This blog post is a continuation for the pipe model that we have developed for the spinning of a created electron-positron pair.
The idea is that the wave function (relevant for the rotation) of the system electron & positron has a 360 degree rotation symmetry. The wave function of the system completes one wavelength when we rotate coordinates by 360 degrees.
The wave function of the system is the product of the wave functions of the electron and the positron.
The individual wave function of a particle does not behave in an intuitive way for our 3-dimensional space: the phase of the wave function is inverted when we rotate coordinates by 360 degrees.
That inversion does not make much sense for a complete quantum system, but does happen if we break up the system into parts.
We may require that the wave function of a complete quantum system must return to the original value after we rotate coordinates by 360 degrees.
If we try to treat the electron as an independent quantum system, we end up with the strange 720 degree rotation symmetry of the spin.
The SU(2) geometry of spinors gets an explanation from this.
What about the gyromagnetic ratio 2? In quantum mechanics, we do not have rules about how to determine the momentum of an individual part of a quantum system. The rotation of the electron lives in a strange 720 degree geometry. How should we map it to the ordinary 3D space to determine the velocity v of the electron and calculate the magnetic force it exerts?
It might be that the observer must treat the electron rotation as having a double velocity compared to the "real" velocity, so that he sees the electron wave function having a 360 degree rotation symmetry. Then he will get a double value for v, and will see the electron exert a magnetic force which is 2X of what we would expect based on the spin angular momentum 1/2 h-bar.
What about the spin angular momentum, the 1/2 h-bar? Why he does not see the electron spin as double? One may conjecture that since quantum mechanics conserves angular momentum, one must see the spin to have its correct value. On the other hand, there is no conservation of magnetic moment. Nothing breaks if an observer sees and feels the magnetic moment of an electron as double.
The spin angular momentum has been measured directly by flipping the spin of many electrons in a block of solid material. We know that the angular momentum is 1/2 h-bar. The flipping obviously has to use the magnetic moment of the electron as a handle which we use to flip the orientation of the spin. Why does the magnetic handle have a double strength, compared to what we would expect?
The Dirac equation offers sort of an explanation. We still need to find the link from the model we tried to sketch, to the Dirac equation.
The Dirac equation does not have a reasonable speed operator v. There is also the mystery of zitterbewegung. These are probably connected to the fact that the electron only forms half of the complete quantum system, the electron-positron pair.
The concept of the reduced mass may offer a clue. A two-particle system is represented by the center of mass and the vector from particle 1 to particle 2. The treatment of the complete electron-positron quantum system might involve similar ideas.
Thursday, March 7, 2019
Thursday, February 28, 2019
Does the Dirac equation work by chance?
Despite three months of hard work we have not been able to find an intuitive physical model which would explain the Dirac equation. Richard P. Feynman wrote that no one has understood the Dirac equation "directly".
Is it possible that the Dirac factorization of the Klein-Gordon operator just by chance adds the necessary degree of freedom that can be used to describe the spin 1/2 of the electron?
Is it possible that the Dirac factorization of the Klein-Gordon operator just by chance adds the necessary degree of freedom that can be used to describe the spin 1/2 of the electron?
Does the Dirac derivation of his equation predict the spin 1/2?
The Klein-Gordon equation is Lorentz covariant, and Dirac derives his own equation in a way which makes it Lorentz covariant, too.
But what properties of the electron does the Dirac equation actually predict?
The eigenfunctions of the equation can be defined as a standard plane wave times a 4-component Dirac spinor.
We then note that if we define an operator S in a way similar to the Pauli equation, S obeys rules which emulate the simple commutation rules we expect from a particle of spin 1/2. That is, if we know the spin projection in the z direction, we know nothing of the projection in the x direction, and so on.
Did the Dirac equation predict the spin 1/2? One might say: no - we just picked an arbitrary operator S which emulates the known rules of spin 1/2. But the Dirac equation made it possible to use a very simple operator S. At least in that sense, the Dirac equation predicted the spin 1/2.
The Dirac hamiltonian H commutes with L + S, where L is the usual orbital angular momentum operator. That fact suggests that S really describes some kind of angular momentum.
