A Magnetic field produced by moving charge in operator form

[dark_matter_is_neat]

TL;DR Summary

In classical mechanics the magnetic field produced by a moving point charge is $\frac{q*\vec{v} \times \vec{r}}{4*\pi*r^{3}}$ but in quantum mechanics $\vec{v}$ is now an operator $\frac{\vec{p}}{m} = \frac{-i*\nabla}{m}$ and the denominator is a function of r and thus could be acted upon by p. Is this term effectively to the left of the denominator and thus acts on the denominator.

I'm wanting to calculate the interaction term of a magnetic moment with the magnetic field of a moving charged particle but I've confused myself about how to treat the magnetic field in quantum mechanics. In classical mechanics the magnetic field is just $\frac{q*\vec{v} \times \vec{r}}{4*\pi*r^{3}}$ but If I want to properly describe this in quantum mechanics $\vec{v} = -i*\nabla /m$ which could in principle act on the denominator of this expression if this term is to the left of the denominator. But does it act on the denominator or does it not act on the denominator and only acts on the wavefunction? Or do I have to express the magnetic field as an anti-commutator of this term with the denominator?

Also just to avoid confusion I'm expressing everything in natural units.

 

[renormalize]

I'm wanting to calculate the interaction term of a magnetic moment with the magnetic field of a moving charged particle but I've confused myself about how to treat the magnetic field in quantum mechanics.

Can you be more explicit? Is your particle $P$ that interacts with the moving-charge spinless (i.e., $P$ has no intrinsic magnetic-moment) or something like an electron (i.e., $P$ has an intrinsic spin magnetic-moment)?

 

[dark_matter_is_neat]

Can you be more explicit? Is your particle $P$ that interacts with the moving-charge spinless (i.e., $P$ has no intrinsic magnetic-moment) or something like an electron (i.e., $P$ has an intrinsic spin magnetic-moment)?

The interacting particle is a spin 1/2 fermion with an intrinsic magnetic moment but no charge. Its mostly just an extraneous detail for my question, since this is just going to lead to an interaction term of form $-\vec{\mu} \cdot \vec{B}$ what I'm mainly concerned with is the form of $\vec{B}$

 

[renormalize]

The interacting particle is a spin 1/2 fermion with an intrinsic magnetic moment but no charge.

OK, so the particle $P$ is something like a neutron. Let's assume that both $P$ and the moving-charge are nonrelativistic. Then the wavefunction $\psi_P$ for $P$ has two components (representing spin up/down) and satisfies the nonrelativistic Schrodinger-Pauli equation:
https://arxiv.org/pdf/2312.01079

(In (1) the $\vec{\sigma}$ are the three $2\times 2$ Pauli matrices.) Now all you have to do (!) is insert your classical expression for the magnetic-field $\vec{\mathcal{B}}$ of the moving charge into (1) and then solve for $\psi_P$ and you're done.

 

[dark_matter_is_neat]

The interacting particle is a spin 1/2 fermion with an intrinsic magnetic moment but no charge.

OK, so the particle $P$ is something like a neutron. Let's assume that both $P$ and the moving-charge move non-relativistically. Then the wavefunction $\psi_P$ for $P$ has two components (representing spin up/down) and satisfies the Schrodinger-Pauli equation:
https://arxiv.org/pdf/2312.01079
View attachment 362245
(In (1) $\vec{\sigma}$ are the three $2\times 2$ Pauli matrices.) Now all you have to do is insert your expression for the magnetic-field $\vec{\mathcal{B}}$ of the moving charge into (1) and then solve for $\psi_P$ and you're done.

What I'm interested in is how to treat this magnetic field when expressed in term of operators. Classically $\vec{B} = \frac{q\vec{v} \times \vec{r}}{4*\pi*r^3}$ but in quantum mechanics $\vec{v} = \vec{p}/m = -i*\vec{ \nabla }/m$, so what I'm wondering is the magnetic field treated as $\frac{1}{4*m*\pi*r^{3}}(-i\vec{\nabla} \times \vec{r})$ or as $(-i\vec{\nabla} \times \vec{r})\frac{1}{4*m*\pi*r^{3}}$. In the first case $\nabla$ doesn't act on the $r^{3}$ term while in the second case it does.

Also, I want the proper treatment on $\vec{B}$ so I can get the proper corresponding interaction term and use this in the first order born approximation to get the scattering cross section for this process (magnetic moment scattering with moving charged particle).

 

[renormalize]

What I'm interested in is how to treat this magnetic field when expressed in term of operators.

I've assumed that the moving charge is treated classically, so:
1) the $v$ in your magnetic-field expression is not an operator.
2) you can use the Schrodinger-Pauli eq.(1) to solve for the quantum scattering of $P$ off of the classical moving charge.

