lecture 2 particle in an isotropic potential
TRANSCRIPT
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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
Τ¦β = Τ¦π Γ Τ¦π
β = πππ£β₯
β’ In an isotropic potential π(π), Τ¦β is a constant of motion. The
particle moves in a plane that contains the centre of the
potential, and the area Τ¦π swept by Τ¦π is swept at a constant rate : Ξ€ππ
ππ‘ = Ξ€β 2π
β’ Total energy of the particle : πΈ =1
2ππ£π
2 +ππ
2ππ2+ π(π)
Veff(π)
The problem is equivalent to a 1D problem, where Τ¦β is set by
some initial conditions
Τ¦β
O
mass π
Τ¦π
Τ¦π
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The effective potential
Total energy of the particle : πΈ =1
2ππ£π
2 +ππ
2ππ2+ π(π)
Veff(π)
Energy
πΈO
Elliptical orbit Τ¦π
π
Apogee
Perigee
π1
π2 > π1
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The orbital angular momentum
Classical approach : Τ¦β = Τ¦π Γ Τ¦π =π₯π¦π§
Γ
ππ₯ππ¦ππ§
Quantum approach : πΏ = π Γ π =πππ
Γππππππ
=πππ β ππππππ β ππππππ β πππ
πΏπ, πΏπ, πΏπ are observables (i.e. hermitian operators, the eigenstates of
which form an orthonormal basis of the Hilbert space)
Commutators : ΰ΅
πΏπ, πΏπ = πβπΏππΏπ, πΏπ = πβπΏππΏπ, πΏπ = πβπΏπ
and πΏ2, πΏ = 0
There exists a set of common eigenstates of π³π and π³πthat form an orthonormal basis of the Hilbert space.
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The orbital angular momentum
Quantum approach : πΏ = π Γ π =πππ
Γππππππ
=πππ β ππππππ β ππππππ β πππ
πΏ operators in cartesian coordinates :
πΏπ =β
ππ¦
π
ππ§β π§
π
ππ¦
πΏπ =β
ππ§
π
ππ₯β π₯
π
ππ§
πΏπ =β
ππ₯
π
ππ¦β π¦
π
ππ₯
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The orbital angular momentum
Quantum approach : πΏ = π Γ π =πππ
Γππππππ
=πππ β ππππππ β ππππππ β πππ
πΏ operators in spherical coordinates :
πΏπ = πβ sinππ
ππ+
cos π
tan π
π
ππ
πΏπ = πβ βcosππ
ππ+
sin π
tan π
π
ππ
πΏπ =β
π
π
ππ
πΏ2 = ββ2π2
ππ2+
1
tan π
π
ππ+
1
π ππ2 π
π2
ππ2
rπ
ππ₯
π¦
π§
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The angular momentum in general
By definition : a set Τ¦π½ =π½ππ½ππ½π
of observables that fulfill the following
commutation rules: ΰ΅
π±πΏ, π±π = πβπ±ππ±π, π±π = πβπ±πΏπ±π, π±πΏ = πβπ±π
Theorems:
1. π½2, Τ¦π½ = 0 There exists a set of common eigenstates of π½2
and π½π that form an orthornormal basis of the Hilbert space.
2. The eigenvalues of π½2 are of the form β2j j + 1 , with π β β βͺ β/2
3. The eigenvalues of π½π are of the form πβ, with π taking
all ππ + π possible values in βπ,βπ + π,β¦ , π β π, π .
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The angular momentum in general
Examples : π = 1 π β {β1, 0, 1}
π =3
2 π β {β
3
2, β
1
2,1
2,3
2}
Exercise : Prove the following properties :
1. π½βπ½+ = π½2 β βπ½π β π½π2, with π½+ = π½π + ππ½π and π½β = π½π β ππ½π
2. π½+π½β = π½2 + βπ½π β π½π2,
3. βπ β€ π β€ π
4. π½βΘπ, π, βπ = 0 and π½+Θπ, π, π = 0
5. For π > βπ,
a) π½βΘπ, π,π is eigenstate of π½2 with eigenvalue π(π + 1)β2
b) π½βΘπ, π,π is eigenstate of π½π with eigenvalue π β 1 β
6. For π < π,
a) π½+Θπ, π,π is eigenstate of π½2 with eigenvalue π(π + 1)β2
b) π½+Θπ, π,π is eigenstate of π½π with eigenvalue π + 1 β
ββ
+β
0
z
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The angular momentum in general
Theorems:
4. The common eigenstates of π½2 and π½π are denoted as Θπ, π,π .
They form an orthonormal basis of the Hilbert space.
π accounts for the degeneracy of sub-eigenspace β° π,π .
π±πΘπ, π,π = π π + π βπΘπ, π,π
π±πΘπ, π,π = πβΘπ, π,π
4. For a given value π, the 2π + 1 sub-eigenspaces β°(π,π) all have
the same degeneracy (π(π), independent of π), and are
connected using the Β« raising Β» and Β« lowering Β» operators :
π½+ = π½π + ππ½π and π½β = π½π β ππ½π
π½+Θπ, π,π =β π π + 1 βπ π + π Θπ, π,π + π
π½βΘπ, π,π =β π π + 1 βπ π β π Θπ, π,π β ππ½+Θπ, π, π = 0π½βΘπ, π, βπ = 0
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Spectrum of the orbital angular momentum
Theorem : The eigenvalues of πΏ2 are β2π π + 1 , with π β β
The common eigenstates of πΏ2 and πΏπ are denoted Θπ,π.