But H does not commute with S if the electron is relativistic. Does that make sense? If we have a free electron, why would its spin change during its flight?
Does the Dirac equation predict the gyromagnetic ratio 2?
The Pauli equation is the non-relativistic limit of the Dirac equation.
In the general form of the Pauli equation, the interaction with a magnetic field B is hidden in the minimal coupling
(σ • (p - qA))^2
kinetic term.
But in the standard form, the interaction is shown explicitly as
σ • B.
The Dirac equation does predict the correct interaction strength and the gyromagnetic ratio 2.
Is it possible that the Dirac equation by chance gets the ratio 2 right, even if the equation does not "really" describe the physical system? That looks unlikely.
The Dirac factorization of the Klein-Gordon operator is a general way to add a spin degree of freedom?
The Klein-Gordon equation with the minimal coupling describes the behavior of a spinless electrically charged massive point particle.
The Dirac factorization trick of the Klein-Gordon operator adds a spin degree of freedom, and furthermore, the minimal coupling gives the right interaction strength with a magnetic field for the spin, too. The minimal coupling term was designed to describe the behavior of a spinless point particle, but it magically produces the right interaction also for the spin angular momentum.
Is it a general rule that factorization of an operator adds a spin-like degree of freedom? If yes, why?
Does the factorization always give a sensible strength for the interaction of the spin with an external field?
Tuesday, February 26, 2019
The spin 1/2 h-bar of an electron is a remnant of the orbital angular momentum of positronium?
In the hydrogen atom, the electron orbital angular momentum in the z direction is an integer multiple of h-bar. Classically (that is, in Newtonian mechanics), almost the entire orbital angular momentum is in the movement of the electron, and only a tiny fraction in the movement of the proton.
In the positronium "atom", the orbital angular momentum is divided evenly between the electron and the positron. This may be the origin of the strange electron spin 1/2 h-bar. An electron and a positron are always created together. If they form some kind of a primitive positronium atom where the particles do not yet possess a spin, then after flying away, the electron and the positron will evenly share the 1 h-bar of orbital angular momentum of the positronium atom. In this model, the electron and the positron would have parallel spins. In the real world, they seem to have opposite spins.
Classically, the distance between the electron and the positron is 2r in the positronium atom, where r is the Bohr radius of the hydrogen atom. Both particles move around the center of mass in a circular orbit whose radius is r. Since the electric pull on the electron in positronium is only 1/4 of the pull in a hydrogen atom, the orbital velocity of the electron has to be 1/2 of that in a hydrogen atom, so that the centripetal acceleration agrees with the electric pull.
The de Broglie wavelength is defined as
λ = h / p.
Since the electron momentum p in a positronium atom is just half of the hydrogen atom, the electron only completes half of a de Broglie wavelength in its circular orbit. In previous blog posts we have said that a "natural" periodic movement of a single particle must contain an integer number of wavelengths, to avoid destructive interference. But we noted that if two particles are moving "in unison", then we may consider them as a single system, and it is enough that the system completes a full number of wavelengths in one period. Now we realize that the positronium atom is just such a system.
In a previous blog post we developed the "pipe model" of an electron-positron pair. The particles rotate in unison at the opposite ends of the pipe. A problem is to find a way how the particles can preserve their tandem movement even when one of the particles is accelerated or its spin is rotated.
If we think of a positronium atom in a laboratory, both the electron and the positron have a spin 1/2 h-bar, and they may also have a multiple of h-bar of orbital angular momentum. How do we model the annihilation then?
In the positronium "atom", the orbital angular momentum is divided evenly between the electron and the positron. This may be the origin of the strange electron spin 1/2 h-bar. An electron and a positron are always created together. If they form some kind of a primitive positronium atom where the particles do not yet possess a spin, then after flying away, the electron and the positron will evenly share the 1 h-bar of orbital angular momentum of the positronium atom. In this model, the electron and the positron would have parallel spins. In the real world, they seem to have opposite spins.
