Are you looking for some short and reliable notes during your CSIR-NET preparations? Then, you have come to a perfect place!
Candidates preparing for their CSIR NET exam might need to get some short study notes and strategies to apply while preparing for the key exam of their life. At this point, We at Byjus Exam Prep come up with short notes on the Perturbation Method, which comes under the Physical Chemistry section of the Chemical Science syllabus.
Our experienced Exam experts have meticulously designed this set of short notes on the Perturbation Method to give you the most standard set of study materials to be focused upon. In this cut-throat competitive world, students need to prepare themselves with the best study materials to help them learn and for their future. So, here we are offering the best study notes that are reliable and can be used by the students during their preparations for the upcoming CSIR-NET 2021 exam.
Study Notes on Perturbation Method
It is one of the Approximate methods.
Let’s derive the equation for a first-order correction to the energy. The problem that we wish to solve is
has been solved exactly, so that the and are known. In order to keep track of the order of our perturbation expansion, it is convenient to introduce a parameter λ into the Hamiltonian operator:
The factor λ is simply a bookkeeping device that will help us identify to what order our resultant perturbation equations are valid. We shall see that terms linear in λ give us what we call first-order corrections, terms in λ2 give us second-order corrections, and so on. At the end of the problem, once we have calculated corrections to the desired order, we shall simply set λ = 1.
The fact that in is of the form means that and En will depend upon λ. We assume that we can express the and En in as power series in λ, so that
If Equations (5) and (6) are to be useful, successive terms in their series must grow progressively less important and we can obtain good approximations to and En with just a few terms.
We substitute Equations 5 and 6 into 1 to obtain-
Each side of this equation is an expansion in λ, which can be written as-
where O(λ3) means terms of order λ3 and higher. Notice that both terms in the first set of parentheses, the coefficient of λ0, are of zero-order, all four terms in the second set of parentheses, the coefficient of λ1, are of the first order, and so on.
Because λ is an arbitrary parameter, the coefficients of each power of λ must be equal to zero separately for Equation 7 to hold. The terms in the first set of parentheses, the coefficient of λ0 cancel because of Equation 3.
Let's look at the coefficient of λ1:
Equation 8 can be simplified considerably by multiplying both sides from the left by integrating over all space. By doing this and then rearranging slightly, we get-
It is convenient (and economical) at this stage to use the bracket notation and write Equation 9 as
The integral in the last term in Equation 10 is unity because we take to be normalized. More importantly, however, the first term on the left side is equal to zero.
because of Equation 3.
Thus, Equation 10 becomes
Second Order Perturbation
It is the correction to the energy i.e. based on the function that is modified during 1st order perturbation energy.
For the second-order perturbation, the energy state can never be equal. The energy correction for the second order is-
If n = m then the value becomes zero.
- Study Notes for CSIR NET Life Science - (Download PDF)
- Study Notes for CSIR NET Chemical Science - Download PDF Here!
Hope the above article was helpful for you. Let me know your feedback in the comments section below!!!
More from us:
- Get Unlimited access to Structured Live Courses and Mock Tests - Online Classroom Program
- Get Unlimited access to 60+ Mock Tests - Buy Test Series
Download the BYJU’S Exam Prep App Now.
The Most Comprehensive Exam Prep App.
App Link: https://bit.ly/3sxBCsm