Recap: The Stern-Gerlach experiment (1921)
Experimental setup
Stern
Gerlach
Silver atoms
Inhomogeneous magnetic field:
!"#
≠ 0 and '$ ≫ ') , '+
!$
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The Stern-Gerlach experiment (single S_G apparatus)
Outgoing beam components?
Predication from CM:
Observation: QM
!≠#
!=#
z-component of the magnetic moment of a
silver atom only has two possible values!
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What Stern & Gerlach observed:
"≠$
Two separate beams:
!
-!
"=$
z-component of the magnetic moment of a
silver atom only has two possible values!
Relation between magnetic moment & spin angular momentum: ! = &
ℏ
Electron: ) = + (ℏ ≅ 6.6×10234 eV 7 8)
'
)
(
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How to interpret the results of the S-G experiments
Relation between magnetic moment & spin angular momentum: ( = )
Electron: , =
ℏ
&
*
,
+
(ℏ ≅ 6.6×10345 eV 8 !)
v Only two possible values of !" are observed:
!" = +
ℏ
&
&
!" = −
ℏ
&
A measurement always causes a quantum system to
jump into an eigenstate of the dynamical variable that
is being measured…
Dirac
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The discovery of electron spin
Timeline
1921
Stern-Gerlach
experiment
Given direct evidence for a two-fold
quantization of the electron magnetic
moment or spin angular momentum!
1925
1928
Notion of electron spin
proposed by
Uhlenbeck & Goudsmit
Electron spin
naturally arises from
the Dirac equation
Note of caution: spin is an intrinsic quantum property of electrons -- you should NOT
think of it as spinning of an electron about its own axis in a class picture!
Exercise: Show that if an electron is considered as classical spinning sphere, then its
equatorial rotational speed is much larger than the speed of light!
A very clever idea which of course has
nothing to do with reality…
Pauli
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Today’s lecture
•
Sequential S-G experiments and their physics indications
•
Construction of mathematical expressions of the spin-1/2 (eigen-)states
in terms of Dirac’s bra-ket notation.
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Sequential S-G experiment #1
Outgoing beam components?
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Sequential S-G experiment #2
Outgoing beam components?
8
Sequential S-G experiment #3
Outgoing beam components?
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Sequential S-G experiment #1
(result & interpretation)
v If a silver atom is initially in the Sz+ spin (eigen-)state, it remains so
after a measurement of Sz.
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Sequential S-G experiment #2
(result & interpretation)
Question: Can we interpret (b) as !"+ beam
50%: #$ + & #% +
50%: #$ + & #% -
Answer: NO!
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Sequential S-G experiment #3
(result & interpretation)
• We cannot determine both Sz and Sx simultaneously [different from classical mechanics
(e.g., spinning top)].
• More precisely, the selection of the Sx + beam by the second apparatus (SGx)
completely destroys any previous information about Sz.
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spin angular momentum of a spin state is
different from the angular momentum of a
classical spinning top
• Sx, Sy, and Sz components of spin angular momentum of a silver atom (electron) cannot be determined simultaneously.
• Lx, Ly, and Lz angular-momentum components of a classical spinning top can be specified simultaneously.
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Mathematical description of the
states of spin angular momentum?
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Analogy between spin-1/2 atom and polarized Light
v Consider the electric field of a linearly polarized monochromatic light wave propagating in the zdirection
§ x-polarized:
§ y-polarized:
v Polarized light subject to filters:
x-polarized
Unpolarized light
x- and y-polarized
beams are orthogonal
• x-filter selects beams polarized in the x-direction
• y-filter selects beams polarized in the y-direction
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Analogy between spin-1/2 atom and polarized Light
•
Analogy: Filter ↔ S-G apparatus:
x-polarized
x- and y-polarized
beams are orthogonal
Unpolarized light
Sz+ and Sz- states are
orthogonal
Sz+ atom ↔ x-polarized light;
Sz- atom ↔ y-polarized light
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Analogy between spin-1/2 atom and polarized Light (cont’d)
•
Analogy: Filter ↔ S-G apparatus:
•
Analogy: Ag atom ↔ polarized light:
x-polarized
x’-polarized
y-polarized
Sz+ atom ↔ x-polarized light;
Sz- atom ↔ y-polarized light
Sx+ atom ↔ x’-polarized light;
Sx- atom ↔ y’-polarized light
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Analogy between spin-1/2 atom and polarized Light (cont’d)
•
Projection of E-field of x’(y’)-polarized light in vector space
with base vector x and y
Using the analogy:
Sz+ atom ↔ x-polarized light;
Sz- atom ↔ y-polarized light
Sx+ atom ↔ x’-polarized light; Sx- atom ↔ y’-polarized light
q Dirac notation:
|$%; ±⟩ is a state (ket) vector
representing Sx± state.
q We represent the spin states of
a silver atom by vectors in an
abstract 2-D vector space.
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Analogy between spin-1/2 atom and polarized Light (cont’d)
o How about the !" ± states?
We need a linear combination of basis vectors Sz+, Sz- with different coefficients from the linear
combinations for !$ ±
o Analogy with (left or right) circularly polarized beam of light:
•
Projection of E-field of left (right)- circularly polarized light in vector space with base vector x and y
Phase difference
&±
Complex form:
%±
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Analogy between spin-1/2 atom and polarized Light (cont’d)
Using the analogy:
Sz+ atom ↔ x-polarized light;
Sz- atom ↔ y-polarized light
Sy+ atom ↔ left circularly polarized light; Sy- atom ↔ right circularly polarized light
(±
q Dirac notation:
|$%; ±⟩ is a state (ket) vector
representing Sy± state.
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Take-home message:
QM states are represented by state vectors in
an abstract complex vector space.
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