Classically, the distance between the electron and the positron is 2r in the positronium atom, where r is the Bohr radius of the hydrogen atom. Both particles move around the center of mass in a circular orbit whose radius is r. Since the electric pull on the electron in positronium is only 1/4 of the pull in a hydrogen atom, the orbital velocity of the electron has to be 1/2 of that in a hydrogen atom, so that the centripetal acceleration agrees with the electric pull.
The de Broglie wavelength is defined as
λ = h / p.
Since the electron momentum p in a positronium atom is just half of the hydrogen atom, the electron only completes half of a de Broglie wavelength in its circular orbit. In previous blog posts we have said that a "natural" periodic movement of a single particle must contain an integer number of wavelengths, to avoid destructive interference. But we noted that if two particles are moving "in unison", then we may consider them as a single system, and it is enough that the system completes a full number of wavelengths in one period. Now we realize that the positronium atom is just such a system.
In a previous blog post we developed the "pipe model" of an electron-positron pair. The particles rotate in unison at the opposite ends of the pipe. A problem is to find a way how the particles can preserve their tandem movement even when one of the particles is accelerated or its spin is rotated.
If we think of a positronium atom in a laboratory, both the electron and the positron have a spin 1/2 h-bar, and they may also have a multiple of h-bar of orbital angular momentum. How do we model the annihilation then?
Tuesday, January 15, 2019
Sabine Hossenfelder on good problems in the foundations of physics
Sabine Hossenfelder has written an interesting post about which problems might be fruitful for research in the foundations of physics. Since our blog is about foundations of physics, let us comment on her program.
Lubos Motl wrote a harsh criticism of Hossenfelder.
The thesis of Hossenfelder is that when experiments conflict theoretical predictions, that is a fruitful experiment-led problem.
If theory itself is inconsistent, that makes a fruitful theory-led problem.
Hossenfelder analyzes 12 problems, if they are fruitful and if they are experiment-led or theory-led. Our blog has touched several of those problems. Let us go through the list of problems.
Dark matter
There is no principle in particle physics which prohibits weakly interacting particles. A priori, the existence of dark matter is more probable than its non-existence. The competing hypotheses, like MOND, suffer from the fact that it is hard to modify newtonian mechanics or general relativity without breaking conservation of energy, momentum, and angular momentum, or equivalence principles. We have not seen anyone developing a MOND model where conservation laws would hold.
Dark energy
Dark energy can be accommodated to general relativity through a cosmological constant. It does not break anything. But what is the origin of the cosmological constant? In our blog, we hold the view that empty space is truly empty - it does not contain energy or vacuum fluctuations. We would not explain dark energy by "vacuum energy" which is present in empty space. Dark energy might be an unknown force.
Hierarchy problem
Why is gravity much weaker than other forces? An anthropic argument is that a strong gravity would make everything collapse into black holes, and humans would not exist.
The weakness itself is not mysterious about gravity. But gravity does have mysterious properties: why does it affect all mass-energy, why does it appear to modify spacetime geometry, why the force is always attractive, and why the gravitating mass is equivalent to the inertial mass?
Grand unification
There is nothing in particle physics that requires the electroweak and strong interactions to be unified at high energies. However, the unification of electromagnetism with the weak interaction hints at that possibility.
Quantum gravity
One of the goals of our optical gravity model, and also our renormalization/regularization study, is to find a way to integrate gravity into ordinary quantum mechanics.
There are problems with the geometry of black holes in classical general relativity. We do not know if the Kerr solution is stable.
We do not know at what speed does information about mass-energy distribution spread in general relativity.
To build a model of quantum gravity, we need to clarify classical general relativity.
Black hole information loss
Our view is that Hawking radiation probably does not exist. Therefore, there is no information loss problem.
In quantum mechanics, systems develop in a unitary way and there is no information loss. The fact that the hypothetical Hawking radiation would break this principle is one of the symptoms which show that Hawking used flawed quantum field theory. Other symptoms include problems with energy conservation, momentum conservation, and the classical limit of his hypothesis.
We do not understand why some physicists hold a religious view that Hawking radiation "must" exist. The derivations of Hawking radiation rest on a very shaky, and probably flawed, use of quantum field theory.
Particle masses
There is no principle in particle physics that requires particle rest masses to have a deeper explanation. But there may exist a model, a string model, for example, which might cast more light on the problem.
