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Atomic and
Answer question 1
![Questions 1 and 2 are based on the following information.
The normalized wave function of a hydogen-like atom is given by
une (r,0,0) = Rne(r)Y (0,0),
where Rne(r) is the radial component of the wave function, defined as
(n-l-1)! (2Z 3/2
2n[(n+1)!]³ nao
Question 1
For the 4p state, construct
-P/2
p/2 p² [²+1 (p),
-20+1
Rne(r)
with ao being the Bohr radius, p=2Zr/nao, the transformed distance from the nucleus, Z, the atomic
number, and L(p) are modified Laguerre polynomials. Back to equation (1), Ym (0,0) are spherical
harmonics, which represent the angular part of the wave function me (r,0,0), with -l≤ m <+l. The
few first modified Laguerre polymials and spherical harmonics are given on page 5.
(1)
(a) the radial wave function Rne(r).
(b) The three components of the normalized wave function for the 4p state, i.e., for m = -1,0,1
Question 2
Find the values of variable r where the 4p wave function has nodes.
HINT: The node of a function is the point where it is zero. Notice that the wave function (r,0,0)
is zero if at least one of its two factors is zero, i.e., Rne(r) = 0 or Ym (0,0) = 0. On the other hand,
Rne(r) = 0 if the corresponding modified Laguerre polynomial is zero, or p = 0.](https://content.bartleby.com/qna-images/question/900ce081-76b5-4728-be49-0a5172963ecc/1e1ee696-cbcf-4d3a-bf97-1d3c80089eaa/w2hubcn_processed.jpeg)
Transcribed Image Text:Questions 1 and 2 are based on the following information.
The normalized wave function of a hydogen-like atom is given by
une (r,0,0) = Rne(r)Y (0,0),
where Rne(r) is the radial component of the wave function, defined as
(n-l-1)! (2Z 3/2
2n[(n+1)!]³ nao
Question 1
For the 4p state, construct
-P/2
p/2 p² [²+1 (p),
-20+1
Rne(r)
with ao being the Bohr radius, p=2Zr/nao, the transformed distance from the nucleus, Z, the atomic
number, and L(p) are modified Laguerre polynomials. Back to equation (1), Ym (0,0) are spherical
harmonics, which represent the angular part of the wave function me (r,0,0), with -l≤ m <+l. The
few first modified Laguerre polymials and spherical harmonics are given on page 5.
(1)
(a) the radial wave function Rne(r).
(b) The three components of the normalized wave function for the 4p state, i.e., for m = -1,0,1
Question 2
Find the values of variable r where the 4p wave function has nodes.
HINT: The node of a function is the point where it is zero. Notice that the wave function (r,0,0)
is zero if at least one of its two factors is zero, i.e., Rne(r) = 0 or Ym (0,0) = 0. On the other hand,
Rne(r) = 0 if the corresponding modified Laguerre polynomial is zero, or p = 0.
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