Lecture 11 Particle on a ring (c) So Hirata, Department of Chemistry, University of Illinois at Urbana- Champaign. This material has been developed and made available online by work supported jointly by University of Illinois, the National Science Foundation under Grant CHE-1118616 (CAREER), and the Camille & Henry Dreyfus Foundation, Inc. through the Camille Dreyfus Teacher-Scholar program. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the sponsoring agencies.
Transcript
Lecture 11Particle on a ring
(c) So Hirata, Department of Chemistry, University of Illinois at Urbana-Champaign. This material has been developed and made available online by work supported jointly by University of Illinois, the
National Science Foundation under Grant CHE-1118616 (CAREER), and the Camille & Henry Dreyfus Foundation, Inc. through the Camille Dreyfus Teacher-Scholar program. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not
necessarily reflect the views of the sponsoring agencies.
The particle on a ring
The particle on a ring is the particle in a box curved with its both ends tied together.
This is a model for molecular rotation, a part of an atom’s electronic wave function, and even a part of a crystal’s electronic wave function.
We introduce a new operator, angular momentum operator, and a new boundary condition, cyclic boundary condition.
Rotational motion
We review the relevant classical physics and mathematical concepts: Angular momentum Cylindrical coordinates
We then switch to quantum mechanics: Angular momentum operator Hamiltonian in cylindrical coordinates Cyclic boundary condition The particle on a ring
The angular momentum
Which is the most effective way of applying force to rotate the gear?
(b); the right angle to the radius
(a) and (c) are wasteful because the effective component is |F| sin θ; (d) is not using leverage.
a
bc
θ
d
The angular momentum
For a given r (position) and p (momentum), its rotational motion is proportional to |r| and |p| sin θ.
Angular momentum J (or l) is a vector vertical to the plane formed by r and p, with length |r||p|sinθ.
θrp
The angular momentum
The mathematical definition of the angular momentum is a vector outer product:
Without losing generality, we can assume r = (rx, 0, 0) and p = (px, py, 0). Then l = (0, 0, lz).
( , , ) ( , , )x y z y z z y z x x z x y y xl l l r p r p r p r p r p r p
l r p
(0,0, ) (0,0, ) (0,0,| || | sin )z x yl r p r pNote that the angular momentum is a vector parallel to z-axis
Cylindrical coordinates
zz
ry
rx
sin
cos
The angular momentum operator
Classical angular momentum
The quantum mechanical operator can be obtained by the conversion
Thereforex
imvp xx
rrx
sincos
rry
cossin zz
ry
rx
sin
cos
The Hamiltonian
zz
ry
rx
sin
cos
The particle on a ring
Cyclic boundary condition
In the cylindrical coordinates, the wave function is Ψ(r,φ,z) at fixed r and z.
The wave function must satisfy the cyclic boundary condition:
After a complete revolution, the function must have the same value and also must trace the function in the previous cycle. Otherwise it would not be single-valued.
The particle on a ring
)()(2 2
2
2
2
Emr
22
2
22
2
kxkx
ikxikx
d ek e
dx
d ek e
dx
Promising
This will have problems with the cyclic boundary condition
ll imlim e
mr
me
mr 2
22
2
2
2
2
22
Energy
The particle on a ring
The cyclic boundary condition
Unlike the particle in a box, we have m = 0. Unlike the particle in a box, we have negative m’s. Energy is doubly degenerate for ml ≥ 1. No zero-point energy; no violation of uncertainty.
,2,1,0 lm
2
22
2mr
ml
The particle on a ring
Box Ring
Doubly degenerate
No zero-point energy
Energy differences in
the microwave range for
molecular rotations
Cyclic boundary condition
Imagine bending the particle in a box and connect the both ends to make a ring. Notice the difference between the boundary conditions used in two problems.
Quantum in nature
How does a microwave oven work?
Rotation of H2O
Quantum in nature
Why is grass green?
Rotation of electrons in chlorophyll
The particle on a ring
Normalization
2
11
2/12
0
2/12
0
2
ddeN lim
The angular momentum operator
Classical angular momentum
The quantum mechanical operator can be obtained by the conversion
Thereforex
imvp xx
rrx
sincos
rry
cossin zz
ry
rx
sin
cos
The angular momentum
Let us act the angular momentum operator on the wave function
The wave function is also an eigenfunction of angular momentum operator (H and lz commute and have the simultaneous eigenfunctions).
Double degeneracy comes from the two senses of rotations.
The uncertainty principle
The complementary observables
The angle and angular moment are the complementary observables.
x xipx
ˆ
t tiE
il x
The particle on a ring
For a state with a definite angular momentum, angle must be completely unknown.
The probability density along the ring is,
completely uniform regardless of ml.
2
1
22*
ll
ll
imim
mm
ee
Cyclic boundary condition and crystals
The crystalline solids have repeated chemical units. We impose cyclic boundary condition on the wave functions of crystals because infinite periodic crystals are mathematically isomorphic to rings of large radius. Crystals’ wave functions as a result have particle on a ring eigenfunctions as their part.
Summary
In the particle on a ring, we have learned a number of important concepts: angular momentum operator, cylindrical coordinates, cyclic boundary condition.
Similarities and differences from the particle in a box: the ring has no zero-point energy, doubly degenerate (|ml| ≥ 1), uniform probability density.
