# QM Sankar 2nd edition -- gram schmidt process on example 1.3.2

• bugatti79
In summary, the conversation was about trying the Gram-Schmidt process on a specific example and discussing the steps involved. The speaker explained the process and asked the listener if they understood it and the theorem behind it. They also discussed the use of different notation and the importance of fully understanding the process before making any changes. The speaker later mentioned that they have studied the topic further and now feel confident in solving the problem.
bugatti79
Hi

I was trying the gram schmidt process on example 1.3.2 page 43 as shown below
u1=(3,0,0), u2=(0,1,2), u3=(0,2,5)v1=u1=(3,0,0)
v2=u2-proj_(w1) u2=(0,1,2)-((<0,1,2><3,0,0>)/9)(3,0,0)

However the top line inner product gives 0...

Lets try to walk you through this.
First do you know what you are actually doing? i.e. explain the gram-schmidt process in a few lines

Second, have you understood the theorem? What does it tell you? (bonus: why do we need it)

And finally why have you used different notation?
I wouldn't advise changing notation before you are certain you really get what is happening.

bugatti79
I have been studying this further. It is clearer to me now and I now know how to proceed with this problem. Thanks

## What is the Gram-Schmidt process?

The Gram-Schmidt process is a mathematical technique used in linear algebra to convert a set of linearly independent vectors into a set of orthonormal vectors. This process is commonly used in applications involving vector spaces, such as in quantum mechanics.

## Why is the Gram-Schmidt process important in quantum mechanics?

In quantum mechanics, it is often necessary to work with orthonormal basis vectors to accurately describe the state of a quantum system. The Gram-Schmidt process allows for the creation of such basis vectors from a set of linearly independent vectors, making it a crucial tool in quantum mechanics.

## Can you provide an example of the Gram-Schmidt process in action?

Sure! In the 2nd edition of QM Sankar, example 1.3.2 walks through the Gram-Schmidt process using a set of three linearly independent vectors in three-dimensional space. The process involves finding the orthogonal projections of each vector onto the span of the previous vectors and then normalizing them to create the orthonormal basis vectors.

## What is the purpose of the Gram-Schmidt process in this specific example?

In example 1.3.2, the Gram-Schmidt process is used to create an orthonormal basis for a three-dimensional vector space. This basis can then be used for various calculations and applications in quantum mechanics.

## Are there any limitations or drawbacks to the Gram-Schmidt process?

One limitation of the Gram-Schmidt process is that it can only be applied to a set of linearly independent vectors. If the initial set of vectors is not linearly independent, the process will fail to produce an orthonormal basis. Additionally, the process can be computationally intensive for large sets of vectors.

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