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Lecture 2

Particle in an isotropic potential

- The angular momentum in Quantum Mechanics -

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The angular momentum in physics

Ԧℒ = Ԧ𝑟 × Ԧ𝑝

ℒ = 𝑚𝑟𝑣⊥

• In an isotropic potential 𝑉(𝑟), Ԧℒ is a constant of motion. The

particle moves in a plane that contains the centre of the

potential, and the area Ԧ𝒜 swept by Ԧ𝑟 is swept at a constant rate : Τ𝑑𝒜

𝑑𝑡 = Τℒ 2𝑚

• Total energy of the particle : 𝐸 =1

2𝑚𝑣𝑟

2 +𝓛𝟐

2𝑚𝑟2+ 𝑉(𝑟)

Veff(𝑟)

The problem is equivalent to a 1D problem, where Ԧℒ is set by

some initial conditions

Ԧℒ

O

mass 𝑚

Ԧ𝑟

Ԧ𝑝

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The effective potential

Total energy of the particle : 𝐸 =1

2𝑚𝑣𝑟

2 +𝓛𝟐

2𝑚𝑟2+ 𝑉(𝑟)

Veff(𝑟)

Energy

𝐸O

Elliptical orbit Ԧ𝑟

𝑟

Apogee

Perigee

𝓛1

𝓛2 > 𝓛1

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The orbital angular momentum

Classical approach : Ԧℒ = Ԧ𝑟 × Ԧ𝑝 =𝑥𝑦𝑧

×

𝑝𝑥𝑝𝑦𝑝𝑧

Quantum approach : 𝐿 = 𝑅 × 𝑃 =𝑋𝑌𝑍

×𝑃𝑋𝑃𝑌𝑃𝑍

=𝑌𝑃𝑍 − 𝑍𝑃𝑌𝑍𝑃𝑋 − 𝑋𝑃𝑍𝑋𝑃𝑌 − 𝑌𝑃𝑋

𝐿𝑋, 𝐿𝑌, 𝐿𝑍 are observables (i.e. hermitian operators, the eigenstates of

which form an orthonormal basis of the Hilbert space)

Commutators : ൞

𝐿𝑋, 𝐿𝑌 = 𝑖ℏ𝐿𝑍𝐿𝑌, 𝐿𝑍 = 𝑖ℏ𝐿𝑋𝐿𝑍, 𝐿𝑋 = 𝑖ℏ𝐿𝑌

and 𝐿2, 𝐿 = 0

There exists a set of common eigenstates of 𝑳𝟐 and 𝑳𝒁that form an orthonormal basis of the Hilbert space.

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The orbital angular momentum

Quantum approach : 𝐿 = 𝑅 × 𝑃 =𝑋𝑌𝑍

×𝑃𝑋𝑃𝑌𝑃𝑍

=𝑌𝑃𝑍 − 𝑍𝑃𝑌𝑍𝑃𝑋 − 𝑋𝑃𝑍𝑋𝑃𝑌 − 𝑌𝑃𝑋

𝐿 operators in cartesian coordinates :

𝐿𝑋 =ℏ

𝑖𝑦

𝜕

𝜕𝑧− 𝑧

𝜕

𝜕𝑦

𝐿𝑌 =ℏ

𝑖𝑧

𝜕

𝜕𝑥− 𝑥

𝜕

𝜕𝑧

𝐿𝑍 =ℏ

𝑖𝑥

𝜕

𝜕𝑦− 𝑦

𝜕

𝜕𝑥

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The orbital angular momentum

Quantum approach : 𝐿 = 𝑅 × 𝑃 =𝑋𝑌𝑍

×𝑃𝑋𝑃𝑌𝑃𝑍

=𝑌𝑃𝑍 − 𝑍𝑃𝑌𝑍𝑃𝑋 − 𝑋𝑃𝑍𝑋𝑃𝑌 − 𝑌𝑃𝑋

𝐿 operators in spherical coordinates :

𝐿𝑋 = 𝑖ℏ sin𝜑𝜕

𝜕𝜃+

cos 𝜑

tan 𝜃

𝜕

𝜕𝜑

𝐿𝑌 = 𝑖ℏ −cos𝜑𝜕

𝜕𝜃+

sin 𝜑

tan 𝜃

𝜕

𝜕𝜑

𝐿𝑍 =ℏ

𝑖

𝜕

𝜕𝜑

𝐿2 = −ℏ2𝜕2

𝜕𝜃2+

1

tan 𝜃

𝜕

𝜕𝜃+

1

𝑠𝑖𝑛2 𝜃

𝜕2

𝜕𝜑2

r𝜃

𝜑𝑥

𝑦

𝑧

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The angular momentum in general

