What does an "intuitive" physical model mean? We have concentrated on studying models which contain one or more point particles. The wave of a classical point particle would be determined using a path integral approach: the lagrangian density over a path would determine the phase of the wave.
The electron as a classical point particle
Maybe the hypothesis that the electron is, in some way, a classical point particle is wrong? We can certainly measure the position of an electron with a very great precision. There are claims on the Internet that experiments set the limit of the electron radius at most to 10^-22 m.
In an earlier blog post we calculated that a point-like electron must move at the speed of light in a circle of radius 2 * 10^-13 m, to explain its spin angular momentum.
The electron in some aspects looks point-like, but its spin and magnetic moment suggest that it should quite a large sphere, or a point particle moving around quite a large circle, to explain its spin and magnetic moment.
A classical point charge involves the paradox of the infinite energy of its electric field. The electric field of the electron contains its mass 511 keV of energy outside the classical radius of the electron 3 * 10^-15 m.
Maybe it is best to reject the idea that the electron is a classical point particle, in any sense.
What is the correct kinetic energy term in the Pauli equation?
The kinetic energy operator in the Pauli equation
(1/ (2m) (σ ∙ (p - qA))^2, + qϕ) ψ
= (i h-bar ∂ / ∂t) ψ
= (i h-bar ∂ / ∂t) ψ
is
(σ ∙ (p - qA))^2,
where σ is the vector of the three Pauli matrices, and the exponent 2 means that we take the product of 2 × 2 matrices, that is, the composite mapping.
One may ask why the operator should not be
(p - qA)^2,
where the exponent 2 means taking the inner product of two vectors? If the vector potential A is zero, then the kinetic energy operator does, indeed, take a simpler form, the Laplace operator,
p^2,
where the exponent 2 means taking the inner product of the vector p with itself.
An inner product p^2 in a 3-dimensional space is simple in the sense that "cross terms" of p_x, p_y of the vector p components in the x, y, z directions do not affect the value of the product. It is just the sum
p_x^2 + p_y^2 + p_z^2
which matters.
However, when A is not zero, the magnetic field determines a preferred coordinate system in space, and it is not at all obvious that the simple inner product form of the kinetic term is the right one then.
In classical electrodynamics with the electron a point particle, in the Hamiltonian formulation, the kinetic term is an inner product also when A is not zero. But in wave mechanics, it might not be the right kinetic term.
The Cauchy stress tensor in classical mechanics is a full 3 × 3 matrix. Elastic energy can be stored in the normal stresses of a solid object, but shear stresses also contribute to elastic energy. That is, "cross terms" in that case are important in the determination of energy of the system.
We conclude that maybe the right kinetic operator in the Pauli equation should also involve cross terms, in one way or another.
If Nature has decided that the kinetic term still should have the form of being a "square" of something, then the "factorization" of p^2 with the Pauli matrices is a good candidate as the kinetic term.
Adding cross terms takes us away from a classic point particle model.
When A is not zero, then operators p_x - qA_x and p_y - qA_y, in general, do not commute. This might be an intuitive reason why the inner product of vectors no longer is the correct kinetic energy operator. In free space, with A zero, operators p_x, p_y, p_z are orthogonal in the sense that they have a common eigenbase, and there is an eigenfunction for which p_x is zero and p_y non-zero, and so on. Maybe it only makes sense to calculate the kinetic energy as a simple inner product if the operators involved are orthogonal? The metric in the underlying space is flat in that case?
If the electron has no point particle model, there are problems for hidden variable interpretations
The de Broglie-Bohm interpretation assumes that particles at all times have a precise position and that they move according to what the "pilot wave" tells them to do. The position of the particle is a hidden variable of the system.
But if the electron does not have an intuitive model where it is a point particle with a precise position, how can we make the de Broglie-Bohm interpretation to work? What are the hidden variables then?
https://en.wikipedia.org/wiki/De_Broglie–Bohm_theory
The Bohm model in the non-relativistic case is simply that we imagine the point particle to be a water molecule which travels along the probability current of the wave function.
According to Wikipedia, the relativistic case is problematic because then in spacetime there are no fixed planes of equal time. What does it then mean that the probability is conserved?
Hrvoje Nicolic and others have developed Bohmian models that handle the Dirac equation.
We may study probability conservation in a future blog post. Probability conservation has further problems in curved spacetime.
The Bohm model in the non-relativistic case is simply that we imagine the point particle to be a water molecule which travels along the probability current of the wave function.
According to Wikipedia, the relativistic case is problematic because then in spacetime there are no fixed planes of equal time. What does it then mean that the probability is conserved?
Hrvoje Nicolic and others have developed Bohmian models that handle the Dirac equation.
We may study probability conservation in a future blog post. Probability conservation has further problems in curved spacetime.
What is the correct kinetic energy term in the Dirac equation?
The Pauli equation comes from a non-relativistic energy momentum relation:
E = p^2 / (2m) + m.
The energy E can be divided in a simple way into the energy of the rest mass m, and the kinetic energy term p^2 / (2m).
The relativistic energy-momentum relation
E^2 = p^2 + m^2
does not admit such a simple formula for the kinetic energy. The formula for E would involve a square root, which is awkward.
The quantum mechanical operator for E involves a partial derivative on time, and p involves partial derivatives on spatial coordinates. A general principle of special relativity is that time and space should be treated in a unified way. That suggest that the temporal and spatial operators should be combined somehow:
E^2 - p^2 = m^2.
https://en.wikipedia.org/wiki/Laplace_operator
The operator on the left side of the equation is a d'Alembert operator or a wave operator. In the case of the Pauli equation, we can split the wave operator into the kinetic energy part (the Laplace operator) and the partial derivative on time. In the relativistic case, such a split may not be possible.
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