Please, carefully read the request's instructions: - Kindly, DO NOT answer this question if you have either already answ
Posted: Tue Apr 26, 2022 7:47 pm
Please, carefully read the request's instructions:
-
Kindly, DO NOT answer this question if you have either
already answered it before or are unable to answer ALL parts of
it.
Respectfully, please only answer if you can SOLVE ALL
PARTS - INCLUDING OPTIONAL PARTS (if any). This is all ONE question
with multiple parts/steps: A - F. Please answer ALL PARTS
(INCLUDING OPTIONAL PARTS, IF ANY). Kindly guide me
through the process. Please be concise, legible and thorough.
.
.
.
.
PLEASE MAKE SURE YOU READ THE INSTRUCTIONS! Answer parts
A - F. NOT JUST A SINGLE PART. PLEASE. DO NOT ANSWER UNLESS YOU CAN
SOLVE ALL PARTS!
.
.
.
.
.
.
Problem 8. [Hydrogen atom.] We define the associated Laguerre polynomials L (x) as dk L (x) = (-1)" d.sk In+k(x). (a) By differentiating the Laguerre differential equation k times d2 d22 d In +(1-2) Ln+nLn = 0 dr = show that L*(x) satisfy the associated Laguerre equation d2 d da2 L'+ (k+1–2)+n1 = 0. dx (b) Define = e k/ -3/2q(k+1)/2L (2) and show that a satisfies the following differential equation n d2 k2 - 1 + 1 2n + k +1 + 4 2.c ) = 0. d.x2 n2 72 4.r2
You will recognize that this is the trick we discussed earlier - it is the redefinition of the function in the second order differential equation which makes the term with the single derivative disappear. (c) Write the general solution of the time-independent hydrogen eigenproblem Ze2 h2 -V– K 2m -U = EU = r by the separation of variables. Use the standard constant labels m and 1(1+1) for the separation of o and 8 dependence. You will recognize that the radial part results in spherical harmonics. After isolating the radial part only you should get: ħ 1 d 2m p2 dr () dR dr Ze2 h2 1(1+1) K R+ R= ER r 2m p2
(d) You are now set to solve the radial part. Change the variables as p= ar (a? = 8m|E|/h?, note that since E is negative |E| = -E) and with 1 = 2mK Ze</(ah?) multiplication by p yields = р 1 pro (no ) + C -*_1971) p<(p) = 0. 10,4" 1d pdp w + PX) = dp 4 = Here x( ) R(p/a) R(r) is just the name for function R with the scaled argument. Compare this expression to the result of part (c) and find the solution to the radial hydrogen equation. Show that the first term is equal to (ex())". (e) Comparing equations in (c) and (d) you should find the expression for px() in terms of . Work through the definitions backwards and write explicitly the solution for R(r) in terms of Le with the appropriate n and k read off from the equation in (d). (f) Put it all back together with the angular part solution, and write the full solution for the hydrogen atom. Did you get it right? Congratulations! Solving this problem is an important stepping stone in the education of every physicist.
-
Kindly, DO NOT answer this question if you have either
already answered it before or are unable to answer ALL parts of
it.
Respectfully, please only answer if you can SOLVE ALL
PARTS - INCLUDING OPTIONAL PARTS (if any). This is all ONE question
with multiple parts/steps: A - F. Please answer ALL PARTS
(INCLUDING OPTIONAL PARTS, IF ANY). Kindly guide me
through the process. Please be concise, legible and thorough.
.
.
.
.
PLEASE MAKE SURE YOU READ THE INSTRUCTIONS! Answer parts
A - F. NOT JUST A SINGLE PART. PLEASE. DO NOT ANSWER UNLESS YOU CAN
SOLVE ALL PARTS!
.
.
.
.
.
.
- Please Carefully Read The Request S Instructions Kindly Do Not Answer This Question If You Have Either Already Answ 1 (76.27 KiB) Viewed 24 times
- Please Carefully Read The Request S Instructions Kindly Do Not Answer This Question If You Have Either Already Answ 2 (91.24 KiB) Viewed 24 times
- Please Carefully Read The Request S Instructions Kindly Do Not Answer This Question If You Have Either Already Answ 3 (128.61 KiB) Viewed 24 times
You will recognize that this is the trick we discussed earlier - it is the redefinition of the function in the second order differential equation which makes the term with the single derivative disappear. (c) Write the general solution of the time-independent hydrogen eigenproblem Ze2 h2 -V– K 2m -U = EU = r by the separation of variables. Use the standard constant labels m and 1(1+1) for the separation of o and 8 dependence. You will recognize that the radial part results in spherical harmonics. After isolating the radial part only you should get: ħ 1 d 2m p2 dr () dR dr Ze2 h2 1(1+1) K R+ R= ER r 2m p2
(d) You are now set to solve the radial part. Change the variables as p= ar (a? = 8m|E|/h?, note that since E is negative |E| = -E) and with 1 = 2mK Ze</(ah?) multiplication by p yields = р 1 pro (no ) + C -*_1971) p<(p) = 0. 10,4" 1d pdp w + PX) = dp 4 = Here x( ) R(p/a) R(r) is just the name for function R with the scaled argument. Compare this expression to the result of part (c) and find the solution to the radial hydrogen equation. Show that the first term is equal to (ex())". (e) Comparing equations in (c) and (d) you should find the expression for px() in terms of . Work through the definitions backwards and write explicitly the solution for R(r) in terms of Le with the appropriate n and k read off from the equation in (d). (f) Put it all back together with the angular part solution, and write the full solution for the hydrogen atom. Did you get it right? Congratulations! Solving this problem is an important stepping stone in the education of every physicist.