This document discusses the Schrodinger wave equation for hydrogen atoms. It begins by presenting the time-independent 3D Schrodinger wave equation and explains how it is converted to polar coordinates due to the radial symmetry of hydrogen atoms. The wave function is assumed to separate into three parts, leading to three equations involving the principal, azimuthal, and magnetic quantum numbers. Quantum numbers and their relationships to orbital shapes are also described. Finally, atomic orbitals are defined as regions of high probability of finding electrons based on the Schrodinger wave equation solution.

1. Quantum Chemistry
B.Sc. SEM-V
Paper-II (Physical Chemistry)
Dr. N. G. Telkapalliwar
Associate Professor
Department of Chemistry
Dr. Ambedkar College, Nagpur

2. Schrödinger wave equation for Hydrogen Atom
The three-dimensional time-independent Schrödinger wave equation is,
Hydrogen atom consists of a centrally located nucleus and an electron revolving
around it. Let us assume that the nucleus is present at origin of coordinate system
and electron is present at point P (x, y, z).
In Hydrogen atom, there are two particles, nucleus and electron, the mass ‘m’ is
replaced by reduced mass ‘’ as;
Also, the potential energy I given by;

3. Hence , the Schrödinger wave equation becomes
• This equation is in Cartesian coordinates (x, y, a coordinates).
• In order to get meaningful information, it is necessary to convert it into polar coordinates (in
terms of r, , ).
•The potential (central force) V(r) depends on the distance r between the proton and electron.
•Transform to spherical polar coordinates because of the radial symmetry.
-------------- (1)

4. This is the Schrödinger wave equation in polar coordinates for hydrogen atom.
In order to solve these equation, let us assume that the wave function ‘’ is a product of
three different functions.
-- (2)
When these values are put in equation (1), we get
-- (3)
-- (4)
-- (5)
When this value is put in equation (2) and three variables are separaed, we get three
different equations as;
Where,
m and  are constants

5. Energy of hydrogen like atoms
The energy of hydrogen-like atoms can be written as-
Where,
Z= Atomic number (nuclear charge)
= reduced mass
e = charge on electron
n= Principal quantum number
H = Planck’s constant

6. Quantum Numbers for Atoms
A set of four numbers which are necessary to locate the energy level or position
of an electron and to specify the size, shape and orientation of the orbital are
called Quantum numbers.

8. The Azimuthal quantum number (angular momentum quantum number) l
determines the shape of an orbital, and therefore the angular distribution.
The number of angular nodes is equal to the value of the angular momentum
quantum number.
Each value of l indicates a specific s, p, d, f subshell (each unique in shape.)
The value of l is dependent on the principal quantum number n.
Unlike n, the value of l can be zero.
It can also be a positive integer, but it cannot be larger than one less than the
principal quantum number (n-1):
l=0,1,2,3,4…,(n−1)(2)(2)l=0,1,2,3,4…,(n−1)
Azimuthal quantum number (l)

10. Magnetic Quantum Number (ml)

11. •The magnetic quantum number describes the energy levels available
within a subshell and yields the projection of the orbital angular
momentum along a specified axis.
•The s subshell (ℓ = 0) contains one orbital, and therefore the mℓ of an
electron in an s subshell will always be 0.
•The p subshell (ℓ = 1) contains three orbitals (in some systems depicted as
three “dumbbell-shaped” clouds), so the mℓ of an electron in a p subshell
will be −1, 0, or 1.
•The d subshell (ℓ = 2) contains five orbitals, with mℓ values of −2, −1, 0, 1,
and 2. The value of the mℓ quantum number is associated with the orbital
orientation.
Magnetic Quantum Number

12. The electron spin quantum number ms does not depend on another quantum
number.
It designates the direction of the electron spin and may have a spin of +1/2,
represented by↑, or –1/2, represented by ↓.
This means that when ms is positive the electron has an upward spin, which can
be referred to as "spin up." When it is negative, the electron has a downward
spin, so it is "spin down."
The significance of the electron spin quantum number is its determination of an
atom's ability to generate a magnetic field or not.
Spin Quantum Number (ms)

13. An atomic orbital is a mathematical term in atomic theory and quantum mechanics
that describes the position and wavelike behavior of an electron in an atom.
According to Heisenberg’s uncertainty principle, it is not possible to locate the
electron exactly.
Quantum mechanics suggests that one can determine the probability of the finding
electron at any point from wave function. The probability is given by 2 .
 In an atom, if we divide the space around nucleus into large number of concentric
shells of thickness dr, the volume of shell between radius ‘r’ and ‘r+dr’ is given by
4r2dr.
 The probability of finding electron in this region is given by 4r2 2 dr.
Concept of atomic orbital
The region of maximum probability is called as atomic orbital.

14. Probability distribution curves
When a graph of 4r2 2 plotted as a function of ‘r’, it is called as radial probability
distribution curve.
The radial distribution functions of various orbital's are called as probability
distribution curves.
4r2 2
4r2 2
4r2 2
4r2 2
4r2 2 4r2 2