By definition : a set Ԧ𝐽 =𝐽𝑋𝐽𝑌𝐽𝑍

of observables that fulfill the following

commutation rules: ൞

𝑱𝑿, 𝑱𝒀 = 𝒊ℏ𝑱𝒁𝑱𝒀, 𝑱𝒁 = 𝒊ℏ𝑱𝑿𝑱𝒁, 𝑱𝑿 = 𝒊ℏ𝑱𝒀

Theorems:

1. 𝐽2, Ԧ𝐽 = 0 There exists a set of common eigenstates of 𝐽2

and 𝐽𝑍 that form an orthornormal basis of the Hilbert space.

2. The eigenvalues of 𝐽2 are of the form ℏ2j j + 1 , with 𝒋 ∈ ℕ ∪ ℕ/2

3. The eigenvalues of 𝐽𝑍 are of the form 𝑚ℏ, with 𝒎 taking

all 𝟐𝒋 + 𝟏 possible values in −𝒋,−𝒋 + 𝟏,… , 𝒋 − 𝟏, 𝒋 .

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The angular momentum in general

Examples : 𝑗 = 1 𝑚 ∈ {−1, 0, 1}

𝑗 =3

2 𝑚 ∈ {−

3

2, −

1

2,1

2,3

2}

Exercise : Prove the following properties :

1. 𝐽−𝐽+ = 𝐽2 − ℏ𝐽𝑍 − 𝐽𝑍2, with 𝐽+ = 𝐽𝑋 + 𝑖𝐽𝑌 and 𝐽− = 𝐽𝑋 − 𝑖𝐽𝑌

2. 𝐽+𝐽− = 𝐽2 + ℏ𝐽𝑍 − 𝐽𝑍2,

3. −𝑗 ≤ 𝑚 ≤ 𝑗

4. 𝐽−ȁ𝑘, 𝑗, −𝑗 = 0 and 𝐽+ȁ𝑘, 𝑗, 𝑗 = 0

5. For 𝑚 > −𝑗,

a) 𝐽−ȁ𝑘, 𝑗,𝑚 is eigenstate of 𝐽2 with eigenvalue 𝑗(𝑗 + 1)ℏ2

b) 𝐽−ȁ𝑘, 𝑗,𝑚 is eigenstate of 𝐽𝑍 with eigenvalue 𝑚 − 1 ℏ

6. For 𝑚 < 𝑗,

a) 𝐽+ȁ𝑘, 𝑗,𝑚 is eigenstate of 𝐽2 with eigenvalue 𝑗(𝑗 + 1)ℏ2

b) 𝐽+ȁ𝑘, 𝑗,𝑚 is eigenstate of 𝐽𝑍 with eigenvalue 𝑚 + 1 ℏ

−ℏ

+ℏ

0

z

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The angular momentum in general

Theorems:

4. The common eigenstates of 𝐽2 and 𝐽𝑍 are denoted as ȁ𝑘, 𝑗,𝑚 .

They form an orthonormal basis of the Hilbert space.

𝑘 accounts for the degeneracy of sub-eigenspace ℰ 𝑗,𝑚 .

𝑱𝟐ȁ𝒌, 𝒋,𝒎 = 𝒋 𝒋 + 𝟏 ℏ𝟐ȁ𝒌, 𝒋,𝒎

𝑱𝒁ȁ𝒌, 𝒋,𝒎 = 𝒎ℏȁ𝒌, 𝒋,𝒎

4. For a given value 𝑗, the 2𝑗 + 1 sub-eigenspaces ℰ(𝑗,𝑚) all have

the same degeneracy (𝑔(𝑗), independent of 𝑚), and are

connected using the « raising » and « lowering » operators :

𝐽+ = 𝐽𝑋 + 𝑖𝐽𝑌 and 𝐽− = 𝐽𝑋 − 𝑖𝐽𝑌

𝐽+ȁ𝑘, 𝑗,𝑚 =ℏ 𝑗 𝑗 + 1 −𝑚 𝑚 + 𝟏 ȁ𝑘, 𝑗,𝑚 + 𝟏

𝐽−ȁ𝑘, 𝑗,𝑚 =ℏ 𝑗 𝑗 + 1 −𝑚 𝑚 − 𝟏 ȁ𝑘, 𝑗,𝑚 − 𝟏𝐽+ȁ𝑘, 𝑗, 𝑗 = 0𝐽−ȁ𝑘, 𝑗, −𝑗 = 0

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Spectrum of the orbital angular momentum

Theorem : The eigenvalues of 𝐿2 are ℏ2𝑙 𝑙 + 1 , with 𝒍 ∈ ℕ

The common eigenstates of 𝐿2 and 𝐿𝑍 are denoted ȁ𝑙,𝑚.

