Evaluate the ground state energy using the variational method

In summary, the conversation is discussing the use of the variational method to evaluate the ground state energy for the function V(x) = k|x|, x \in [-a,a], V(x) = \infty, x \notin [-a,a]. The homework equation a = \infty and \psi = \frac{A}{x^{2}+c^{2}} is used to normalize the trial function, and the attempt at a solution involves integrating the function 1/((x^2+c^2)^2) from -a to a. The speaker also mentions using Mathematica, which gave them unexpected results. The conversation also touches on the use of imaginary values for c and the instruction to assume a=\
  • #1
Urvabara
99
0

Homework Statement



[tex]V(x) = k|x|, x \in [-a,a], V(x) = \infty, x \notin [-a,a][/tex]. Evaluate the ground state energy using the variational method.

Homework Equations



[tex]a = \infty[/tex] and [tex]\psi = \frac{A}{x^{2}+c^{2}}[/tex].

The Attempt at a Solution



[tex]1 = |A|^{2}\int_{-a}^{a}\frac{1}{(x^{2}+c^{2})^{2}}\,\text{d}x.[/tex] Is this a correct way to start? How can I calculate this? I used Mathematica, but it only gives some weird-looking answers.
 
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  • #2
Yes, that's how you normalize the trial function. And what do you mean by "weird"?

Also, why do you say a=infty?
 
  • #3
Gokul43201 said:
Yes, that's how you normalize the trial function. And what do you mean by "weird"?

This is what I put:
In[2]:=
\!\(∫\_\(-a\)\%a\( 1\/\((x\^2 + c\^2)\)\^2\) \[DifferentialD]x\)

This is what I got:
Out[2]=
\!\(2\ a\ If[Im[c\/a] ≥ 1 || Im[c\/a] ≤ \(-1\) || Re[c\/a]
≠ 0, \(c\/\(a\^2 + c\^2\) + ArcTan[
a\/c]\/a\)\/\(2\ c\^3\),
Integrate[1\/\((c\^2 + \((a - 2\ a\ x)\)\^2)\)\^2, {x, 0, 1}, \
Assumptions \[Rule] Re[c\/a] \[Equal] 0 && \(-1\) < Im[c\/a] < 1]]\)

I do not understand. What are those imaginary things?

Gokul43201 said:
Also, why do you say a=infty?

In the paper, it says: "Assume that [tex]a=\infty[/tex] and use the trial [tex]\psi(x)=...[/tex]."
 
  • #4
Urvabara said:
This is what I put:
In[2]:=
\!\(∫\_\(-a\)\%a\( 1\/\((x\^2 + c\^2)\)\^2\) \[DifferentialD]x\)

This is what I got:
Out[2]=
\!\(2\ a\ If[Im[c\/a] ≥ 1 || Im[c\/a] ≤ \(-1\) || Re[c\/a]
≠ 0, \(c\/\(a\^2 + c\^2\) + ArcTan[
a\/c]\/a\)\/\(2\ c\^3\),
Integrate[1\/\((c\^2 + \((a - 2\ a\ x)\)\^2)\)\^2, {x, 0, 1}, \
Assumptions \[Rule] Re[c\/a] \[Equal] 0 && \(-1\) < Im[c\/a] < 1]]\)

I do not understand. What are those imaginary things?
I can't read that easily, but I believe it allows for values of c that are not real. For the problem, you could chose to limit yourself to real c.

In the paper, it says: "Assume that [tex]a=\infty[/tex] and use the trial [tex]\psi(x)=...[/tex]."
Then please write this down as part of the question.Always write down the complete question. Do not summarize or reword in any way.
 

FAQ: Evaluate the ground state energy using the variational method

1. What is the variational method and how does it work?

The variational method is a mathematical technique used to approximate the ground state energy of a quantum mechanical system. It works by finding an approximate wave function that minimizes the energy expectation value, taking into account the uncertainty principle.

2. How is the variational method used to evaluate the ground state energy?

The variational method is used by finding a trial wave function that depends on one or more adjustable parameters. These parameters are then varied to minimize the energy expectation value, giving an approximation of the ground state energy.

3. What are the advantages of using the variational method?

The variational method is advantageous because it provides a lower bound on the ground state energy, meaning that the actual ground state energy will always be equal to or lower than the calculated value. It also allows for the inclusion of different physical effects and interactions, making it a versatile tool for studying quantum systems.

4. Are there any limitations to the variational method?

One limitation of the variational method is that it relies on the choice of a suitable trial wave function. If the chosen function is not a good approximation of the true ground state wave function, the calculated energy may not be accurate. Additionally, the method can become computationally intensive for complex systems.

5. How does the variational method compare to other methods for evaluating the ground state energy?

The variational method is a simple and intuitive approach that can provide accurate results for a wide range of systems. However, it may not be as accurate as other more advanced methods such as the Hartree-Fock method or coupled cluster theory. These methods are often more computationally demanding and are better suited for studying larger and more complex systems.

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