Wave Function

When Classical mechanics failed to apply in microscopic particles like electrons, protons, atoms, molecules, etc. since it ignored the concept of dual behaviour of matter, especially for subatomic particles and the uncertainty principle, the branch of science that deals with the dual behaviour of matter and interaction of subatomic particles, Quantum mechanics was introduced. 

It is a theoretical conceptual science that deals with the study of the mathematical description of the motion and interaction of subatomic particles, incorporating the concepts of quantization of energy, wave-particle duality, the uncertainty principle, and the correspondence principle. In 1926 the field of quantum mechanics was developed independently by two famous scientists Werner Heisenberg and Erwin Schrödinger. The fundamental equation of quantum mechanics was developed by Schrödinger; he won the Nobel Prize in Physics in 1933. The equation that incorporates wave-particle duality of the matter was proposed by de Broglie. 

  The mathematical representation of the quantum state of an isolated quantum is known as a wave function. It is a complex-valued probability amplitude, and the probabilities for the possible results of measurements made on the system can be derived from it. The Greek letters ψ and Ψ are the most common symbols for the representation of a wave function. 

SCHRODINGER EQUATION    

• In 1926, Schrödinger published the famous wave equation now named after him, the Schrödinger equation. This equation was based on classical conservation of energy using quantum operators and the de Broglie relations and the solutions of the equation are the wave functions for the quantum system.

• The Schrödinger equation describes the wave properties of an electron in terms of position, mass, energy.

• The equation determines how wave functions are related with time, and a wave function behaves qualitatively with other waves, such as water waves or sound waves. 

• It is a typical wave equation. This equation satisfies the property of “wave function”, and explains the duality of wave character.

• Schrödinger gives a recipe for constructing this operator from the equation for the total energy of the system. The total energy of the system takes into account the addition of kinetic energies of all the subatomic particles (electrons, nuclei), with attractive electric potential energy between the electrons and nuclei and repulsive electric potential energy among the electrons and nuclei individually.

• The simplest form of time-independent Schrodinger equation is 

  H ψ = E ψ

Where

 E = Total energy of electron

 ψ = wave function of electron

 H = Hamilton operator

In one dimension

 Ĥ = −ħ²2m d²dx² + V(x) 

Where, 

  V(x) = potential energy, (V = − Ze²/4πr)

In three dimensions, 

 Ĥ = −ħ²2m (²x²+²y² + ²z²) + V(x,y,z)

 (Hamiltonian operator) (Eigenfunction) = (Eigenvalue) (Eigenfunction)

 H ψ = E ψ

➢ Eigenfunction is the wave function of an electron corresponding to the energy E

.➢ Eigenfunction is different for each eigenvalue.

 ➢ By solving the Schrödinger equation one can find the wave functions (eigenfunctions) and the corresponding allowed energies(eigenvalue) 

SOLUTION OF SCHRODINGER EQUATION

There are many solutions to the above equation. However, the acceptable solutions must satisfy the following conditions.

1. Ψ must be a single value function

2. Ψ must be continuous.

3. Ψ must be finite.

4. The first derivative of Ψ w.r.t. its variables must be continuous.

5. Condition of Orthogonality 

If ψ1 and ψ2 are two acceptable wave functions, they are orthogonal.

  −+𝜓1𝜓2 𝑑𝜏 = 0

6. Probability of finding particles over the whole space must be unity (Normalisation Condition)

  −+𝜓2 𝑑𝜏 = 1 (𝑑𝜏 gives the volume element given by dx, dy, and dz) 

If ψ is a complex function −+𝜓1𝜓2 𝑑𝜏 = 1

SOME IMPORTANT POINT  

• The solution gives the possible energy levels the electron can occupy and the corresponding wave function of the electron associated with each energy level. For an atom, several wave functions (1, 2, 3) will satisfy these conditions, and each of these has corresponding energy (E1, E2, E3)

• The wave function is a mathematical function whose value depends upon the coordinates of the electron in the atom and does not carry any physical meaning. Such wave functions of hydrogen or hydrogen-like species with one electron are called atomic orbitals.

•  An orbital is a one-electron wave function.

• The quantized energy states and corresponding wave functions which are characterized by a set of three quantum numbers (principal quantum number n. azimuthal quantum number l and magnetic quantum number ml) arise as a natural consequence in the solution of the Schrödinger equation.

• When an electron is in any energy state, the wave functions. corresponding to that energy state contains all information about the electron. 

• The probability of finding an electron at a point within an atom is proportional to the 𝛗²at that point.

•  has no physical quantity.

IMPORTANT FEATURE OF QUANTUM MECHANICAL MODEL OF ATOM

1. The power of electrons in atoms is calculated by numbers (that is, they can have only certain values), for example when electrons are bound to the nucleus in atoms.

2. The presence of calculated energy levels is a direct result of electron-like electromagnetic structures and is approved by Schrödinger’s wavelength calculations.

3. Both the exact position and the exact electron velocity of an atom cannot be determined simultaneously (the Heisenberg uncertainty system). Therefore, the path of the electron atom cannot be determined or known precisely.

4. Atomic orbital is the activity of electron waves in atoms. Whenever an electron is defined by a wave function, we say the electron takes that orbital.

5. Since most such wave activity is possible with an electron, there are many atomic orbitals in the atom. These “one-electron orbital wave function” or orbitals form the basis of the electronic structure of atoms.

6. In each orbital, the electron has a direct force. The orbital cannot contain more than two electrons. In an atom with many electrons, electrons are filled with various orbitals in the order of magnitude. In each electron atom there are many electrons, therefore, there will be an orbital wave activity in it.

7. All the electron information in the atom is stored in its orbital wave function and quantum mechanics help extract this information.

8. The probability of finding an electron in the interior of an atom is equal to the square of the orbital wave activity or 𝛗² at that moment.

9. 𝛗² is known as overcrowded and always prone to overcrowding.

10. 𝛗² When looking at different numbers of different points within an atom, it is possible to predict the area around the nucleus where the electrons will be most present.

11. If the value is 0 then it is known as the node and the finding on an electron for that reason is also 0.

12.   has no significant value.

CONCLUSION 

  Wave function helps us to determine the position of the electron structure of orbitals, it helps to understand the quantum mechanical model of the structure of the atom.