spdf3

Possible Quantum Numbers Diagram for Fourth Shell (n=4): 

spdfhandwritten

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