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Siddharth Sahu

In this lesson we are going to develop mathematical  basics needed for better understanding  of quantum  mechanics and also beneficial for all GATE exam.

Hindi

(Hindi) Quantum Mechanics: GATE (Physics)

In today's lecture we talk about Linear Vector Spaces and look at some example of them. LVS can be said a building block of high quantum Tower. So it's important for us to understand it.

Linear Vector Spaces (in Hindi)

Here We get a idea of dual vector space. It is important to know about them so that we can define inner product of two vectors with more genrality.

Dual Vector Spaces (in Hindi)

Here I'm going to give you a more General idea of dot product  that is called inner product of vector. After that I discuss the concept of length of a vector (Norm)  and  some theorem related to it.

Inner Product and Norm of a Vector (in Hindi)

Here another important topic from Linear vector space and that is Independent and dependent vectors. So take into what are linearly independent vectors and what linearly dependent vectors and work out some examples.

Linearly Independent Vectors (in Hindi)

In today's lecture we talk about the dimensionality of a vector space and then go on to Concept lf basis in Vector Space that take us to orthonormal Basis.

Dimension of a Vector Space and Orthonormal Basis (in Hindi)

You will not always get the orthonormal basis but You can make them orthonormal basis How?  Here the Gram-Schmidt Orthonormalization to Normalize the oblique vectors.

Gram-Schmidt Orthonormalization (in Hindi)

Problem Solution Of Lecture 06

Here How we can expand any 2x2 matrix in pauli matrix and Identity element.

Pauli Matrix As Basis

Think Matrix as Linear transformation is building block of Linear algebra.

Linear Transformation as Matrix

Here I talk about some example of Linear transformation.

Matrix as Linear Transformation : 02

In this video We will talk about Linear operator and take a look on some of their properties and example.

Linear Operator

In today's lecture we talk about projection operator and it's properties.

The Projection Operator

Linear operator in n dimensional vector space can be represent as nxn matrix take a look How?

Matrix Elements Of Linear Operator

Let's How to provide an basis to Linear operator and How to expand it in this basis.

Operator Spaces

Operator take ket to ket,  adjoint operator take bra to bra. Let's see How?

Adjoint Of An Operator

Vector can only write in the linear combination of those vector which are linearly independent and span the lvs.  What does Span mean?  Let's see.

Span of LVS

In this video we talk about Hermition And Unitary Operators and see some of properties of them.

Hermition, Anti-Hermition And Unitary Operators

Unitary Operator preserve the inner product in lvs,  Let's see proof.

Problem Solutions Lec. 17

You can say a same thing in two different way,  in the similar manner you can transform a. Vector to another. Vector in two different ways and that is active transformation and passive transformation in this lecture we talked about active transformation and passive transformation.
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Active And Passive Transformation

In this lecture we learn about some special vectors called Eigen vector that have special property that they remain in same diagonal line after the Transformation.

Eigen Vector And Eigen Values

In this video we generalize the Eigen value problem , Talk about characteristic equation,  An Example and Degeneracy.

The Eigen Value Problem

In today's lecture we talk 3 theorem from your linear algebra course about the eigen values of Hermition Operators. We are not going to their rigorous proofs but take it as given and analyze it.

EigenValues Of Hermition Matrices

In Today's lecture we talk about degeneracy and How to deal with degeneracy.

Degeneracy

In this Lecture we talk about three statements of spectral theorem of unitary operators & see a proof and a trivial example.

Spectral Theorem Of Unitary Operators

In this video we talk about how to change you basis to someother one.

Change Of Basis

In this video we talk about two very important topic that are eigen basis and diagonalization of a matrix.

Diagonalization Of Matrix & Eigen Basis

In this video we see that diagonalization of Hermition matrix is always possible and see the simultaneous diagonalization of Hermition Operator.

Diagonalization Of Hermition Matrix

In quantum mechanics and quantum field theory, the propagator is a function that specifies the probability amplitude for a particle to travel from one place to another in a given time, or to travel with a certain energy and momentum. In Feynman diagrams, which serve to calculate the rate of collisions in quantum field theory, virtual particles contribute their propagator to the rate of the scattering event described by the respective diagram. These may also be viewed as the inverse of the wave operator appropriate to the particle, and are, therefore, often called (causal) Green's functions (called "causal" to distinguish it from the elliptic Laplacian Green's function).[1][2]

The Propagator

The Propagator Part 2

We know about scalar function that take scalar as input and give an scalar as output.  In video we discuss the concept of functions of operators.

Functions Of Operators

Ever think how the action operator change with some parameter? To do that we use the idea of differential but how you differentiate the operator?  Is they behave same as function? If not when they do?  These questions might occurring in your mind , let's answer them.

Derivatives Of Operators

In this video ,we consider infinite dimensional linear vector spaces which by definition  are those whose number of linearly independent vectos is not bounded.

Lessons

32 lessons • 4h 43m

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Linear Vector Spaces (in Hindi)

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Dual Vector Spaces (in Hindi)

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Inner Product and Norm of a Vector (in Hindi)

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Linearly Independent Vectors (in Hindi)

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Dimension of a Vector Space and Orthonormal Basis (in Hindi)

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Gram-Schmidt Orthonormalization (in Hindi)

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Problem Solution Of Lecture 06

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Pauli Matrix As Basis

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Linear Transformation as Matrix

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Matrix as Linear Transformation : 02

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