Their associated wave functions in positions space are denoted𝜓 𝑟, 𝜃, 𝜑 = 𝑅 𝑟 𝑌𝑙

𝑚(𝜃, 𝜑)

Solve ൝𝐿2 𝑌𝑙

𝑚 𝜃, 𝜑 = 𝑙(𝑙 + 1)ℏ2 𝑌𝑙𝑚 𝜃, 𝜑

𝐿𝑍 𝑌𝑙𝑚 𝜃, 𝜑 = 𝑚ℏ 𝑌𝑙

𝑚 𝜃, 𝜑

Solutions : 𝑌𝑙𝑙 𝜃, 𝜑 = 𝑐𝑙 𝑠𝑖𝑛

𝑙𝜃 𝑒𝑖𝑙𝜑 with 𝑐𝑙 =(−1)𝑙

2𝑙 𝑙!

2𝑙+1 !

4𝜋

𝑌𝑙𝑚−1 𝜃, 𝜑 = 𝐿− 𝑌𝑙

𝑚 𝜃, 𝜑 /(ℏ 𝑙 𝑙 + 1 −𝑚(𝑚 − 1))

Radial part Angular part

« Spherical harmonic »න0

∞

𝑟2 𝑅 𝑟 2𝑑𝑟 = 1

න0

2𝜋

𝑑𝜑න0

𝜋

𝑑𝜃 sin 𝜃 𝑌𝑙𝑚 𝜃, 𝜑 2 = 1

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Spherical Harmonics

l = 0, m = 0

𝑌00 𝜃, 𝜑 =

1

4𝜋

Plot of 𝑌𝑙𝑚 𝜃, 𝜑

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Spherical Harmonics

l = 1, m = 0 l = 1, m = 1

𝑌10 𝜃, 𝜑 =

3

4𝜋cos 𝜃 𝑌1

1 𝜃, 𝜑 = −3

8𝜋sin 𝜃 𝑒𝑖𝜑

Plot of 𝑌𝑙𝑚 𝜃, 𝜑

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Spherical Harmonics

l = 2, m = 0 l = 2, m = 2l = 2, m = 1

𝑌20 𝜃, 𝜑 =

5

16𝜋(3𝑐𝑜𝑠2 𝜃 − 1) 𝑌2

1 𝜃, 𝜑 = −15

8𝜋sin 𝜃 cos 𝜃 𝑒𝑖𝜑 𝑌2

2 𝜃, 𝜑 =15

32𝜋𝑠𝑖𝑛2 𝜃 𝑒𝑖2𝜑

Plot of 𝑌𝑙𝑚 𝜃, 𝜑

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Spherical Harmonics

l = 3, m = 0 l = 3, m = 2l = 3, m = 1 l = 3, m = 3

Plot of 𝑌𝑙𝑚 𝜃, 𝜑

𝑌𝑙𝑚 𝜃, 𝜑 = −1 𝑚

2𝑙 + 1

4𝜋

𝑙 − 𝑚 !

𝑙 + 𝑚 !𝑃𝑙𝑚 cos 𝜃 𝑒𝑖𝑚𝜑

Legendre function : 𝑃𝑙𝑚 𝑢 = 1 − 𝑢2 𝑚 𝑑𝑚

𝑑𝑢𝑚𝑃𝑙 𝑢 , −1 ≤ 𝑢 ≤ 1

Legendre polynomial : 𝑃𝑙 𝑢 =−1 𝑙

2𝑙 𝑙!

