CHEM 442 Lecture 17 Problems
17-1. Write the Coulomb potential energy between a nucleus with Z protons and an
electron separated by distance r in vacuum. Write down the force the electron
experiences. What is the direction of the force?
17-2. Consider a classical system of two masses m1 and m2 at Cartesian coordinates x1
1
and x 2 , respectively, interacting through a Coulomb potential
. The total energy
x1 - x 2
1
1
1
can be written as m1x12 + m2 x 22 +
, where x (the dot) means time derivative.
2
2
x1 - x 2
1
1
1
Show that the energy can be rewritten as MX 2 + mx 2 + , where M = m1 + m2 (total
2
2
x
m1m2
m x + m2 x 2
mass), X = 1 1
(center-of-mass coordinates), m =
(reduced mass), and
m1 + m2
M
x = x1 - x 2 (relative coordinates).
17-3. Show that the 6-dimensional Schrödinger equation of a hydrogenic atom,
2
2
æ
Ze2 ö
2
2
Ñ
Ñ
ç
÷ Y X,x = EY X,x , is subject to separation of center-ofX
2m x 4pe 0 x ø
è 2M
mass coordinates X and relative coordinates x.
(
)
(
)
17-4. The hydrogenic atom’s Schrödinger equation in the spherical coordinates centered
at the nucleus can be written as
2 ì 2
2 ¶ 1 æ 1 ¶2
1 ¶
¶ ö üï
Ze2
ï¶
+
+
+
sinq ÷ ý Y r,q ,j Y r,q ,j
í
2 m îï ¶r 2 r ¶r r 2 çè sin 2 q ¶f 2 sin q ¶q
¶q ø þï
4pe 0r
(
)
(
)
= EY ( r,q ,j )
Show that the separation of variables can be achieved in the form,
Y r,q ,j = R r Y q ,j , where Y q ,j is furthermore a spherical harmonics.
(
)
() ( )
( )
17-5. In the radial part of the hydrogenic Schrödinger equation,
2
æ ¶2 2 ¶ ö
Ze2
l(l +1) 2
+
RR+
R = ER , suggest the physical interpretation of
2m çè ¶r 2 r ¶r ÷ø
4pe 0r
2 mr 2
the last term in the left-hand side.
17-6. Verify that the (n,l) = (1,0) solution, R = Ce- r /2 , satisfies the radial equation,
-
1
¶2 R 2 ¶R R l(l +1)
- +
R = E ¢R , where E ¢ = - 2 .
2
2
4n
r ¶r r
¶r
r
17-7. Verify that the (n,l) = (2,1) solution, R = C re- r /4 , satisfies the radial equation.