Download Chapter 9 - Lecture 3

Chap 12
Quantum Theory: techniques and
applications
Objectives:
Solve the Schrödinger equation for:
•
•
•
Translational motion (Particle in a box)
Vibrational motion (Harmonic oscillator)
Rotational motion (Particle on a ring & on a sphere)
Rotational Motion in 2-D
Fig 9.27 Angular momentum of a particle of
mass m on a circular path of radius r in xy-plane.
Classically,
angular momentum:
Jz = ±mvr = ±pr
and
J2z
E
2I
Where’s the quantization?!
Fig 9.28 Two solutions of the Schrödinger equation
for a particle on a ring
• For an arbitrary λ, Φ is
unacceptable: not single-valued:
Φ = 0 and 2π are identical
• Also destructive interference of Φ
This Φ is acceptable:
single-valued and
reproduces itself.
Acceptable wavefunction
with allowed wavelengths:
2r
 
ml
h
h
Apply de Broglie relationship:   mv  p
Now:
Jz = ±mvr = ±pr
Jz  
hr

As we’ve seen:
Gives:
Finally:
2r
 
ml
Magnetic
quantum number!
Jz  ml  where ml = 0, ±1, ±2, ...
J2z
m2l  2
E

2I
2I
Ψml (φ) 
eimlφ
(2π)1/ 2
Fig 9.29 Magnitude of angular moment for a particle on a ring.
Ψml (φ) 
eimlφ
(2π)1/ 2
Right-hand
Rule
Fig 9.30 Cylindrical coordinates z, r, and φ. For a
particle on a ring, only r and φ change
Let’s solve the Schrödinger equation! 
Fig 9.31 Real parts of the wavefunction for a
particle on a ring, only r and φ change.
As λ decreases,
|ml| increases
in chunks of h
Fig 9.32 The basic ideas of the vector representation
of angular momentum:
Vector orientation
Angular momentum
and
angle
are complimentary
(Can’t be determined
simultaneously)
Fig 9.33 Probability density for a particle in a definite
state of angular momentum.
Probability = Ψ*Ψ
with Ψml (φ) 
eimlφ
(2π)1/ 2
Gives:
*
Ψml Ψml
 e imlφ  eimlφ 
1






 (2π)1/ 2  (2π)1/ 2  2π



Location is completely
indefinite!
Rotation in three-dimensions: a particle on a sphere
Schrodinger equation
Hamiltonian:

Laplacian
2 2
H 
 V
2m
2 
2
x
2

2
y
2

2
z 2
V = 0 for the particle and r is constant, so Ψ(θ, φ)

2 2
H 
 Ψ  EΨ
2m
By separation of variables:
Ψ(θ, φ)  Θ(θ)Φ(φ)
Fig 9.35 Spherical polar coordinates. For particle on
the surface, only θ and φ change.
Fig 9.34 Wavefunction for particle on a sphere must
satisfy two boundary conditions
Therefore:
two quantum numbers
l and ml
where:
l ≡ orbital angular
momentum QN = 0, 1, 2,…
and
ml ≡ magnetic QN =
l, l-1,…, -l
Table 9.3 The spherical
harmonics Yl,ml(θ,φ)
Fig 9.36 Wavefunctions for particle on a sphere
+
Sign of Ψ
+
-
+
-
Fig 9.38 Space quantization of angular momentum
for
l=2
Because ml = -l,...+l,
the orientation of a
rotating body
is quantized!
Problem: we know θ, so...
we can’t know φ
Permitted values of ml
θ
Fig 9.39 The Stern-Gerlach experiment confirmed
space quantization (1921)
Classical
expected
Observed
Ag
Inhomogeneous B field
Classically: A rotating charged body has a magnetic
moment that can take any orientation.
Quantum mechanically: Ag atoms have only two spin
orientations.
Fig 9.40 Space quantization of angular momentum
for
l = 2 where φ is indeterminate.
θ