Chapter 12 Spin (??)

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Chapter 12 Spin (??)

1. The Mathematical Description of Spin
2. Wave Function with Spin
3. The Pauli Equation

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Experimental demonstration of spin
(1)
(2)
The doublet splitting of sodium atom. The
transition of valence electron from the first
excited state to the ground state (2p?2s) lead
to two adjacent spectral lines of 589.0 nm and
589.6 nm.
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The magnitude of the magnetic moment of the
electron caused by the orbital motion,
The z component of the orbital angular momentum
is quantized by means of
For each angular momentum l?, there are 2l1
possibilities. For the doublet of sodium atom,
2ls1 2, so
Spin angular momentum is only
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1. The Mathematical Description of Spin
Spin is an angular momentum, and its description
is analogous to the orbital angular momentum. Its
three components are
Their commutation relations are
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Their commutation relations are
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In its eigenrepresentayion (????), the
eigenvalues of ?z is 1, and they can be written
by the form of diagonal matrix, namely
Since the unit matrix unchanged when the
representation changes, so
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According to
We get
Using the same method, we can obtain the
relations of other components
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The Pauli matrices are anticommuting (???).
set
Using the their anticommuting relations of Pauli
matrices, we get
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So
Since the Pauli matrix must be Hermitian, namely
So we obtain a21a12
So
(? is real)
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The total spin is
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Example 1 In the presentation ?z , solve the
eigenfunction and eigenvalue of ? x.
Solution
In the presentation ?z ,
Set the eigenfuction of ? x can be written by
Its corresponding eigenvalue is ?, therefore we
get
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namely
We obtain
When ? 1, a b. According to the condition of
normalization, we get the eigenfunction
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When ? -1, a -b. According to the condition
of normalization, we get the eigenfunction
Problem In the presentation ?z , solve the
eigenfunction and eigenvalue of ? y
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So
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According to the same method, we get
Therefore we obtain
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Example 3 spin angular momentum projects (??)
of an electron in the direction of (sin?cos?,
sin?sin?, cos?) is
Problem solve the eigenfunction and eigenvalue
of Sn
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So
Since the values which the electron spin projects
in arbitrary direction have only two values,
namely , the eigenvalue of Sn is
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Set the corresponding eigenfunction is
So we get
According to the normalized condition, we get
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When
When
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2. Wave Function with Spin (?????)
After considering spin, the wave function of a
particle can be written by
Since the spin component Sz can take only two
values, namely Sz½h
So the spin wave function has only two
components, i.e.
The complete spin wave function is described by
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where
? indicate only the state of the spin, namely,
spin up or spin down.
is the probability finding an electron with spin
up at r and t.
is the probability finding an electron with spin
down at r and t.
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3. The Pauli Equation
In the absence of spin, the Hamiltonian for the
motion of an electron in an electromagnetic field
can be written
Spin interacts with the magnetic field, and the
magnetic moment is
where
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The potential in magnetic field is
So
The Schrodinger equation of a particle with spin
is
where
? is called the spinor wave function.
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4. The simple Zeeman effect
In a weak magnetic field B, the orbital angular
moment and spin will interact with magnetic. This
interaction potential can be written
Where M is magnetic moment. M is generally
written as
Where J denotes orbital angular moment or spin, q
is the charge, g is Lande factor. For the orbital
angular moment, we have g1. However g 2 for the
spin.
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Set
So we get
Where H0 is the Hamiltonian in the absence of
magnetic field. The wave function is
?1 and ?2 are the spatial parts of the total wave
function respectively, and they can be described
by
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Where ?L is the Larmor frequency,
So we obtain
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According to the following equations
So we get
for
for
In the magnetic field, the energy level will
split, and degeneracy is missing. For the s
state, which have no orbital angular moment, the
energy level splits into two level.
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Example Only considering spinor motion, set an
electron is in magnetic field B(0, B, 0). When t
0, the electron is in the state of spin up,
namely, its spinor wave function is
Problem (1) solve the spinor function in case of
t gt0
(2) How long does the time go through when the
state of the electron turns from spin up to
spin down?
(3) At the time t, the probability of finding an
electron in the state of spin up or spin down
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Solution (1) In the case of only considering
spinor motion, Schrodinger equation satisfies
Where Hs is the Hamiltonian of the spinor motion,
and it can be written by
Where ?B is Bohr magneton, B is the magnitude of
magnetic field.
At the time t, set the spinor function of an
electron is
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After the above differential equation are solved,
we get
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According to the initial condition,
We obtain
(2) From the spinor wave function ?(t), we can
easily get
Spin up
Spin down
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Spin up
Therefore, when the period
The direction of spin of the electron changes,
namely from spin up to spin down or spin
down to spin up. The spinor state of the
electron oscillates between ?1/2 and ?-1/2, the
oscillation period is
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(3)
Therefore the probability of finding an electron
in the state of spin up is
The probability of finding an electron in the
state of spin down is
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Exercise
1. Set operators A and B commute with ?
respectively, prove the following expression.
2. Only considering spinor motion, set an
electron is in magnetic field B(0, 0, B). When t
0, the electron is in the state of spin down,
namely, its spinor function is
solve the spinor function in case of t gt0
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3. Only considering spinor motion, when tgt0, the
direction of magnetic field is along z, namely B0
(0, 0, B0). When tgt 0, another magnetic field
B1(t) is applied to the system, its direction is
perpendicular to that of B0, namely
Problem (1) solve the spinor function in case of
t gt0
(2) How long does the time go through when the
state of the electron turns from spin up to
spin down?