Find the normalisation constant using a trial wavefunction

[martinhiggs]

Homework Statement

a particle of mass m, confined to a one dimensional infinite potential of
0$$\leq$$x$$\leq$$1 V(x) = 0
elsewhere V(x) = $$\infty$$

Homework Equations

Choose as a trial wavefunction

$$\Psi$$(x) = Nx[1 - $$\alpha$$x + ($$\alpha$$ - 1)x$$^{2}$$]

Verify that

N$$^{2}$$ = $$\frac{K}{16 - 11\alpha + 2\alpha^{2}}$$

The Attempt at a Solution

1 = <$$\Psi$$|$$\Psi$$>

1 = $$\int^{1}_{0}$$Nx[1 - $$\alpha$$x + ($$\alpha$$ - 1)x$$^{2}$$] Nx[1 - $$\alpha$$x + ($$\alpha$$ - 1)x$$^{2}$$] dx

1 = N$$^{2}$$ $$\int^{1}_{0}$$ x$$^{2}$$[1 - $$\alpha$$x + ($$\alpha$$ - 1)x$$^{2}$$]$$^{2}$$

Is this right so far?? I'm not sure how to carry on. Should I expand the brackets??

 

[jdwood983]

Homework Statement

a particle of mass m, confined to a one dimensional infinite potential of
$$0\leq x \leq 1$$, $$V(x)=0$$
elsewhere $$V(x)=\infty$$

Homework Equations

Choose as a trial wavefunction

$$\Psi(x) = Nx(1 - \alpha x + (\alpha - 1)x^{2})$$

Verify that

$$N^{2} = \frac{K}{16 - 11\alpha + 2\alpha^{2}}$$

The Attempt at a Solution

$$1 = \langle\Psi|\Psi\rangle$$

$$1 = \int^{1}_{0}Nx(1 - \alpha x + (\alpha - 1)x^2)\cdot Nx(1 - \alpha x + (\alpha - 1)x^{2}) \,dx$$

$$1 = N^{2}\int^{1}_{0} x^{2}(1 - \alpha x + (\alpha - 1)x^{2})^{2}\,dx$$

Is this right so far?? I'm not sure how to carry on. Should I expand the brackets??

Yes and Yes. Well, you could possibly integrate it as is, but it'd be easier to see after expanding the parenthesis.

PS: Rather than typing {tex}code{/tex} for every variable, just write the whole thing in tex, it'll look a bit nicer.

 

[martinhiggs]

When I expand the brackets and multiply by x^2

I get the following:

x$$^{6} + x^{2} + 2 \alpha^{2} - \alpha^{4} - 2 \alpha x^{3} + \alpha^{2}x^{4} - \alpha x^{5}$$


This seems totally wrong when I look at what I am supposed to get for N^2...

 

[jdwood983]

When I expand the brackets and multiply by x^2

I get the following:

$$x^{6} + x^{2} + 2 \alpha^{2} - \alpha^{4} - 2 \alpha x^{3} + \alpha^{2}x^{4} - \alpha x^{5}$$


This seems totally wrong when I look at what I am supposed to get for N^2...

I think you expanded incorrectly. You should get

$$\left(\alpha-1\right)^2x^6-2\alpha(\alpha-1)x^5+2(\alpha-1)x^4+\alpha^2x^4-2\alpha x^3+x^2$$

When you integrate this over the range 0 to 1, you should get the answer.

 

[jdwood983]

If it helps see a little bit more clearly, $$K=210$$. Not sure why your question put $$K$$ rather than $$210$$, but it is what it is.

Also, you could fully expand what I wrote down and then integrate, but you'll have a few extra terms to work with; it's usually easier to expand and regroup after integration.

 

[martinhiggs]

Ah, excellent! Thank you SO much for your help! :)