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.

Acceptable wavefunction

with allowed wavelengths:lmr2

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 9.35 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 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.

Fig 9.40 Space quantization of angular momentum

for l = 2 where φ is indeterminate.

θ