Normalizing Trial Function with 2 Normalization Factors

In summary, the conversation discusses the normalization of a trial function in the context of using the variational method. The trial function is given and has two unknown normalization factors, N1 and N2. The problem is how to normalize a sum of linear equations with their own normalization factors. Various attempts at solving this problem are discussed, including integrating the function and solving simultaneously for the unknown factors. The conversation concludes with the suggestion to perform the integration and use one of the N factors as a free parameter to solve for the ground state energy.
  • #1
timman_24
52
0

Homework Statement


I am given a trial function and before I use the variational method, I need to normalize the trial function. This is easy usually, but I don't know what to do in this specific case:

The trial function is:
X[x]=N1(1-x^2)+N2(x-x^3)
Domain: -1<x<1

N1 and N2 are the normalization factors that I need to solve for.

My problem is that I do not know how to normalize a sum of linear equations each with their own normalization factor. The problem says nothing about weighing one trial term more over another. I thought about normalizing each separately setting them to 50% weight, but I thought that was a cop out.

2. The attempt at a solution

Another thing I tried was writing two equations and solving for the 2 unknown N1 and N2. I split the domain and integrated X[x]^2 from -1 to 0 in one equation and then from 0 to 1 in the other. I then solved the equations simultaneously. However, one N always goes to zero which I do not want.

Does anyone know the proper way to do this?

Thanks
 
Physics news on Phys.org
  • #2
The normalization condition is still just

[tex]1=\int_{-\infty}^{\infty}|\psi(x)|^2dx=\int_{-1}^{1}|X(x)|^2dx[/tex]

perform the integration and you will get a relationship between [itex]N_1[/itex] and [itex]N_2[/itex], leaving you with one free parameter to vary.
 
  • #3
Okay, so one of the N's will be the free parameter I solve for to get the ground state energy?

Thanks for the help.
 

1. What is "Normalizing Trial Function with 2 Normalization Factors"?

"Normalizing Trial Function with 2 Normalization Factors" is a method used in scientific research to adjust for variations in experimental conditions or data. It involves using two normalization factors to standardize the data and make it comparable across different trials or samples.

2. Why is normalizing trial function important?

Normalizing trial function is important because it allows for accurate and fair comparisons between different samples or trials. It takes into account any variations or differences in experimental conditions that may affect the data, ensuring that the results are not biased or misleading.

3. How does normalizing trial function with 2 normalization factors work?

The first normalization factor is used to adjust the data to a common scale, while the second normalization factor is used to correct for any systematic variations between the samples or trials. Together, these factors help to standardize the data and make it more reliable for analysis and comparisons.

4. When should I use normalizing trial function with 2 normalization factors?

Normalizing trial function with 2 normalization factors should be used when there are multiple samples or trials with varying experimental conditions, and you want to compare the data across these different conditions. It is also useful when conducting statistical analyses or making conclusions based on the data.

5. Are there any limitations to normalizing trial function with 2 normalization factors?

While normalizing trial function with 2 normalization factors can be a useful tool, it is important to note that it may not be appropriate for all types of data. It also relies on the assumption that the two normalization factors are independent and accurately reflect the variations in the data. It is always important to carefully consider the appropriateness and limitations of this method before applying it to your research.

Similar threads

  • Advanced Physics Homework Help
Replies
1
Views
498
  • Advanced Physics Homework Help
Replies
3
Views
848
  • Advanced Physics Homework Help
Replies
10
Views
322
  • Advanced Physics Homework Help
Replies
19
Views
325
Replies
1
Views
758
  • Advanced Physics Homework Help
Replies
2
Views
790
  • Advanced Physics Homework Help
Replies
7
Views
1K
  • Advanced Physics Homework Help
Replies
7
Views
949
  • Advanced Physics Homework Help
Replies
1
Views
955
Replies
16
Views
397
Back
Top