Quantum field theory
Our hypothesis is that both the infrared and ultraviolet divergences of Feynman integrals are a result of a wrong integration order. We will study that hypothesis in spring 2019.
The Landau pole means that higher order Feynman diagrams will contribute more to the process than lower order diagrams. It is a complexity explosion. The energy is so high that a black hole would form before Landau pole energies are reached. The black hole may save us from a Landau pole.
The measurement problem
Our view is that the many worlds interpretation, where the "branch" for an observing "subject" is chosen with the Bohmian hidden variable method, is the most sensible interpretation of quantum mechanics.
It is not clear if we can ever devise experiments which would differentiate between interpretations. The problem may remain a philosophical one.
The flatness problem
If empty space is truly empty of energy, then flatness is expected.
In optical gravity, we have a hypothesis that the true geometry of spacetime is the flat Minkowskian geometry. But we would need a model to explain the Big Bang. If spacetime is flat, why does the universe appear to expand?
Magnetic monopoles
Some GUTs imply the existence of magnetic monopoles. However, in ordinary particle physics there is no principle that would dictate that they should exist.
A deeper understanding of quantum electrodynamics may resolve this problem. An electron is a source of the electric field. Why there is no source particle for the magnetic field?
Baryon asymmetry
We pointed out that if there are superheavy particles and antiparticles, then a single particle might decay into a whole visible universe which contains just matter. The asymmetry is probably just a local phenomenon.
Why is the cosmic microwave background so isotropic?
Why is the temperature so uniform in areas which are not causally connected in a standard Big Bang model? There may be unknown laws of physics which create a nearly uniform energy distribution in a phase change of the universe. There is no need for the patches to be causally connected if the same mechanism creates the mass-energy in each patch.
The inflation hypothesis explains the uniformity, but Paul Steinhardt has criticized it because it requires fine-tuning which may be even harder than the problem it tries to explain.
Another explanation would be a Big Bounce model. But we do not know laws which would cause the universe to contract after a Big Bang.
Sunday, January 13, 2019
The gyromagnetic ratio is 2 because the spin lives in a 1+2-dimensional space?
In the Pauli equation, the effect of the vector potential can be modeled with the flow vector of water.
In our example, an electron circles in a plane close to the wire. There are 1+2 dimensions. If the spin would live in a 1+1-dimensional system, then the effect of B on the kinetic energy of a stationary solution might be 2X in the spin case.
https://en.wikipedia.org/wiki/Pauli_matrices#Eigenvectors_and_eigenvalues
If we stretch the plane by 1 % in the direction of the x axis, then a vector which is at a 45 degree angle to the x axis will get stretched by 0.5 %. Many angles between the eigenvectors of various σ_i are 45 degrees or 135 degrees. If the effect of a magnetic field would be to stretch one eigenvector by 1%, then the volume of a 3D cube might grow by 2 %. This might be the origin of the gyromagnetic ratio g = 2.
Close to the wire carrying the current, water flows to the same direction as the current. The velocity vector of water is the vector potential A.
---> ---> A
------> ----->
--------------------- wire
I -->
If we have an electron doing a circle in the plane of the page anti-clockwise, then the flow of the water "helps" it in its movement close to the wire. The water flow will inflate the wavelength of the electron close to the wire.
Suppose that we have a stationary wave solution in a 2-dimensional (the time is the 3rd dimension) round cavity. Suppose that we increase the area of half of the cavity by 1% and decrease the other half by 1%. This corresponds to modifying the wave equation slightly inside the cavity. The wavelength in one half will decrease 0.5%. That corresponds to a kinetic energy increase of 1%.
But if the cavity would be 1-dimensional, and we would decrease the length of one half by 1%, that would correspond to a 2% increase in kinetic energy.
The angular momentum of the translational movement of an electron,
r × v,
lives in a 1+3-dimensional space. There are three orthogonal axes of rotation. But the spin angular momentum seems to live in a 1+2-dimensional space. The eigenvectors of the Pauli matrices are in a 2-dimensional space.
r × v,
lives in a 1+3-dimensional space. There are three orthogonal axes of rotation. But the spin angular momentum seems to live in a 1+2-dimensional space. The eigenvectors of the Pauli matrices are in a 2-dimensional space.
