Elementary Ideas from Quantum Mechanics and Its Model of the Atom

The quantum mechanical model of the atom is based on Schrödinger’s equation and the solutions derived from it. In the quantum mechanical model of the atom, the dual nature of the particle is considered. Dual nature includes particle nature and wave nature. 

Rather than certainty, the basis of the model is probability. It is difficult to execute the quantum mechanical model of the atom for a multi-electron system because it is difficult to solve Schrödinger’s equation for multi-electron species.

Quantum model definition

Schrödinger wave equation

• Erwin Schrödinger formed the quantum mechanical model of the atom in 1926. He took Bohr’s model of an atom a little further by considering the dual nature of electrons.
• He considered electrons as waves with quantised energy levels, i.e. the magnitude of energy is pre-defined.
• The quantised energy of an electron is a valid solution of Schrödinger’s equation, which has been derived as a result of the wave nature of the electron.

Physical significance of Ψ and 2

• Ψ is a wave function and it can be defined as the magnitude of the electron wave. It has no physical significance. It can have positive, negative, or imaginary values.
• 2, also known as probability density, expresses the probability of finding an electron at a specific point in an atom.

Quantum numbers and shapes of orbitals

• Quantum numbers are a set of defined variables that are used to describe the position and energy of electrons in an atom. We have a set of four quantum numbers in the quantum mechanical model of the atom:

1. Principle quantum number (n)
2. Azimuthal quantum number (l)
3. Magnetic quantum number (m)
4. Spin quantum number (s)

Principal quantum number (n) 

• The Principal quantum number is represented by letter ‘n’.
• It expresses the electron shell of an atom.
• Since principle quantum numbers express the distance between nucleus and electron, the greater the principal quantum number, the greater is the distance between the nucleus and its electron in an atom.
• The value of the principal quantum number can be any positive integer (n=1,2,3,4……..) with shell n=1 being closest to the nucleus. Hence it cannot have negative values or be equal to zero.
• When electrons are found in the excited state, i.e. with a high energy level, they tend to move to a lower energy state; from a shell with a high quantum number to a shell with a low quantum number. Shells with high principal quantum numbers have higher energy than a shell with a relatively low principal quantum number.
• Angular momentum can also be calculated using the principal quantum number, by using the following formula:

                 mvr=nh2

Azimuthal quantum number (l)

• It was given by Arnold Sommerfeld and is represented by the letter ‘l’.
• It defines the shape of the orbital and is equal in magnitude to the number of angular nodes present in the orbital.
• The azimuthal quantum number can express different subshells such as s,p,d, and f, which are different in shape and size.
• Its value depends on the principal quantum number and it ranges from 0 to (n-1).

Magnetic quantum number (m)

• The magnetic quantum number was given by Zeeman and is represented by the letter ‘m’.
• It defines the number of orbitals and their orientation in an electron shell.
• Its value depends on the azimuthal quantum number and for every value of azimuthal quantum number (l), it ranges from -l to +l.
• Evaluation of the number of orbitals in a shell can be done by using the formula (2l+1), where l is the azimuthal quantum number.

Spin quantum numbers (s)

• It was proposed by Goldshmidt and Ulen Back, and is indicated by the letter ‘s’.
• There are two values of ‘s’, which are given as + ½ and −½.
• The spin may be anti-clockwise or clockwise.
• A positive value of spin quantum number indicates upward spin, whereas a negative value denotes downward spin.
• The value of spin quantum determines whether the atom can produce a magnetic field or not. 

Features of the quantum mechanical model of atom

• The energy of an electron is quantised, i.e. predefined values of energy are derived for an electron.
• The quantised value of energy is the solution of Schrödinger’s equation, as a result of the wave nature of electrons.
• Following Heisenberg’s uncertainty principle, only the probability of the presence of an electron can be calculated, which is determined by 2 at a specific point. ψ represents wave function, which has no physical significance.
• From different values of 2 in different regions of an atom, it is possible to determine the region where the probability of finding an electron is high.
• 2 represents the probability of finding electrons always and has a positive value.
• An atomic orbital is the wave function of an electron in an atom. An electron occupies an atomic orbital when defined by a wave function.

Conclusion

The quantum mechanical model of the atom is based on Schrödinger’s equation and the solutions derived from it. Schrödinger’s equation is used mainly for single-electron species. There are four main quantum numbers associated with the quantum mechanical model of atoms.

  and 2 are important symbols in the quantum mechanical model of atoms. They represent the magnitude of electron waves and the probability of finding electrons respectively. The quantum mechanical model of the atom reflects several features, including quantised energy levels.