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Basic properties of electron spin. Spinors.

Magnetism of electron gas

Abstract: This page describes the basic properties of the electron spin. It describes what is the spin direction, spin projection and spin precession and how they should be treated in a description of electron gas.



Content

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1 Spin as a measure of breaking of the time- inverse symmetry for an electron

2.Spinor

3. Spinor & Spin direction

4.How to calculated the electron wavefunction from a known spinor or a known spin direction?

5.Why Pauli matrixes are used to describe the spin?

6. Spin Projection

7. Spin in a magnetic field

8. Spin direction vs. spin projection?

9. Quantum states and Spin

 

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The page was created in 2014. It describes my personal view and my findings on the topic.

Results in short : Spin is a measure of breaking of the time- inverse symmetry for an electron

Spin is a measure of breaking of the time- inverse symmetry for an electron

Breaking of the Time-inverse symmetry is only parameter described by the spin

Magnetic moment of electron due to spin, spin properties of electron and fermion nature of electron, all these features are consequence of breaking of the time inverse symmetry

click on image to enlarge it

 

(fact) The electron spin describes the breaking of the time inverse symmetry. All spin properties are determined by features of the time-inverse symmetry.

(fact) The broken time-inverse symmetry and correspondingly the electron spin are described by the spinors.

 

Each property of an elementary particle (like charge, momentum, energy) corresponds to breaking of one specific symmetry of our Universe. The spin describes the breaking of the time- inverse symmetry.

 

 

 

 


Spinor

To note
An electron is a quantum mechanical object and its spin should be treated quantum -mechanically The conventional representation of spin as a 3D spatial vector may tempt to apply the operational rules of a geometrical vector to spin, which is wrong for the most cases.
When working with spin, it is important to understand that spin is not a conventional 3D spatial vector. Spin is described by a spinor and it possesses its own specific properties.

(fact): The spin and therefore spinor describes the all features originated from the breaking of the time- inverse symmetry (See here for more details)

(fact): The electron wavefunction of an electron is changed when the direction of time flow is changed. For example, the electron wavefunction is Ψ1, but it becomes Ψ2 when the time direction is reversed. Under external force (e.g. under external magnetic field) the electron wavefunction can change from Ψ1 to Ψ2. In general the electron wavefunction is the linear combination of Ψ1 and Ψ2:,. The coefficients a1 and a2 are called the spinor .

(fact): Due the relativistic connection between time coordinate t and spacial coordinate (x,y,z), the influence of time inverse depends on the direction (x,y,z) in space. As a result, the spin has a direction (x,y,z) and the spinor is transformed when the coordinate system (x,y,z) is rotated. See Eq.d6, which describes the spinor transformation under an axis rotation

(fact): Features of the time-reversal symmetry for an electron are fully described by Pauli matrixes. The spinors are the eigenvectors of the Pauli matrixes

 
To read facts about spinors, click here to expand

In contrast to a particle with zero spin, which is described by a single wavefunction, the properties of an electron, which has ½ spin, are described by a linear combination of two wave functions (for example ). The coefficients in this combination depend on the direction of spin. For a given direction of spin, the wavefunction of an electron will be

The spinor S describes the coefficients in (1) as


Examples (conventional basis for )_:

When the spin is parallel and anti parallel to the x-axis, spinors will be

When spin is parallel and anti parallel to the y-axis, spinors will be

When spin is parallel and anti parallel to the z-axis, spinors will be

The spinors are the eigenvectors of the Pauli matrixes

For the spin-up spinors the eigenvalue is +1 and for the spin-down spinors the eigenvalue is -1.


 

Spin-up and Spin-down electrons

Spinors for spin-up and spin-down electrons are orthogonal, therefore their dot product is zero. For example, for spinors

the dot product is

Eq. (6b) means that a spinor, which describes an electron with spin directed in some direction, can be decomposed as a sum of spinors corresponding to opposite spins.

Also, the scalar product of spinors is important (See. Ref. 1), because it is the only bilinear form obtained from two spinors which is invariant to the rotation of the coordinate system.

The scalar product of spinors for spin-up and spin-down electrons is equal to 1:

[1] Landau and Lifshitz "Quantum Mechanics. Non-relativistic Theory" page 208 (file page 222). Djvu-file can be downloaded here or the book can be read online here.

Spinors in Wikipedia

 

 

 


All properties of an electron or a group of electrons are described by a spinor.

