Mathematical steps in the variational method derivation

[mody mody]

Hello
i read a quantum mechanics part and found a part of mathematics that i am not a familiar with it so i need a help how to get the equation at the bottom of the attached photo (the mathematics of applying variational method )

 

[BvU]

Hello mody²

The one numbered (2) ? Or the one ending in $\sqrt\alpha$ ?
Starting from where ?
What makes you use the term 'variational method' ? Any context ? Where was it mentioned before ?

 

[mody mody]

i want second equation that ended by root of alpha
starting by just following the sequence given at the photo

that was mentioned here
https://arxiv.org/pdf/1510.07813.pdf

 

[mike1000]

Gamma function?

https://en.wikipedia.org/wiki/Gamma_function

 

[BvU]

starting by just following the sequence given at the photo

I mean where, at what point, at what level, do I have to start ? Do you have any experience at all with a QM course, the hydrogen atom wave functions, the $Y_{lm}$ ? If they are not the 'usual spherical harmonics' to you, I suggest you take a QM textbook and see how far you can come.

Where, exactly, do you get stuck when you work out this <H> ? Show your steps in detail and someone will help you put the last pieces of the puzzle together. Starting from 1+1=2 is really too tedious.