𝑑𝑙

𝑑𝑢𝑙1 − 𝑢2 𝑙

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The Zeeman effect

• The orbital magnetic moment

Classical mechanics

Current : 𝐼 = 𝑞𝑣

2𝜋𝑅

Magnetic moment : ℳ = 𝐼 Ԧ𝒜 =1

2𝑞𝑅 × Ԧ𝑣

𝓜=𝒒

𝟐𝒎𝒆𝓛

Magnetic interaction : −𝓜.𝑩

Quantum mechanics

Magnetic moment : M =𝒒ℏ

𝟐𝒎𝒆L/ℏ

Magnetic interaction : H = −M .𝑩 (normal Zeeman effect)

• The normal Zeeman effect predicts that a B-field (along 𝑧) lifts

the degeneracy of the 2𝑙 + 1 sub-states : 𝐸𝑛𝑙𝑚 = 𝐸𝑛 −𝑚𝜇𝐵B(−l ≤ 𝑚 ≤ 𝑙).

Proton

𝑚𝑃 ≫ 𝑚𝑒

Electron 𝑞,𝑚𝑒

𝑅Area Ԧ𝒜

M = 𝜇𝐵L/ℏ

𝜇𝐵 =𝒒ℏ

𝟐𝒎𝒆: Bohr’s magneton

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The Stern & Gerlach experiment

The experiment (1922)

Oven

Ag atoms

𝑧

Strong magnetic

gradient along 𝑧

beam

Ag atoms are neutral

no Laplace forceAg atoms are paramagnetic

Permanent magnetic moments ℳ, oriented randomly

Force : Ԧ𝐹 = Grad 𝓜.𝑩 𝐹𝑧 =ℳ𝑧𝜕𝐵𝑧

𝜕𝑧 deviation along 𝑧

Expected result : one spot, symmetric with respect to 𝑧 = 0

Actual result : 2 spots, at ± 𝝁𝑩𝝏𝑩𝒛

𝝏𝒛

The electron has an intrinsic angular momentum 𝑺 (not of orbital nature),

with eigenvalue ℏ𝟐𝒔 𝒔 + 𝟏 where 𝒔 = Τ𝟏 𝟐

This spin is associated to an intrinsic magnetic moment 𝑀𝑆 = 𝟐𝜇𝐵S/ℏ

Screen

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The Stern & Gerlach experiment

Otto Stern, Nobel Prize 1943

Walther Gerlach

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The Stern & Gerlach experiment

Quantization of the

components of the

intrinsic angular

momentum (spin) of

the electron

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19B. Friedrich and D. Herschbach, Phys. Today 56, 53 (2003)

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To read more …

• On the angular momentum in quantum mechanics : CDL1, chapter VI

• On the spherical harmonics : CDL1, compl. AVI

• On Bohr’s model : CPP1, chapters I and VI

• On the spin of the electron and the Stern & Gerlach experiment :

TB, chapter VI; CDL1, chapters IV and IX; CPP1, chapter X

CDL1 : Cohen-Tannoudji, Diu, Laloë, Quantum Mechanics, volume 1TB : Tualle-Brouri, Introduction à la Mécanique Quantique, Cours 1A de l’IOGS

CPP1&2 : Cagnac, Pebay-Peyroula, Atomic physics, volumes 1 & 2

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… about observables that commute.

Take two observables that commute : 𝐴, 𝐵 = 0

In other words, eigenspace 𝓔𝒂 is globally invariant under action of 𝐵

Theorem 1 : ȁ𝜓 is eigenstate of 𝐴 with eigenvalue 𝑎

𝐵ȁ𝜓 is eigenstate of 𝐴 with eigenvalue 𝑎

Some useful theorems …

ȁ𝜓

𝐵ȁ𝜓

𝓔𝒂

Theorem 2 : Take two eigenstates ห𝜓1 and ห𝜓2 of 𝐴 with eigenvalues 𝑎1 and 𝑎2 (𝑎2 ≠ 𝑎1).

Then, 𝜓1 𝐵 𝜓2 = 0

In other words, 𝐵 does

not couple different

eigenspaces

𝐵 =

⋯ ⋯⋯ ⋯

0 00 0

0 00 0

0 00 0

⋯ ⋯⋯ ⋯

0 00 0

0 00 0

0 00 0

⋯ ⋯⋯ ⋯

𝓔𝒂𝟏

𝓔𝒂𝟐

𝓔𝒂𝟑

𝐵𝓔𝒂𝟐𝐵𝓔𝒂𝟏 𝐵𝓔𝒂𝟑

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… about observables that commute.

Take two observables that commute : 𝐴, 𝐵 = 0

Some useful theorems …

Theorem 3 : There exists a set of common eigenstates of 𝐴 and 𝐵 that form an orthonormal

basis of the Hilbert space.