Eigenvector Application

In modern physics, the application of eigenvectors is very important. The exact location of the electron can be found with the help of the eigenfunction. In geometrical application, the eigenvector representing nonzero real numerical eigenvalue points in a direction that shows the stretched transformation, and the eigenvalue is the factor by which it is stretched. In 1926 the field of quantum mechanics was developed independently by two famous scientists Werner Heisenberg and Erwin Schrödinger. Schrödinger developed the fundamental equation of quantum mechanics. He used this concept in quantum mechanics to find the position of electrons.

 The eigenvector is not rotated in the multidimensional vector space. Eigenvectors are prominent in the analysis of linear transformations. The prefix” eigen” comes from the German word eigen, which means “own.” It is originally used to study the principal axis of the rotational dynamics of rigid bodies. Eigenvectors have numerous applications in the field of atomic orbitals, vibration analysis, facial recognition, matrix diagonalization and stability analysis.

Consider an eigenvector V whose linear transformation is T, a nonzero vector. When the transformation T is applied to it, it does not change direction. Applying T to the eigenvector we found the eigenvector by the scalar value λ, called an eigenvalue.

Mathematically it can be written as:

       T (V) =  (V)

It is known as the eigenvalue equation .λ may be any scalar it may be negative, positive, zero or complex.

Application

Many complex theories can be simplified with the help of eigenvectors. Eigenvectors find numerous applications in physics and chemistry. The most common application of eigenvectors is in geometric transformation, Schrodinger equation, molecular orbitals, etc.

Schrödinger equation

A classic example of an eigenvector and eigenvalue equation is the transformation T represented in terms of a time-independent differential operator is the Schrödinger equation in quantum mechanics.

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 Schrödinger equation is 

          H ψ = E ψ

Where

       E = Total energy of electron

       ψ = wavefunction of electron is its eigenfunctions corresponding to the eigenvalue

       H = Hamiltonians which, is a second-order differential operator

      In one dimension

                                             Ĥ = −ħ²2m d²/dx² + V(x) 

Where, 

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

In three dimensions, 

            Ĥ = −(ħ²/2m) (∂²/∂x² + ∂²/∂y² + ∂²/∂z²) + V(x,y,z)

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

                                                                  H ψ = E ψ

Here the eigenvector of eigenvalue Ψ must be a single value function, continuous and finite. The first derivative of Ψ w.r.t. its variables must be continuous.

It obeys the Condition of Orthogonality,

if ψ1 and ψ2 are two acceptable eigenfunctions in the form of wave functions, they are orthogonal.

                             ∫𝜓1𝜓2 𝑑𝜏 = 0

It also obeys the Normalisation Condition, which is the probability of finding particles over the whole space must be unity.

  ∫𝜓1𝜓2 𝑑𝜏 = 1 (𝑑𝜏 gives the volume element given by dx, dy, and dz) 

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.

It is known as overcrowded and always prone to overcrowding 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. If the value is 0, then it is known as the node, and the finding on an electron for that reason is also 0.

Communication systems

Claude Shannon used the eigenvalues to estimate the theoretical conceptual limit as to how much information can be transmitted through a communication medium like mobile line, radio signal or air. He did this by applying eigenvectors and eigenvalues to calculate these by the communication channel expressed as a matrix and then evaluated on the eigenvalues.    

Geology and Glaciology

 In the field of geology, mainly in the study of glacial till, eigenvector application and eigenvalues are used as a special method by which a huge amount of information of a clast fabric’s constituents’ orientation and dip can be summarised in a 3-D space with the help of  six numbers. A geologist collects all such data for hundreds or thousands of clasts in a soil sample that can only be compared graphically by a Tri-Plot (Sneed and Folk) diagram, or as a Stereonet on a Wulff Net.

Mechanical engineering 

In mechanical engineering the application of eigenvectors allows us to reduce and simplify a linear operation into separate, more straightforward problems. For example, if an external deforming force is applied to a plastic body, this deformation can be dissected into principal directions whose direction is in  the highest deformation. The vectors in the principal directions are the eigenvectors and the percentage deformation in each principal direction is the corresponding eigenvalue.

Another application of eigenvectors of the moment of inertia defines the principal axes of a rigid body. The tensor of the moment of inertia is a key quantity that is required to evaluate the rotation dynamics of a rigid body around its centre of mass.

Vibration analysis

Eigenvector is applied in the vibration analysis process of mechanical structures with many degrees of freedom. The eigenvalues are the natural frequencies (eigenfrequencies) of vibration or oscillation, and the eigenvectors are represented by the shapes of those vibrational modes. Mathematically, it is represented by an equation 

                (w2m + wc + k ) x = 0 

This property of orthogonality of these eigenvectors gives decoupling of the differential equations so that the system can be shown as a linear algebraic summation of the eigenvectors. This eigenvalue problem of complex structures is sometimes solved using the process of finite element analysis but probability gives rise to a solution of scalar-valued vibration problems.

 In the structure of the bridge, this natural frequency of the bridge is the eigenvalue of the smallest magnitude of that system, which is the model of the bridge.

Conclusion

The eigenvalues are not only used to discover new and better designs for the future but also used to explain natural occurrences. Some of the results are quite surprising to the field of physics. Eigenvectors have a wide range of applications in various fields like physics, chemistry, geology for solving many natural problems and explaining many natural events. Some of their applications in quantum mechanics, communication systems, mechanical engineering, vibrational analysis, etc. are explained in the article.