23.2: The Quantum Mechanical Model of the Atom
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 is the imaginary number,
is Planck’s constant divided by
is the wave function
is the mass of the particle
is the Laplacian operator, (these refer to partial second derivatives)
is the potential energy influencing the particle
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 , , and .
Solutions to Schrödinger’s equation involve four special numbers called quantum numbers . (Three of the numbers, , , and , 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 statement that no two electrons can have the same four quantum numbers is known as the Pauli exclusion principle .
The principal quantum number, , is a positive integer that indicates the main energy level of an electron within an atom. According to quantum mechanics, every principal energy level has one or more sublevels within it. The number of sublevels in a given energy level is equal to the number assigned to that energy level. That is, principal energy level 1 will have 1 sublevel, principal energy level 2 will have two sublevels, principal energy level 3 will have three sublevels, and so on. In any energy level, the maximum number of electrons possible is . Therefore, the maximum number of electrons that can occupy the first energy level is . For energy level 2, the maximum number of electrons is , and for the 3rd energy level, the maximum number of electrons is . The table below lists the number of sublevels and electrons for the first four principal quantum numbers.
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 , , , , , , , and so on. Every energy level will have an sublevel, but only energy levels 2 and above will have sublevels. Similarly, sublevels occur in energy level 3 and above, and sublevels occur in energy level 4 and above. Energy level 5 could have a fifth subenergy level named , but all the known atoms can have their electrons described without ever using the sublevel. Therefore, we often say there are only four subenergy levels, although theoretically there can be more than four sublevels.
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 , the sublevel = .
When the azimuthal quantum number , the sublevel = .
When the azimuthal quantum number , the sublevel = .
When the azimuthal quantum number , the sublevel = .
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 sublevel has only one orbital. Each of the sublevels has three orbitals. The sublevels have five orbitals, and the sublevels have seven orbitals. If we wished to assign the number of orbitals to the unused sublevels, would have nine orbitals and would have eleven. You might note that the number of orbitals in the sublevels increases by odd numbers . As a result, the single orbital in energy level 1 is the orbital. The four orbitals in energy level 2 are a single orbital and three orbitals. The nine orbitals in energy level 3 are a single orbital, three orbitals, and five orbitals. The sixteen orbitals in energy level 4 are the single orbital, three orbitals, five orbitals, and seven orbitals.
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 chart above shows the relationship between (the principal quantum number), the number of orbitals, and the maximum number of electrons in a principal energy level. Theoretically, the number of orbitals and number of electrons continue to increase for higher values of . However, no atom actually has more than 32 electrons in any of its principal levels.
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 sublevel is a sphere. The probability patterns for the three orbitals in the sublevels are shown below. The three images on the left show the probability pattern for the three orbitals in each of the three dimensions. On the far right is an image of all three orbitals together. These orbitals are said to be shaped like dumbbells (named after the objects weight lifters use), water wings (named after the floating balloons young children use in the swimming pool), and various other objects.
The probability patterns for the five orbitals are more complicated and are shown below.
The seven orbitals shown below are even more complicated.
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, , , and , 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, , is a positive integer that indicates the main energy level of an electron within an atom.
Practice
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 ?
Review
 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 .
 What is the total number of electrons in the orbitals of the second energy level?
 How many energy levels are necessary to contain the first 10 electrons in an electron cloud?
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Learning Objectives
 Describe the quantum mechanical model of the atom.
 Indicate the possible values for the quantum number and indicate what information the quantum number gives about the electron.
 Indicate the possible values for the quantum number and indicate what information the quantum number gives about the electron.
 Indicate the possible values for the quantum number and indicate what information the quantum number gives about the electron.
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Date Created:
Jun 26, 2013Last Modified:
Jan 08, 2015Vocabulary
 quantum: A quantum is a discrete packet of energy.
 photon: A photon is a quantum of electromagnetic radiation.
 probability distribution: In quantum mechanics, the probability distribution is a mathematical function that gives the probability of finding a particle in any small region of space.
 Schrodinger’s equation: An equation describing the state and evolution of a quantum mechanical system, given boundary conditions. Different solutions to the equation are associated with different wave functions, usually associated with different energy levels. This equation is fundamental to the study of wave mechanics.
 orbital: The wave function of an electron in an atom or molecule, indicating the electron's probable location.
 quantum number: A number used when describing the energy levels available to atoms and molecules. An electron in an atom or ion has four quantum numbers to describe its state.