The difference of dimensionality may be the origin of the strange gyromagnetic ratio g = 2 for the spin. If the effect of a magnetic field is to change the volume (or the area in 2D or the length in 1D) of a half of the the stationary wave system by a ratio which depends on B, then its effect on the kinetic energy depends on the dimensionality of the system.
https://en.wikipedia.org/wiki/Pauli_matrices#Eigenvectors_and_eigenvalues
If we stretch the plane by 1 % in the direction of the x axis, then a vector which is at a 45 degree angle to the x axis will get stretched by 0.5 %. Many angles between the eigenvectors of various σ_i are 45 degrees or 135 degrees. If the effect of a magnetic field would be to stretch one eigenvector by 1%, then the volume of a 3D cube might grow by 2 %. This might be the origin of the gyromagnetic ratio g = 2.
Why does the spin live in 1+2 dimensions?
Why does the spin live in one less spatial dimensions than the ordinary angular momentum? Maybe some uncertainty relation drops one spatial dimension from the description of a 1/2 h-bar angular momentum?
Friday, January 11, 2019
Is there a "quantum state" of an individual electron in a multiple electron system
The Pauli exclusion principle claims that in a non-hydrogen atom, each electron occupies a separate "quantum state". Similarly, in a piece of metal, free electrons fill a "Fermi sea" of states, each falling to its own pigeonhole which is determined by the "quantum state" of the electron.
We criticized the Pauli exclusion principle because there is no definition of what that "quantum state" is, or means.
Let us look at the helium atom. The usual way of modeling it is to assume that we have a single particle moving in a 6-dimensional space. There is a central potential due to the nucleus and a "planar potential" which has a very high energy when the single particle position is equivalent to the two electrons being very close.
If we find a solution of the Schrödinger equation, is there any way to "factorize" it into two parts where each part describes the state of a single electron?
If we have two electrons in different hydrogen atoms, there exists such a factorization. It is trivial.
Let us look at a simpler problem. Suppose that we have two particles in an external potential well in a space with a time dimension and one space dimension. There is a repulsion between the particles. The Schrödinger equation then is about a single particle in a 1+2-dimensional space.
The external potential makes a square well for the single particle. In addition to that, there is the interaction potential of the original two particles. That potential is concentrated on the line y = x, where x and y are the spatial coordinates of the single particle.
________
| / |
| / |
| / |
|/______|
^ y
|
|
---------> x
The diagram is not in scale. The rectangle depicts the square potential well. The diagonal depicts the interaction potential wall of the original two particles.
A solution of the Schrödinger equation is a stationary wave inside the rectangle. It is like a rectangular drum skin vibrating in a resonance pattern within that square.
Now, is there any reason why the solution could be factorized into solutions of two individual particles?
If the particles do not interact, then the factorization is trivial. If there is a weak interaction, we may get some results with perturbation methods. But what if the interaction is strong?
Let us look at a simpler problem. Suppose that we have two particles in an external potential well in a space with a time dimension and one space dimension. There is a repulsion between the particles. The Schrödinger equation then is about a single particle in a 1+2-dimensional space.
The external potential makes a square well for the single particle. In addition to that, there is the interaction potential of the original two particles. That potential is concentrated on the line y = x, where x and y are the spatial coordinates of the single particle.
________
| / |
| / |
| / |
|/______|
^ y
|
|
---------> x
The diagram is not in scale. The rectangle depicts the square potential well. The diagonal depicts the interaction potential wall of the original two particles.
A solution of the Schrödinger equation is a stationary wave inside the rectangle. It is like a rectangular drum skin vibrating in a resonance pattern within that square.
Now, is there any reason why the solution could be factorized into solutions of two individual particles?
If the particles do not interact, then the factorization is trivial. If there is a weak interaction, we may get some results with perturbation methods. But what if the interaction is strong?
Assume that each solution is determined uniquely by a set of quantum numbers of each electron
Let us assume that we have a strongly interacting electron system. Let us assume that each solution of the Schrödinger equation is uniquely (except by a phase factor) determined by some "quantum numbers" that we attach to each individual electron.
