Simple question on minimising the trial wavefunction

[rwooduk]

Homework Statement

After a calculation of the lowest energy using two variational parameters a and b it is found that: $$E_{T}(a,b) = 2a^{2} + 16b^{2}+a$$
What is the optimal (minimum) value of $$E_{T}$$

Homework Equations

It's just derivation.

The Attempt at a Solution

$$\frac{\delta E_{T}}{\delta a} = 4a + 1 = 0$$

therefore a= -1/4

$$\frac{\delta E_{T}}{\delta b} = 32b = 0$$

therefore b=0

when the values are put into E_T I get zero?

$$E_{T}(a',b') = 2 (-\frac{1}{4})^{2} - \frac{1}{4} = 0$$

why would it be zero?

also were were told to take the second derivative to find the inflection? why would we do this? what does it tell us?

Thanks in advance for any help

 

[Orodruin]

It is not zero. You did a computational error:

2(1/4)^2 -1/4 = 2/16 - 1/4 = 1/8 -2/8 = -1/8

 

[rwooduk]

It is not zero. You did a computational error:

2(1/4)^2 -1/4 = 2/16 - 1/4 = 1/8 -2/8 = -1/8

hm that would explain it, many thanks for pointing this out, appreciated!

Any idea as to why the second derivative is taken and what it would tell us about the system?

Thanks for the reply!

 

[rwooduk]

i'm probably being really stupid here but what would I do with something like:

E_T (a,b,c) = (a+b)² - ab + c⁴

which gives:

dE/da = 2a + b = 0
dE/db = 2b + a = 0

i can see it has solutions but what values should I use?