Physics 741 – Graduate Quantum Mechanics 1
Solution Set P
1.
[30] In class we solved the hydrogen atom, which is a spherical 1/
r
potential.
Consider another spherically symmetric potential, namely, the spherical
harmonic oscillator
2
22
1
Hm
m
ω
=+
P
Q
This potential is most easily solved by separation of variables, but it is very
helpful to take advantage of the spherical symmetry to find solutions.
(a) [3] Factor eigenstates of this Hamiltonian into the form
()
(
)
(
)
,,
,
m
l
rR
r
Y
ψ
θφ
=
.
Find a differential equation satisfied by the radial
wave function
R r
.
We are attempting to find eigenstates of the Hamiltonian, that is, solutions of
HE
=
.
Since we have spherical symmetry, we expect their angular dependence to
take the form of spherical harmonics, so that
( ) ( ) ( )
,
m
l
r
Y
=
.
Plugging into
Schrödinger’s equation, we see from the lecture notes (8.45) that we have
() () ()
2
2
2
2
2
2
1
,
2
21
dl
l
ERr
rRr
Rr
VrRr
m r dr
r
mE
d
l
l
m
rdr
r
⎧⎫
+
=−
−
+
⎡⎤
⎨⎬
⎣⎦
⎩⎭
+
+
+
=
==
(b) [5] At large
r
, which term besides the derivative term dominates?
Show that
for large
r
, we can satisfy the differential equation if
( )
2
1
2
exp
R
rA
r
−
∼
,
and determine the factor
A
that will make this work.
For Hydrogen, the potential term vanished at infinity, but in this case, the
potential becomes large at infinity, and cannot be ignored.
The leading terms will then
be the derivative term acting both times on
R
(
r
) and the potential term, so we are
approximately trying to satisfy the equation
2
2
dm
R
rr
R
r
dr
∼
=
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 Spring '04
 Unknow
 Physics, mechanics, Angular Momentum, Atomic orbital, wave function, Rotational symmetry

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