Can we derive the Pauli exclusion principle, for example, from the antisymmetricity of the fermion wave function? The antisymmetry means that the sign of the wave function is flipped if we replace coordinate values of, say, x_1, y_1, z_1 with x_2, y_2, z_2, and conversely.
Now, if electrons 1 and 2 have the exact same quantum numbers, we have:
Ψ_switched = Ψ_original,
because the sequence of quantum numbers specifying Ψ did not change.
But, the antisymmetry of the fermion wave function implies
Ψ_switched = -Ψ_original.
We have that Ψ must be zero. We can derive the Pauli exclusion principle from the assumptions:
1. The wave function solution is uniquely determined by a set of "quantum numbers" which can be "assigned" to the coordinate triplet of each electron.
2. The wave function is antisymmetric under the switch of two coordinate triplets.
Is there a mathematical proof that helium atom solutions have property 1 above?
A brief Internet search does not lead us to any such proof. A related question is in which cases a wave equation has a discrete spectrum of stationary states or "resonant" states.
https://en.wikipedia.org/wiki/Spectral_theorem
The spectral theorem states that all the solutions of the Schrödinger equation can be written as sums of eigenfunctions of the hamiltonian. Each eigenfunction is associated with an energy eigenvalue.
In which cases is the spectrum of energy eigenvalues discrete?
Does the spectral theorem imply anything about a multiple electron system? Could the theorem give some factorization of the solution for individual electrons?
Now, if electrons 1 and 2 have the exact same quantum numbers, we have:
Ψ_switched = Ψ_original,
because the sequence of quantum numbers specifying Ψ did not change.
But, the antisymmetry of the fermion wave function implies
Ψ_switched = -Ψ_original.
We have that Ψ must be zero. We can derive the Pauli exclusion principle from the assumptions:
1. The wave function solution is uniquely determined by a set of "quantum numbers" which can be "assigned" to the coordinate triplet of each electron.
2. The wave function is antisymmetric under the switch of two coordinate triplets.
Is there a mathematical proof that helium atom solutions have property 1 above?
A brief Internet search does not lead us to any such proof. A related question is in which cases a wave equation has a discrete spectrum of stationary states or "resonant" states.
https://en.wikipedia.org/wiki/Spectral_theorem
The spectral theorem states that all the solutions of the Schrödinger equation can be written as sums of eigenfunctions of the hamiltonian. Each eigenfunction is associated with an energy eigenvalue.
In which cases is the spectrum of energy eigenvalues discrete?
Does the spectral theorem imply anything about a multiple electron system? Could the theorem give some factorization of the solution for individual electrons?
Friday, December 28, 2018
The minimal coupling p - qA
UPDATE Jan 11, 2019: The Landau quantization
https://en.wikipedia.org/wiki/Landau_quantization
solves the problem of an electron in a uniform magnetic field. The solution hangs on the fact that the hamiltonian only depends on coordinate x in the Wikipedia article. The solution is then symmetric on translations along the y axis, which means that p_y must be constant, an eigenvalue of the p_y operator.
The operator
p_y = -i d/dy
gives the canonical momentum which is not the measured kinetic momentum. Our earlier discussion in the text below mixed the canonical momentum with the kinetic momentum. This led to confusion in our text.
https://physics.stackexchange.com/questions/281687/why-is-p-y-conserved-in-the-landau-gauge-when-we-know-the-electron-moves-in-ci
At the Physics Stack Exchange there are several questions that rise from the confusion.
If we break the symmetry along the y axis by adding an electric potential which confines the electron into some y interval (y_0, y_1), what happens then?
---
The Pauli equation contains a hamiltonian where the kinetic energy term is something like
(p - qA)^2 / (2m).
There p is the momentum vector, q is the charge, and A is the magnetic vector potential.
If we have a wire with a current, then the vector potential points to the direction of the current and is less when we go farther from the wire.
We assume that the wire has no electric field. Its magnetic field lines are circles around the wire.
If we let a charge fly freely closer to the wire, the magnetic force is at a right angle relative to the kinetic momentum of the electron.
