1. Determine (rop|E3,2-1) using the above information. 2. Show that your solution is normalized, i.e., I| (rep|E3,2,-1) (E3,2,-1|rO4) r² sin® dr d@ dø where r is integrated from 0 to o, 0 from 0 to TT, and ø from 0 to 27n.
1. Determine (rop|E3,2-1) using the above information. 2. Show that your solution is normalized, i.e., I| (rep|E3,2,-1) (E3,2,-1|rO4) r² sin® dr d@ dø where r is integrated from 0 to o, 0 from 0 to TT, and ø from 0 to 27n.
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![The eigenvectors in position space (spherical coordinates) of the Hamiltonian operator H
that represent the hydrogen atom system are very complicated:
3
(п —1- 1)!
2n(n + 1)!
2
{r®¢\En!m) =
e-r/(na)
na
A {L (2r/(na)]}Y(0, 4)
'n-l-1
па,
where the integers:
n = 1,2, ..., 0
l = 0,1, ...,n – 1
-l < m sl,
the value a = 0.529 × 10-10m is known as the Bohr radius, the associated Laguerre
polynomial is:
d
E(x) = (-1)" () Lp+q(x)
where
ex
Lq (x) =
q! ldx) (e-*x9),
\dx,
and the spherical harmonic function:
|(21 + 1) (1 – Iml)! eimo pm (cos 0)
Y" (0, 4)
= E
i(lu| + 1)
where
S(-1)",
m 2 0
m < 0'
E =
(1,
d Im/
P" (x) = (1 – x² )Im/21) P.(x),
and
P:(x) :
d
(x² – 1)'.
2'1! \dx)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbbeeb476-a64f-4459-b8af-acf5dbdffec4%2F6222104c-0ba5-4410-8718-3bb7777a0680%2F2cpm54j_processed.png&w=3840&q=75)
Transcribed Image Text:The eigenvectors in position space (spherical coordinates) of the Hamiltonian operator H
that represent the hydrogen atom system are very complicated:
3
(п —1- 1)!
2n(n + 1)!
2
{r®¢\En!m) =
e-r/(na)
na
A {L (2r/(na)]}Y(0, 4)
'n-l-1
па,
where the integers:
n = 1,2, ..., 0
l = 0,1, ...,n – 1
-l < m sl,
the value a = 0.529 × 10-10m is known as the Bohr radius, the associated Laguerre
polynomial is:
d
E(x) = (-1)" () Lp+q(x)
where
ex
Lq (x) =
q! ldx) (e-*x9),
\dx,
and the spherical harmonic function:
|(21 + 1) (1 – Iml)! eimo pm (cos 0)
Y" (0, 4)
= E
i(lu| + 1)
where
S(-1)",
m 2 0
m < 0'
E =
(1,
d Im/
P" (x) = (1 – x² )Im/21) P.(x),
and
P:(x) :
d
(x² – 1)'.
2'1! \dx)
Transcribed Image Text:1.
Determine (ro4 E3,2,-1) using the above information.
2.
Show that your solution is normalized, i.e.,
I| (reo\E3,2-1) (E3,2,–1|r04) r² sin0 dr d0 dø
where r is integrated from 0 to o, 0 from 0 to T, and o from 0 to 27n.
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