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Transcript
Applying the Correspondence Principle
to the Three-Dimensional Rigid Rotor
David Keeports
Mills College
[email protected]
Quantum Mechanical
Correspondence Principle
No system strictly obeys
classical mechanics
Instead, all systems are
quantum systems, but …
“Quantum systems appear to be
classical when their quantum
numbers are very large.”
The Instructional Challenge in
Presenting the Correspondence Principle
Consider “obviously classical”
systems and show that they are
really quantum systems
Correspondence Principle Applied to
Fundamental Quantum Systems
Particle in 1-Dimensional Box
Particle in 3-Dimensional Box
Harmonic Oscillator
2-Dimensional Rigid Rotor
3-Dimensional Rigid Rotor
Hydrogen Atom
Particle in 1-Dimensional Box
æ2ö
æ np x ö
y (x) = ç ÷ sin ç
÷
èLø
è L ø
1/2
if n ® ¥
uniform probability distribution
from x = 0 to x = L
Particle in 3-Dimensional Box
1
2
æ 8 ö
æ nx p x ö æ nyp y ö æ nz p z ö
y (x, y, z) = ç
÷ sin ç
÷ sin ç
÷
÷ sin ç
è abc ø
è a ø è b ø è c ø
if nx , ny , nz ® ¥
uniform probability distribution
within 3-dimensional box
Harmonic Oscillator
yv (x) = Nv H v (a x)e
-(a x)2 /2
as v ®¥
probability is enhanced at turning points
2-Dimensional Rigid Rotor
y (f ) =
1
p
sin ( mf )
y (f ) =
1
p
cos ( mf )
if m ® ¥ all angles become equally probable
In each case as a quantum
number increases by 1,
DE / E @ 0
System energy appears to be
a continuous function, i.e.,
quantization not evident
A Classical Three-Dimensional Rigid Rotor
Consider a rigid rotor of binary star
dimensions rotating in xy-plane
Assume both masses are solar masses M
and separation is constant at r = 10 AU
2
M
M
m=
=
M +M 2
I = mr = 2.226 x 10 kg m
2
54
2
Gm
mv
m(r / 2) w
From F = 2 =
=
,
r
(r / 2)
(r / 2)
2
2
w = 2.815 x 10 /s
-7
2
2
L = Iw = 6.266 x 10 kg m /s
47
2
1 2 L
K = K rot = Iw = = 8.820 x 10 40 J
2
2I
2
U = constant º 0 during rotation
E = K +U = 8.820 x 10 J
40
But does this 3-D rotor really
obey classical mechanics?
No, it is a quantum system that only
appears to obey classical mechanics
because its quantum numbers are very large!
Why Are Quantum Numbers Large?
Eigen-Operators for 3-D Rigid Rotors
2 ù
é 1 ¶ æ
ö
¶
1 ¶
Ĥ = - ê
çsinq ÷ + 2
2ú
2I ë sinq ¶q è
¶q ø sin q ¶f û
2 ù
é
æ
ö
1
¶
¶
1
¶
L̂2 = - 2 ê
ç sinq ÷ + 2
2ú
¶q ø sin q ¶f û
ë sinq ¶q è
2
¶
L̂z = -i
¶f
Spherical Harmonics Are Eigenfunctions
Eigenvalues of Operators
J(J +1)
ĤYJ,M J (q, f ) =
2I
2
YJ,M J (q, f )
L̂2YJ,M J (q, f ) = J(J +1) 2YJ,M J (q, f )
L̂zYJ,M J (q, f ) = M J YJ,M J (q, f )
J = 0, 1, 2, ...
M J = - J, - J +1, ... , + J
For assumed orbit in the xy-plane, angular
momentum and its z-component are
virtually indistinguishable, so …
L @ Lz
J = MJ ® ¥
The Size of J = MJ
J(J +1)
E=
2I
2
40
= 8.820 x 10 J
J = 5.94 x 10
Large!
81
Energy and the Correspondence Principle
Suppose that J increases by 1:
DE E(J +1) - E(J) 2
-82
=
= = 3.37 x 10
E
E(J)
J
Energy quantization unnoticed
Rotor Orientation From
Spherical Harmonic Wavefunctions
YJ,M J (q, f ) = QJ,M J (q )FM J (f )
QJ,M J (q ) = N J,M J PJ,M J (q )
1
F0 (f ) =
2p
1
F M J ,c (f ) =
cos(M J f )
p
1
F M J ,s (f ) =
sin(M J f )
p
QJ,M J (q ) can be complex
(-1) 2J +1 (J - M J )! M J d
(sin 2 J q )
QJ,M J (q ) = J
sin q
J+ M J
2 J!
2 (J + M J )!
[d(cosq )]
J+ M J
J
But when J = M J , QJ,M J (q ) is very simple:
QJ,M J (q ) = N J sin J q
Because QJ,M J (q ) = N J sin q
J
If J ® ¥,
probability ® 0 unless q =
p
2
probability ® 0 outside xy-plane
F M J ,c (f ) =
1
p
cos(M J f )
2M J angular nodes and
2M J angular antinodes
If M J ® ¥, probability is proportional to Df
No f is favored
Localization of axis at a particular f
requires superposition of wavefunctions
with a range of angular momentum values
Uncertainty principle: Angular certainty
comes at the expense of
angular momentum certainty
The Hydrogen Atom Problem
in the Large Quantum Number Limit:
Consider Earth-Sun System
GmS m
Ñ yy = Ey
2m
r
2
2
-
2
Ñ y2
2me
e
2
y
4pe0 r
= Ey
Results for Quantum Earth
n =1, 2,3,...
l = 0,1, 2,..., n -1
ml = 0,±1,±2,...,±l
Gmm 1
E =2
2
2
n
L = l(l +1)
2
3
Lz = ml
2
S
Assumed circular orbit implies
n @ l = ml ® ¥
consistent with correspondence principle
ynlm (r, q, f ) = Rnl (r)Qlm (q )Fm (f )
With n @ l = ml ® ¥,
ynlm (r, q, f ) implies that Earth’s
l
l
l
l
spatial probability distribution is
y
0
Earth is in a hydrogen-like
orbital characterized by
huge quantum numbers
x
Quantum Mechanical Earth:
Where Orbitals Become Orbits.
European Journal of Physics,
Vol. 33, pp. 1587-98 (2012)
End