What is the spinor?

A. The spinor gives the electron wave function for a given direction of the electron spin. For a single electron, the spinor contains two wave functions. One wavefunction describes the electron wavefunction for one direction of its spin (for example, when the spin direction is along the z axis) and the second wavefunction describes the electron wavefunction for opposite direction of the spin (for example, when the spin direction is opposite to the z axis).

The spinor is a mathematical description of time- inverse symmetry. The spinor of an electron describes the features associated with the breaking of the time-inverse symmetry for the electron.

Precession of electron spins.

 

The electron wave function is a scalar or a spinor?

A. The wavefunction of an electron is a scalar, which describes the probability to find the electron at each point in space. The spinor describes, which scalar wave function the electron has for each direction of its spin. Since the spin direction may change in time, it is convenient to describe an electron by a spinor.

 

Can the spin of one electron vary in space and in time?

A. The spin is intrinsic feature of an electron. One electron has one spin value and one spin direction over its all spacial size. In a system of several electrons there may be a spacial variation of spin value and spin direction. For example, in an atom the nuclear and electrons have different spin directions and different spin value.

The spin direction of an electron may change in time. The rotation of the spin direction in a magnetic field is called the spin precession.




Spinor & Direction of Spin

see more about spin direction here SpinorVsSpinDirection.pdf
Note: an electron always has a defined direction of its spin. This basic property of an elementary particle came from the fact that the time inversion symmetry is broken for the electron (See here).
Precession of magnetization M around the z axis
Fig.5 Magnetization precession. The θ is the precession angle, which is a constant in time. The φ is the oscillating angle of the precession..
click on image to enlarge it

How to find spinor from a known spin direction. Basic procedure:

A quantum state of an electron, which spin direction is at angles θ and φ as shown in Fig.5, is described by a spinor S, which is an eigenvector of the following Pauli matrix:

Calculation of the eigenvector of (a1) gives the spinor of an electron, which spin direction is at angles θ and φ as shown in Fig.5, as

or

example 1: Spin is in the xy-plane θ= π/2

When the spin direction is in the xy-plane and φ is the angle of spin with respect to the x-axis, the Pauli matrix (a1) becomes

The spinor for the example 1 (eigenvector of (d1) is calculates as

example 2:Spin is in the xz-plane φ= 0

The Pauli matrix (a1) becomes

The spinor for the example 2 (eigenvector of (d3) is calculates as

or

 


How to calculated the electron wavefunction from a known spinor or a known spin direction?

In order to find the wavefunction for any spin direction, it is necessary to to know the wavefunction at least for two directions. For example, if the wavefunction is , when the spin along the z-axis, and the wavefunction is , when the spin opposite to the z-axis, the wavefunction for other direction can be calculated as follows:

along z- axis

spinor:

wavefunction:

-----

opposite to z- axis

spinor:

wavefunction:

-----

along x- axis

spinor:

wavefunction:

-----

opposite to x- axis

spinor:

wavefunction:

-----

 


Why Pauli matrixes are used to describe the spin? Why the Pauli matrixes are so special?

Three Pauli matrixes are only matrixes, which satisfy the following conditions:

(condition 1) They should be 2x2 matrixes, because there are only two independent wavefunctions, which correspond to the direct and reversed time flow.

(condition 2) They should be an unit matrixes. Obviously, the spin rotation should not change the number of electrons.

(condition 3) Double rotation over 90 degrees around any axis should reverse the spin or the same should give the time-inverse wavefunction.

 


Spin projection

It is possible to find a projection of spin onto any direction. It is also possible to decompose spin directed in one direction as a sum of spins directed parallel and anti parallel to some other direction (for example, a sum of spin-up and spin-down wavefunctions).

Spin projection (p) (click here to expand)

It should be noted that spinors of two spins, which are directed parallel and antiparallel to some axis, are orthogonal. Therefore, their dot product equals zero

Using the property (p1), a spin can be represented as a sum of two spins of opposite directions as

 

Example 1 Spin directed along the x-axis is decomposed as a sum of opposite spins directed along the z-axis.

It should be noted that this is impossible for a conventional spacial vector. From (2) and (5), we have

 

Example 2 spin directed in the xy-plane is decomposed as a sum of opposite spins directed along the x-axis.

