@Carl Love 

Oh okay, I was being a little too ambitious in my hope of Maple solving those solutions exactly.
I suppose you could add an additional constraint for b in terms of a, but anyhow wouldn't really know how to justify it.

Once again thanks for the effort you put into helping, I am still fairly new to using the software but the process has been a learning experience.

Regards, 

Billy.

@Carl Love 

My mistake, I meant to say 

amin := solve(D2(a,b), a);

bmin := solve(D3(a,b), b);

where D2 is the derivative dE/da and D3 is defined to be dE/db.

Here is the worksheet:

Minimum_Energy_Sheet_Y0_complicated.mw

Update:

I have extended the function so that it is of two variables (a,b), but when trying to use solve(D2,a) where D2(a,b), it just comes up with a warning: solutions may have been lost. For some reason all I can solve for is m.

Warning: http://imgur.com/V7Hr8aP

Solving for m: http://imgur.com/N8i8qcZ

Any ideas as to how to solve for both derivatives = 0 to find amin and bmin such that:

amin := solve(D2(a,b), a);

bmin := solve(D2(a,b), b);

Thanks in advance.

@Carl Love 

Thanks once again for your help, it really was top notch. Managed to fix the work sheet up so that I can start using other more complicated wavefuntions.

Many thanks, Billy.

@Carl Love 

Here is the sheet for Maple.

If you could show me how to auto select the correct minimum a value as well that would be really helpful as well.

Many thanks.

@Carl Love thanks for the help, it did the evaluation. However it gave out 3 solutions, any idea why?

The first one is the one I need, so I can manually select it. But when I sub in a_min into the equation for E(a), it refuses to evaluate all the a's (although it still evaluates some...). 

Here is the image: http://i.imgur.com/Rfjj0ZB.png