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: [tex]E_{T}(a,b) = 2a^{2} + 16b^{2}+a[/tex]
What is the optimal (minimum) value of [tex]E_{T}[/tex]

Homework Equations

It's just derivation.

The Attempt at a Solution

[tex]\frac{\delta E_{T}}{\delta a} = 4a + 1 = 0[/tex]

therefore a= -1/4

[tex]\frac{\delta E_{T}}{\delta b} = 32b = 0[/tex]

therefore b=0

when the values are put into E_T I get zero?

[tex]E_{T}(a',b') = 2 (-\frac{1}{4})^{2} - \frac{1}{4} = 0[/tex]

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?

 

[paranormal]

I would like to clarify a few things about the content provided. Firstly, it is important to note that the provided function E_{T}(a,b) is not a trial wavefunction, but rather an expression for the total energy of a system with two variational parameters a and b.

In order to minimize E_{T}, we can use the method of taking the derivative with respect to each parameter and setting it equal to zero. This will give us the optimal values for a and b that minimize E_{T}. However, it is important to note that this method only gives us a local minimum, and there may be other values of a and b that result in a lower total energy. Therefore, it is important to also consider the physical meaning of the parameters and the system being studied in order to determine if the obtained values are truly optimal.

Regarding the question about why E_{T} would be zero when the obtained values for a and b are plugged in, it is likely due to a mistake in the calculation. When a=-1/4 and b=0 are substituted into the original expression for E_{T}, the result is E_{T} = -1/4, not zero.

Finally, the reason for taking the second derivative to find the inflection point is to determine whether the obtained values for a and b correspond to a minimum or a maximum. A positive second derivative indicates a minimum, while a negative second derivative indicates a maximum. This information can help in determining the reliability of the obtained values and whether further optimization is necessary.