Rewriting (d2) and (3)

where phi is the angle of spin with respect to the x-axis. The spinor of spin directed in the xy-plane is

From Eq.(p1)

or

the probability to be in spin up and spin down states will be

in the case of small phi Eq. (p7) is simplified to

 

 


 

 


only-possible states of spin in a magnetic field
Temporally stable states Temporally unstable states
spin along field spin-inactive
When spin is aligned along magnetic field, the electron energy is smallest Two electrons of opposite spins, which occupy one quantum state, are spin-inactive, because the time-inverse symmetry is not broken for them. As a result, they do not interact with the magnetic field at all. Since the time-inverse symmetry is not broken, there is no spin direction and the cases of left figure and right figure are fully identical. There is no direction for this state.
spin opposite to magnetic field spin precession
Emission of circularly- polarized photon, at first, creates the spin precession and finally the spin alignment along the magnetic field. The direction of the magnetic moment is alternating with the precession frequency. It results in emission of circularly- polarized photons and the spin damping. The spin damping aligns the spin along the direction of the magnetic field.
click on image to enlarge it

 

Spin in a magnetic field

 

Below it is shown how to describe the Zeeman effect and the Larmor precession of spin in a magnetic field using spinors.

 

How it is done?
1. The spinor is calculated for the case electron spin is either parallel or anti parallel to magnetic field.
2. The spin precession is calculated for the case when the spin direction is different from the direction of the magnetic field.

 

Step 1. The calculation of the spinor for the case when the spin direction is either parallel or antiparallel to the magnetic field.

In a magnetic field the electrons, whose spin directed along and opposite to the magnetic field, have different energy (Zeeman effect). The difference in energy is given as

where g is the g- factor, is the Bohr magneton.

For example, in the case when a magnetic field H is applied along the z-axis, the electrons with spin directed parallel and anti parallel to the z-axis have energy

where E0 is the electron energy in the absence of a magnetic field.

From Eq.(21), the wavefunction of spin-down and spin-up electrons can be expressed as

or

Eqs. (23) describe electron wavefunction for two directions of the spin by taking the projection on the z axis of following spinor:

 

Step 2. The calculation of the spinor for the case when the spin direction is either parallel or antiparallel to the magnetic field.

The Eqs. (23) describe the case when both the spin direction and the direction of the magnetic field are along the z-axis. When the spin direction turns away from the z-axis through angle theta, the electron wavefunction should still be described by Eq.(23) but using the spinor corresponding to the turned spin. Substituting Eq. (d.4) for the spinor turned by angle theta away from the z-axis into Eq(23) gives

Eq. (24) can be simplified to

where S(t) is the time-dependent spinor

which describes an electron, whose spin direction has time-independent angle theta with respect to the z-axis and the time –dependent angle phi in the xy- plane. Comparing Eqs. (d5) and (26), the angle of spin in the xy-plane with respect to the x-axis is

Therefore, Eq (25) described a precession of spin around the magnetic field with the Larmor frequency

Fig 3. Animated figure. Precession of spins of a an conduction electron (blue arrows) due to the exchange interaction with the local d-electrons. The precession axis is along the spin direction of the local d-electrons (red arrow). The angle between spin directions of sp- and d-electrons is 70 deg.

Eq.(24) and Eq (25) are equivalent. In Eq (24) the spinor is time-independent and it describes two electron states of different energy. In contrast, in Eq. (25) the spinor is time-dependent and it describes electron states of the same energy, but with spin precession.

Therefore, there are two fully equivalent representations of spin in a magnetic field. In the first representation the spin state is described as a superposition of two spin states with different energy and opposite time-independent spin-directions. In the second representation the spin state is described as a state with a constant energy, but with time-dependent spinor. It corresponds to the time-dependent spin direction or the spin precession.

 

The spin precession is not specific for the interaction of spin with magnetic field, but is a general property of spin.


 

Spin direction vs. spin projection?

Along any direction (for example, the direction of the magnetic field, spin can have only two projections ½ and -½). In contrast, spin can have any direction in the 3D space. This looks like as two controversial statements and sometimes it confuses the understanding of spin properties.

In fact, this is just two identical representations of a "spin" state in a magnetic field. In the first representation a "spin" state is represented as a sum of two projections with different energies. The energies are different for the projection with spin parallel or anti parallel to the magnetic field. There is a finite probability for a “spin” state to be in a state with either higher energy or lower energy. In the second representation, the “spin” state has a defined energy, but there is spin precession around the magnetic field. It should be noted that the precession frequency (the Larmor frequency) does not depend on the angle between the spin direction and the magnetic field and there is a spin precession even in the case when the spin parallel to the magnetic field.