---
NOTE: p is the canonical momentum. It does not change. The kinetic momentum p - qA does change, as it should.
https://en.wikipedia.org/wiki/Landau_quantization
solves the problem of an electron in a uniform magnetic field. The solution hangs on the fact that the hamiltonian only depends on coordinate x in the Wikipedia article. The solution is then symmetric on translations along the y axis, which means that p_y must be constant, an eigenvalue of the p_y operator.
The operator
p_y = -i d/dy
gives the canonical momentum which is not the measured kinetic momentum. Our earlier discussion in the text below mixed the canonical momentum with the kinetic momentum. This led to confusion in our text.
https://physics.stackexchange.com/questions/281687/why-is-p-y-conserved-in-the-landau-gauge-when-we-know-the-electron-moves-in-ci
At the Physics Stack Exchange there are several questions that rise from the confusion.
If we break the symmetry along the y axis by adding an electric potential which confines the electron into some y interval (y_0, y_1), what happens then?
---
The Pauli equation contains a hamiltonian where the kinetic energy term is something like
(p - qA)^2 / (2m).
There p is the momentum vector, q is the charge, and A is the magnetic vector potential.
If we have a wire with a current, then the vector potential points to the direction of the current and is less when we go farther from the wire.
We assume that the wire has no electric field. Its magnetic field lines are circles around the wire.
If we let a charge fly freely closer to the wire, the magnetic force is at a right angle relative to the kinetic momentum of the electron.
---
NOTE: p is the canonical momentum. It does not change. The kinetic momentum p - qA does change, as it should.
The curl of the magnetic vector potential
The magnetic vector potential A is defined as the vector field whose curl is the magnetic field B:
∇ × A = B.
The component of the curl pointing to the direction of the thumb at a point x is defined as the path integral of A on the circular path pointing to the direction of the fingers of the right hand, divided by the area enclosed by the path.
Let us consider an electric wire of a finite length:
• • •
• • • • • • •
-----------------------------
I -->
The dots • mark magnetic field B lines pointing out of the page.
A vector potential A which reproduces the field B is like:
--> --> -->
----> ----> ----> A
--------> -------->
--------------------------------
I -->
|A| is larger close to the wire. A points to the direction of the electric current.
Now, if we have an electron approaching the wire in the diagram from up, the electron will draw a path like:
e- ^
\ /
\ ____/
-----------------------------
I -->
^ y
|
|
------> x
^ y
|
|
------> x
The electron will turn counterclockwise in the magnetic field. Since the magnetic force
F = q v × B
is orthogonal to the velocity vector v, it will not change the absolute value of v but its direction. The electron will leave the magnetic field with the same absolute velocity at which it arrived.
In the diagram above, the quantity
H = (p - eA)^2 / (2m)
is not larger when the electron is close to the wire because p is not the kinetic momentum but the canonical momentum. Note that the charge e of the electron is negative.
Hamilton's equations
The equations are
dp/dt = -dH/dq
dq/dt = dH/dp,
where p is the canonical momentum and q is the (canonical?) position.
where p is the canonical momentum and q is the (canonical?) position.
The lagrangian
The hamiltonian is derived from the lagrangian. Let us check if the lagrangian gives the correct circular orbit for the electron in our example case.
The action is defined as the path integral of the lagrangian over a time interval [t1, t2] over a path q(t). If the path is a correct time evolution of the system, then the action should remain constant under "very small changes" of the path.
The path integral, of course, suffers from the fact that the set of allowed paths does not have an exact mathematical definition. Do we allow fractals? Let us consider just paths q(t) which are built from intervals of analytical functions.
Does the action change under a "very small" deformation of the circle orbit q(t)? We assume that t_1, t_2, q(t_1), and dq(t_1)/dt remain fixed.
https://en.wikipedia.org/wiki/Lagrangian_mechanics#Electromagnetism
The lagrangian of a massive charged particle with charge e in magnetic field is
L = m/2 * v^2 + e v • A,
where v is the velocity vector and A is the magnetic vector potential. The dot marks the inner vector product.
Let us consider a circular path in the diagrams above. Let us assume that the magnetic field B is constant and points out of the page.