Their associated wave functions in positions space are denotedπ π, π, π = π π ππ
π(π, π)
Solve ΰ΅πΏ2 ππ
π π, π = π(π + 1)β2 πππ π, π
πΏπ πππ π, π = πβ ππ
π π, π
Solutions : πππ π, π = ππ π ππ
ππ ππππ with ππ =(β1)π
2π π!
2π+1 !
4π
πππβ1 π, π = πΏβ ππ
π π, π /(β π π + 1 βπ(π β 1))
Radial part Angular part
Β« Spherical harmonic Β»ΰΆ±0
β
π2 π π 2ππ = 1
ΰΆ±0
2π
ππΰΆ±0
π
ππ sin π πππ π, π 2 = 1
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Spherical Harmonics
l = 0, m = 0
π00 π, π =
1
4π
Plot of πππ π, π
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Spherical Harmonics
l = 1, m = 0 l = 1, m = 1
π10 π, π =
3
4πcos π π1
1 π, π = β3
8πsin π πππ
Plot of πππ π, π
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Spherical Harmonics
l = 2, m = 0 l = 2, m = 2l = 2, m = 1
π20 π, π =
5
16π(3πππ 2 π β 1) π2
1 π, π = β15
8πsin π cos π πππ π2
2 π, π =15
32ππ ππ2 π ππ2π
Plot of πππ π, π
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Spherical Harmonics
l = 3, m = 0 l = 3, m = 2l = 3, m = 1 l = 3, m = 3
Plot of πππ π, π
πππ π, π = β1 π
2π + 1
4π
π β π !
π + π !πππ cos π ππππ
Legendre function : πππ π’ = 1 β π’2 π ππ
ππ’πππ π’ , β1 β€ π’ β€ 1
Legendre polynomial : ππ π’ =β1 π
2π π!
ππ
ππ’π1 β π’2 π
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The Zeeman effect
β’ The orbital magnetic moment
Classical mechanics
Current : πΌ = ππ£
2ππ
Magnetic moment : β³ = πΌ Τ¦π =1
2ππ Γ Τ¦π£
π=π
ππππ
Magnetic interaction : βπ.π©
Quantum mechanics
Magnetic moment : M =πβ
πππL/β
Magnetic interaction : H = βM .π© (normal Zeeman effect)
β’ The normal Zeeman effect predicts that a B-field (along π§) lifts
the degeneracy of the 2π + 1 sub-states : πΈπππ = πΈπ βπππ΅B(βl β€ π β€ π).
Proton
ππ β« ππ
Electron π,ππ
π Area Τ¦π
M = ππ΅L/β
ππ΅ =πβ
πππ: Bohrβs magneton
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The Stern & Gerlach experiment
The experiment (1922)
Oven
Ag atoms
π§
Strong magnetic
gradient along π§
beam
Ag atoms are neutral
no Laplace forceAg atoms are paramagnetic
Permanent magnetic moments β³, oriented randomly
Force : Τ¦πΉ = Grad π.π© πΉπ§ =β³π§ππ΅π§
ππ§ deviation along π§
Expected result : one spot, symmetric with respect to π§ = 0
Actual result : 2 spots, at Β± ππ©ππ©π
ππ
The electron has an intrinsic angular momentum πΊ (not of orbital nature),
with eigenvalue βππ π + π where π = Ξ€π π
This spin is associated to an intrinsic magnetic moment ππ = πππ΅S/β
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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To read more β¦
β’ On the angular momentum in quantum mechanics : CDL1, chapter VI
β’ On the spherical harmonics : CDL1, compl. AVI
β’ On Bohrβs model : CPP1, chapters I and VI
β’ 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, LaloΓ«, Quantum Mechanics, volume 1TB : Tualle-Brouri, Introduction Γ la MΓ©canique Quantique, Cours 1A de lβIOGS
CPP1&2 : Cagnac, Pebay-Peyroula, Atomic physics, volumes 1 & 2
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β¦ about observables that commute.
Take two observables that commute : π΄, π΅ = 0
In other words, eigenspace ππ is globally invariant under action of π΅
Theorem 1 : Θπ is eigenstate of π΄ with eigenvalue π
π΅Θπ is eigenstate of π΄ with eigenvalue π
Some useful theorems β¦
Θπ
π΅Θπ
ππ
Theorem 2 : Take two eigenstates ΰΈ«π1 and ΰΈ«π2 of π΄ with eigenvalues π1 and π2 (π2 β π1).
Then, π1 π΅ π2 = 0
In other words, π΅ does
not couple different
eigenspaces
π΅ =
β― β―β― β―
0 00 0
0 00 0
0 00 0
β― β―β― β―
0 00 0
0 00 0
0 00 0
β― β―β― β―
πππ
πππ
πππ
π΅ππππ΅πππ π΅πππ