 


 

 

Fig 1. Three possibilities of occupation of an electron state. Only a state, which is filled by one electron, has a defined spin direction. For the spin- inactive state (also called "full" state) and the "empty" state, the time-inverse symmetry is not broken. As a result, they both do not have a spin direction.

Quantum states and Spin.

1. The conduction electrons can be in "spin", spin- inactive state and "empty" states. Only "spin" states have spin directions. The spin of spin- inactive state and "empty" states is zero and these states do not have a spin direction.

2. There are two electrons of opposite directions in a spin- inactive state, but the information about the spin direction of electrons before been scattered into a "full" state is not memorized in the "full" state. Therefore, the two electrons of opposite spin directions can have spin directed in any direction with equal probability.

3. "spin" states have a spin direction. spin- inactive state and "empty" states do not have a spin direction

4. There are two identical representations of a "spin" state in a magnetic field. In the first representation a "spin" states is represented as a sum of two projections with different energies. The energies are different for the projections with spin parallel are antiparallel to the magnetic field. In the second representation, there is a spin precession around the magnetic field. The energy of the "spin" state is constant.

 

States filled by one or two electrons or not filled

An electron quantum state can either be filled by no electrons or one electron or two electrons of opposite spins. We define these three states as “full”, “empty” and “spin” states (See Fig.1).

Only the "spin" state can have a spin direction. The spin of "full" and "empty" states is zero and these states do not have a spin direction. This can be easily understood by comparing an electron with a photon. The spin direction can be thought of as analogous to the polarization direction of a photon. The analog of a "spin" state is linearly polarized light. It always has a defined direction of polarization (it could be s- or p- polarized or somewhere in between). The analog of a "full" state is unpolarized light (light of equal p- and s- polarizations). The polarization direction in this case can not be defined (The intensity of light is equal for any polarization direction).

1. "spin" state

"spin" state is an electron state in which one of the two possible places is occupied by an electron. The wavefunction for a "spin" state is

The time inverse symmetry is broken for a "spin" state. When the time inverse symmetry is broken, there is always a direction along which it is broken. Of course, it is the spin direction.

The "spin" state always has a spin direction along which the time inverse symmetry is broken.

A change of the spin direction can be described by a corresponding change of the coefficients a1 and a2 in the spinor (7).

 

2. spin- inactive state or "full" state

A spin-inactive state (also called a "full" state) is a electron state, in which both of two possible places for an electron are occupied. The spin-inactive state state has spin "0" and it is described by a scalar wavefunction, which remains unchanged when the coordinate system is rotated.

Such a wavefunction can be expressed as the scalar product of spinors for spin-up and spin-down electrons.

It is important to emphasize that the wave functions (8) is the same for electrons with any spin direction. This means that the wavefunction (8) is the same for any combination of the spinors of opposite spins.

The time inverse symmetry is not broken for a spin-inactive state. A wavefunction of a spin-inactive state is invariant under axis rotation.

 

 

Fig 2.A spin-inactive state state. The time-inverse symmetry is not broken for this electron state. As a result, the spin-inactive state does not have spin direction. Therefore, all these representation of the spin-inactive state are identical and indistinguishable. A wavefunction of a spin-inactive state is invariant under axis rotation.

 

 

Important note

The spin-inactive state can not have any direction!!! For example, three spin-inactive states shown in Fig. 2 are indistinguishable.

Note: A wavefunction of a particle with zero spin is invariant under axis rotation. Therefore, all properties of the particles should also be invariants under axis rotation

3. "empty" state

"empty" is a electron state in which both of the two possible places for an electron are unoccupied.

Similar to the spin-inactive state, the time-inverse symmetry is not broken for the "empty" state. As a result, the " empty" state has no spin direction.


 

Only "spin" states may contribute to magnetic properties of conduction electrons. The spin-inactive state and "empty" states have zero spin and they do not contribute to the magnetic properties.

This could be understood by a comparison with EPR, where only unpaired electrons contribute to the EPR signal (Details, see here)

 

 

 

 

 

 

 


 

The same content can be found in V. Zayets JMMM 356 (2014)52–67 (click here to download pdf)

 

 

 

 

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