Chap 12Chap 12Quantum Theory: techniques and Quantum Theory: techniques and
applicationsapplications
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
I2J
E2z
Where’s the quantization?!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.
Apply de Broglie relationship:
Now: Jz = ±mvr = ±pr
As we’ve seen:
Gives: where ml = 0, ±1, ±2, ...
Finally:
ph
mvh
I2m
I2J
E22
l2z
hrJ z
lmr2
lz mJ
Magneticquantum number!
2/1
im
m)2(
e)(Ψ
l
l πφ
φ
Fig 9.29 Magnitude of angular moment for a particle on a ring.
Right-handRule
2/1
im
m)2(
e)(Ψ
l
l πφ
φ
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| increasesin chunks of h
Fig 9.32 The basic ideas of the vector representation
of angular momentum:
Vector orientation
Angular momentumand
angleare complimentary
(Can’t be determinedsimultaneously)
Fig 9.33 Probability density for a particle in a definite
state of angular momentum.
Probability = Ψ*Ψ
with 2/1
im
m)2(
e)(Ψ
l
l πφ
φ
Gives:
πππ
φφ
2
1
)2(
e
)2(
eΨΨ
2/1
im
2/1
im
m*
m
ll
ll
Location is completelyindefinite!
Rotation in three-dimensions: a particle on a sphere
Hamiltonian:
Schrodinger equation
Vm2
H 22
2
2
2
2
2
22
zyx
Laplacian
V = 0 for the particle and r is constant, so ),(Ψ φθ
ΨEΨm2
H 22
)(Φ)(Θ),(Ψ φθφθ
By separation of variables:
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
Fig 9.38 Fig 9.38 Space quantizationSpace quantization of angular momentum of angular momentum
for for l l = 2= 2
Problem: we know Problem: we know θθ, so..., so...
we can’t know we can’t know φφ
θBecause mBecause mll = - = -ll,...+,...+ll,,the the orientationorientation of a of a
rotating bodyrotating bodyis quantized!is quantized!
Permitted values of ml
Fig 9.39 The Stern-Gerlach experiment confirmed
space quantization (1921)
Ag
Classicalexpected Observed
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.