Question

List and describe the significance of the quantum numbers needed to specify the internal state of a hydrogenic (i.e. meaning it only has one electron) atom.

Answer 1

There are four quantum numbers used to describe the distribution of electron density and the values of other conserved quantities in the dynamics of a quantum system (e.g. an atom) :

1. Principal Quantum Number ()

2. Angular Momentum Quantum Number or Azimuthal Quantum number ()

3. Magnetic Quantum Number ()

4. Electron Spin Quantum Number ()

These quantum numbers are derived from the mathematical solutions to Schrödinger’s Wave Equation :

Hydrogenic atoms are basically one-electronics systems. Thus in Quantum Mechanics, it is similar to the situation where an electron is trapped inside a sphere. Therefore it will have standing waves very similar to an electron bound to a positively charged nucleus by electrostatic attraction. Now I will describe the significance of quantum numbers needed to quantify the possible standing waves or states of an electron trapped by its electrostatic attraction to a positively charged nucleus.

0.at the centre in this one-electronic situation.

1. Principal Quantum Number () :

The principal quantum number designates the size of the orbitals (also called shells). It has integral values like = 1, 2, 3...so on. As n increases, the electron orbital becomes larger and the orbital electron spends more time farther from the nucleus. For a single-electron Hydrogen atom, = 1. Thus it generally has its first shell filled with one electron with rest of the shells being void. The energy of the Hydrogen atom also depends on only :

2. Angular Momentum Quantum Number or Azimuthal Quantum number () :

The azimuthal quantum number has integral values of  = 0 to = - 1 for each value of . This quantum number defines the shape of the orbital.

For Hydrogen = 1, therefore there will be only one possible value of , which is 0.

Orbital names are defined by azimuthal quantum number. For example,

= 0 denoted s-orbital

= 1 denotes p-orbital

= 2 denotes d-orbital

= 3 denotes f-orbital.....so on.

Thus for Hydrogen atom with = 1 and = 0, we say that the electron is in the 1s state or 1s orbital.

3. Magnetic Quantum Number () :

For an integral value, the magnetic quantum number has integral values of = - to + including 0. It helps to distinguish electrons at the same orbital having same and values.

Thus for Hydrogen atom with = 1 and = 0, will also be 0.

4. Electron Spin Quantum Number () :

The electron spin quantum number has only two possible values => +1/2 or -1/2. It helps to describe whether it goes up or down in a spatially varying magnetic field.