The Hydrogen Atom

https://isaacphysics.org/questions/differentiation_6_8?board=aa336c58-d5c6-408e-9a6c-772bbfe10b17&stage=a_level

I tried differentiating and substituting in for the expression, and simplifying, but then I realised you can't use "E" in the answer.

Then my thought process was ok it says "lowest energy" so maybe I differentiate E with respect to r and make that 0?

But that hasn't worked out either - im not sure if im using the right method or if I made a mistake in my working?

My working looks really messy and is virtually undecipherable so im not gonna post it but help would be much appreciated whether my method is right.

Reply 1

mqb2766

Original post by mosaurlodon

https://isaacphysics.org/questions/differentiation_6_8?board=aa336c58-d5c6-408e-9a6c-772bbfe10b17&stage=a_level
I tried differentiating and substituting in for the expression, and simplifying, but then I realised you can't use "E" in the answer.
Then my thought process was ok it says "lowest energy" so maybe I differentiate E with respect to r and make that 0?
But that hasn't worked out either - im not sure if im using the right method or if I made a mistake in my working?
My working looks really messy and is virtually undecipherable so im not gonna post it but help would be much appreciated whether my method is right.

Not worked it through, but hint 2 suggests substituting the wave function into the ODE and noting that the constant terms sum to zero and the (1/r) terms sum to zero as well. I presume thats enough to get a0 and E?

I expect A comes from elsewhere as it can be cancelled from the ODE.

(edited 12 months ago)

Reply 2

mosaurlodon

Somehow I just brute forced the calculation and managed to get it.

I cancelled the A pretty early but the struggle was simplifying the derivative of E with respect to r (and also forgetting the odd minus signs here and there) but the answer was relieving.

Thank you!