# Evaluate the ground state energy using the variational method

## Homework Statement

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

## Homework Equations

$$a = \infty$$ and $$\psi = \frac{A}{x^{2}+c^{2}}$$.

## The Attempt at a Solution

$$1 = |A|^{2}\int_{-a}^{a}\frac{1}{(x^{2}+c^{2})^{2}}\,\text{d}x.$$ Is this a correct way to start? How can I calculate this? I used Mathematica, but it only gives some weird-looking answers.

Last edited:

Gokul43201
Staff Emeritus
Gold Member
Yes, that's how you normalize the trial function. And what do you mean by "weird"?

Also, why do you say a=infty?

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?

Also, why do you say a=infty?

In the paper, it says: "Assume that $$a=\infty$$ and use the trial $$\psi(x)=...$$."

Gokul43201
Staff Emeritus
Gold Member
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 $$a=\infty$$ and use the trial $$\psi(x)=...$$."
Then please write this down as part of the question.Always write down the complete question. Do not summarize or reword in any way.