<-- v e-
______
/ \ --> --> -->
| |
\______/ -----> -----> A
^ y
|
|
------> x
The electron e- does a perfect circle. It leaves position x at a velocity v at time t_1, and returns back at time t_2.
We assume that there is no electromagnetic radiation out of the system. The radius of the circle is determined by the absolute speed |v| of the electron and the strength of the magnetic field B = ∇ x A.
What happens to the path integral of the lagrangian L over the time interval [t_1, t_2]? The action is
t_2
S = ∮ m/2 * v^2 + e v • A,
t_1
where the path integral is done counter-clockwise.
We may assume that |A| is zero at the upmost point of the circle. Let us increase the speed of the electron by 1% for the whole circle, so that its radius is 1% larger. The starting point of the electron is kept constant. How does the action S change?
The kinetic part grows 2 %. The circle extends further down. In the lower half of the circle, |A| is 1% larger and |v| is too. The contribution is negative and grows by about 2%. There is no contradiction in these numbers. The lagrangian seems to work ok.
https://en.wikipedia.org/wiki/Lagrangian_mechanics#Electromagnetism
The lagrangian of a massive charged particle with charge e in magnetic field is
L = m/2 * v^2 + e v • A,
where v is the velocity vector and A is the magnetic vector potential. The dot marks the inner vector product.
Let us consider a circular path in the diagrams above. Let us assume that the magnetic field B is constant and points out of the page.
<-- v e-
______
/ \ --> --> -->
| |
\______/ -----> -----> A
^ y
|
|
------> x
The electron e- does a perfect circle. It leaves position x at a velocity v at time t_1, and returns back at time t_2.
We assume that there is no electromagnetic radiation out of the system. The radius of the circle is determined by the absolute speed |v| of the electron and the strength of the magnetic field B = ∇ x A.
What happens to the path integral of the lagrangian L over the time interval [t_1, t_2]? The action is
t_2
S = ∮ m/2 * v^2 + e v • A,
t_1
where the path integral is done counter-clockwise.
We may assume that |A| is zero at the upmost point of the circle. Let us increase the speed of the electron by 1% for the whole circle, so that its radius is 1% larger. The starting point of the electron is kept constant. How does the action S change?
The kinetic part grows 2 %. The circle extends further down. In the lower half of the circle, |A| is 1% larger and |v| is too. The contribution is negative and grows by about 2%. There is no contradiction in these numbers. The lagrangian seems to work ok.
If the energy would vary, could a hamiltonian work for the Pauli equation?
If the energy of the electron would be larger in the lower part of the circle in the diagram, then the phase of its wave function would rotate faster in the lower part. The phase difference between different points in the circle would tend to infinity, which would mean an infinite momentum.
The Schrödinger equation under a scalar potential works beautifully because the total energy is constant, and the phase of the wave function rotates at the same rate everywhere.
The Biot-Savart law and magnetic interaction
For slowly moving charges, the magnetic force is
F = μ_0 / (4π) * q_1 q_2 / r^2 * v_1 x (v_2 x r),
where r is the vector from charge q_1 to q_2. The formula is a consequence of the Biot-Savart law.
To which direction does the force point? In our example above, we have electrons moving to left inside the wire, and our test electron does a circular path. That is, the force is always orthogonal to the velocity vector v_2 of the test electron.
Can we model the force simply as a force between two point objects? The forces on the charges q_1 and q_2 will generally exert a torque on the system q_1 & q_2. Conservation of angular momentum requires that there is an opposite torque on something. Apparently, the magnetic field can store angular momentum.
If we stop both charges with some device, the magnetic field disappears. If there is no radiation out, then the angular momentum stored in the magnetic field has to be returned back to the system q_1 & q_2 & the device.
The minimal coupling is approximately right for the classical electron?
In our example case of an electron circling in a loop, some angular momentum is stored in the combined electromagnetic field of the wire and the electron. Can we neglect this effect in the classical treatment of the electron?
Probably yes, because a hypothetical self-interaction of a single electron is much weaker than the the force which is caused by the external magnetic field B.
Subscribe to:
Posts (Atom)