Quantum Talk

I think they are on a break.....

Ended up overexaggerating😛😛😛😛😛

The level of focusssssss

The man, the myth, the legendddd

Conservation of momentum is a very powerful tool in classical mechanics that have helped understand several of the properties of forces and is extensively used in engineering. One of the best example of the practical applicability of conservation of momentum is calculating the thrust of a rocket. This post shows the mathematical formulations of the rocket thrusts.

In multivariable calculus, there are three important and independent operators that can be combined in a variety of ways to give other operators. The three main operators as discussed in the previous post are Gradient, Divergence and Curl. These operators can be combined in ways that give other operators of which the best example is Laplacian. Laplacian is basically the divergence of gradient of a scalar field. Laplacian is defined for vector valued functions as well and in this case, each components is treated as a scalar and taken Laplacian of. The end result is again a vector. This post talks about them in detail.

Continuation of the differentiation of a single valued and single variable function to a multivariable function can be done in a variety of ways but the must applicable and logically consistent operations are Gradient, Divergence and Curl. Here, I have made a node of all the important things that one need to get started with these operations.

One of the most important concepts in modern physics is Eigenvalues and Eigenvectors of a matrix. Eigenvectors are also termed as the invariant vectors i.e. they are not changed under matrix transformation other than their length. This property of invariance makes it very useful for understanding Eintein's theory of relativity and quantum mechanics.

Another important term used in matrix operation is its rank. The rank of a matrix is the number of independent Eigenvectors it has. It is also defined as the high order non-zero minor of the matrix.

Unitary Matrix: In the realm of linear algebra, a unitary matrix holds a distinct significance. It is a square matrix that maintains a unique property: its conjugate transpose is identical to its inverse. This property imbues unitary matrices with intriguing properties, particularly in the domain of Quantum Mechanics, where they play a pivotal role in various mathematical formulations and quantum operations.

Hermitian Matrix: Among the array of matrices with specialized properties, the Hermitian Matrix stands out prominently, especially within the context of Quantum Mechanics. A Hermitian Matrix possesses a defining characteristic: its complex transpose coincides with the matrix itself. As a consequence, the elements residing along the main diagonal of a Hermitian Matrix are guaranteed to be real numbers, adding an additional layer of structure and meaning to these matrices within quantum theoretical frameworks.

Today, I attended my first lecture on Matrices (Mathematics) and here are the basic properties of orthogonal matrices that we learnt.

Definition of Orthogonal Matrix
A matrix whose inverse is equal to its transpose are termed as Orthogonal matrices.

Special Properties of Orthogonal Matrices
1. The determinant of an orthogonal matrix is either 1 or -1.
2. The product of two orthogonal matrices is also orthogonal.
3. The inverse of an orthogonal matrix is also orthogonal.

The following images provide the basic proofs of these properties which uses properties of matrix algebra (associativity) and transpose of matrices. Hope you find that informative and useful.