Erwin Schrödinger (1887 – 1961) was an Austrian physicist who achieved fame for his contributions to quantum mechanics, especially the Schrödinger equation, for which he received the Nobel Prize in 1933. His development of what is known as Schrödinger's wave equation was made during the first half of 1926. It came as a result of his dissatisfaction with the quantum condition in Bohr's orbit theory and his belief that atomic spectra should really be determined by some kind of discrete energy value.
The Quantum Mechanical Model of the Atom
Most definitions of quantum theory and quantum mechanics offer the same description for both. These definitions essentially describe quantum theory as a theory in which both energy and matter have characteristics of waves under some conditions and characteristics of particles under other conditions.
Quantum theory suggests that energy comes in discrete packages called quanta (or, in the case of electromagnetic radiation, photons). Quantum theory has some mathematical development, often referred to as quantum mechanics, that offers explanations for the behavior of electrons inside the electron clouds of atoms.
The waveparticle duality of electrons within the electron cloud limits our ability to measure both the energy and the position of an electron simultaneously. The more accurately we measure either the energy or the position of an electron, the less we know about the other. Our fact that we cannot accurately know both the position and the momentum of an electron at the same time causes an inability to predict a trajectory for an electron. Consequently, electron behavior is described differently than the behavior of normal sized particles. The trajectory that we normally associate with macroscopic objects is replaced for electrons in electron clouds, with statistical descriptions that show, not the electron path, but the region where it is most likely to be found. Since it is the electron in the electron cloud of an atom that determines its chemical behavior, the quantum mechanics description of electron configuration is necessary to understanding chemistry.
The most common way to describe electrons in atoms according to quantum mechanics is to solve the Schrodinger equation for the energy states of the electrons within the electron cloud. When the electron is in these states, its energy is welldefined but its position is not. The position is described by a probability distribution map called an orbital.
Schrodinger’s equation is shown below.
where
It should be quite clear that without years of high level mathematics, just seeing the equation is no value at all. Without understanding the math, the equation makes no sense.
The stable energy levels for an electron in an electron cloud are those that have integer values in three positions in the equation. Schrodinger found that having integer values in these three places in the equation produced a wave function that described a standing wave. These three integers are called quantum numbers and are represented by the letters
Solutions to Schrödinger’s equation involve four special numbers called quantum numbers. (Three of the numbers,
The principal quantum number,
Principal Quantum Number  Number of SubLevels  Total Number of Electrons 
1  1  2 
2  2  8 
3  3  18 
4  4  32 
The largest known atom contains slightly more than 100 electrons. Quantum mechanics sets no limit as to how many energy levels exist, but no more than 7 principal energy levels are needed to describe all the electrons of all the known atoms. Each energy level can have as many sublevels as the principal quantum number, as discussed above, and each sublevel is identified by a letter. Beginning with the lowest energy sublevel, the sublevels are identified by the letters
The principal energy levels and sublevels are shown in the following diagram. The principal energy levels and sublevels that we use to describe electrons are in red. The energy levels and sublevels in black are theoretically present but are never used for known atoms.
The subenergy levels are identified by the azimuthal quantum number,
When the azimuthal quantum number
When the azimuthal quantum number
When the azimuthal quantum number
When the azimuthal quantum number
Quantum mechanics also tells us how many orbitals are in each sublevel. In Bohr’s model, an orbit was a circular path that the electron followed around the nucleus. In quantum mechanics, an orbital is defined as an area in the electron cloud where the probability of finding the electron is high. The number of orbitals in an energy level is equal to the square of the principal quantum number. Hence, energy level 1 will have 1 orbital (1^{2}), energy level 2 will have 4 orbitals (2^{2}), energy level 3 will have 9 orbitals (3^{2}), and energy level 4 will have 16 orbitals (4^{2}).
The
Principal Energy Level (n) 
Number of Orbitals Present

Total Number of Orbitals (n^{2})  Maximum Number of Electrons (2n^{2}) 
1 

1  2 
2 

4  8 
3 

9  18 
4 

16  32 
The Table above shows the relationship between
Each orbital will also have a probability pattern that is determined by interpreting Schrödinger’s equation. The 3dimensional probability pattern for the single orbital in the
The probability patterns for the five
The seven
You should keep in mind that no matter how complicated the probability pattern is, each shape represents a single orbital, and the entire probability pattern is the result of the various positions that either one or two electrons can take.
Summary
 Energy comes in discrete packages called quanta.
 Electrons in the electron cloud of atoms have a limited number of energy levels due to the quantization of energy.
 The most common way to describe electrons in atoms according to quantum mechanics is to solve the Schrodinger equation for the energy states of the electrons within the electron cloud.
 When the electron is in these states, its energy is welldefined but the electron position is not.
 The position is described by a probability distribution map called an orbital.
 Solutions to Schrödinger’s equation involve four special numbers called quantum numbers. (Three of the numbers,
n ,l , andm , come from Schrödinger’s equation, and the fourth one comes from an extension of the theory.)  These four numbers completely describe the energy of an electron.
 Each electron has exactly four quantum numbers, and no two electrons have the same four numbers.
 The principal quantum number,
n , is a positive integer(1,2,3,…n) that indicates the main energy level of an electron within an atom.
Practice
Questions
http://www.youtube.com/watch?v=IsA_oIXdF_8
Follow up questions:
 Who first suggested that matter also might exhibit the properties of both particles and waves?
 When the concept of describing the trajectory of an electron was given up, what description of the electron replaced it?
 If the shape of the electron orbital is spherical, how many values are possible for the quantum number
l ?
Review
Questions
 If we were dealing with an atom that had a total of 20 electrons in its electron cloud, how many of those electrons would have quantum number
n=1 .  What is the total number of electrons in the
p orbitals of the second energy level?  How many energy levels are necessary to contain the first 10 electrons in an electron cloud?