 

[dark_matter_is_neat]

I am interested in calculating the scattering cross section for the 2-to-2 scattering of a moving charge (Such as proton) with a 0-charge particle with a magnetic moment, So I want the interaction term associated with the moving proton's magnetic field treated using quantum mechanically, so I can get the contribution to the matrix element from this interaction. Essentially what I want is the form of $\vec{\mu} \cdot \vec{B}$ so I can use it to calculate $\bra{\psi_{i}}\vec{\mu} \cdot \vec{B} \ket{\psi_{f}}$

 

[renormalize]

I am interested in calculating the scattering cross section for the 2-to-2 scattering of a moving charge (Such as proton) with a 0-charge particle with a magnetic moment, So I want the interaction term associated with the moving proton's magnetic field treated using quantum mechanically, so I can get the contribution to the matrix element from this interaction. Essentially what I want is the form of μ→⋅B→ so I can use it to calculate ⟨ψi|μ→⋅B→|ψf⟩

1) So is the neutral particle stationary/fixed or can it rebound after the scattering?
2) Is the impact parameter of interest large enough that you can ignore the strong interaction? Or is the neutral particle not a nucleon? If not, what is it?
3) Do the initial and final wavefunctions $\psi_i,\psi_f$ represent just the proton or just the neutron or are they 2-particle wavefunctions?

 

[PeterDonis]

the magnetic field of a moving charged particle

A proper quantum mechanical treatment of this requires quantum field theory. Normally QFT is considered overkill for this type of problem; you just model the magnetic field as an ordinary classical potential in the Schrodinger Equation. But if you're not satisfied with that, you need QFT.

 

[dark_matter_is_neat]

1) So is the neutral particle stationary/fixed or can it rebound after the scattering?
2) Is the impact parameter of interest large enough that you can ignore the strong interaction? Or is the neutral particle not a nucleon? If not, what is it?
3) Do the initial and final wavefunctions $\psi_i,\psi_f$ represent just the proton or just the neutron or are they 2-particle wavefunctions?

The wavefunctions refer to the two-particle systems (for the most simple case we can treat the two particles as starting and ending as free particles, so the initial and final wavefunctions are just products of plane-wave states). Neither of the particles are fixed. The neutral particle of interest isn't actually a nucleon I just used a neutron as an example of a neutral particle with a magnetic moment, basically the particle only interacts with the moving charged particle (say a proton) via its magnetic moment.

 

[dark_matter_is_neat]

A proper quantum mechanical treatment of this requires quantum field theory. Normally QFT is considered overkill for this type of problem; you just model the magnetic field as an ordinary classical potential in the Schrodinger Equation. But if you're not satisfied with that, you need QFT.

So could I just treat the velocity as not an operator, if I want to calculate this without QFT (Just keep treating $\vec{B} = \frac{\vec{v} \times \vec{r}}{4*\pi*r^{3}}$ as having $\vec{v}$ as a parameter and not an operator)?

 

[renormalize]

...basically the particle only interacts with the moving charged particle (say a proton) via its magnetic moment.

Take a look at MAGNETIC SCATTERING OF NEUTRONS. It considers the scattering of a neutron from a (charged) electron and considers the following interaction Hamiltonian:

It should be applicable to your problem with three provisos:
1) You'll have to flip $e\rightarrow -e$ to accommodate your $+$-charged particle.
2) You can ignore $\vec{A_e}$ since you deal with only a single charged particle and there is (presumably) no external magnetic field.
3) Equation (4.1.5) ignores terms of order $(m_e/m_N)^2$. You need to specify $m_+/m_0$ to see if this approximation is valid in your case.

 

[div_grad]

I'm wanting to calculate the interaction term of a magnetic moment with the magnetic field of a moving charged particle (...)

In non-relativistic quantum mechanics based on the Schrödinger equation, you can only consider the electromagnetic fields as external fields. The most familiar example of this fact is the quantum-mechanical treatment of the hydrogen atom - you do not consider the H atom as a proper 2-particle system, but rather you model it as a single electron moving in the external electromagnetic field produced by the (stationary) proton (note that, in this case, the electromagnetic potentials are chosen in such a gauge that only the Coulomb potential enters the Schrödinger equation; in order to consider also the transverse vector potential of this field, you must go over to relativistic quantum field theory).

Also, if you would like to calculate the effect of the magnetic field produced by a moving particle on the magnetic moment of this same particle, then not only would you have to use QFT for that - you will also encounter enormous mathematical difficulties, because of the self-interaction of the particle with its own electromagnetic field.

Assuming that you actually want to calculate the interaction energy of the particle's magnetic moment with some external magnetic field $\mathbf{B} = \nabla \times \mathbf{A}$, then you simply change the kinetic energy operator for the particle as follows:
$$
\frac{\mathbf{P}^2}{2m} \longrightarrow \frac{(\mathbf{P} - q\mathbf{A})^2}{2m} \rm{,}
$$
where $q$ is the charge of the particle (for the electron $q=-|e|$, for the proton $q=+|e|$, etc.).

I am interested in calculating the scattering cross section for the 2-to-2 scattering of a moving charge (Such as proton) with a 0-charge particle with a magnetic moment (...)

Then in this case, if your 0-charge particle with a magnetic moment is heavier than the moving charge (such as proton), you can move to the rest-frame of this heavier 0-charge particle, i.e., you put this particle in the center of your coordinate system. Then with respect to this coordinate system you may consider the magnetic field produced by the moving charge as a classical external field (no operators involved).

 

[dark_matter_is_neat]

Assuming that you actually want to calculate the interaction energy of the particle's magnetic moment with some external magnetic field $\mathbf{B} = \nabla \times \mathbf{A}$, then you simply change the kinetic energy operator for the particle as follows:
$$
\frac{\mathbf{P}^2}{2m} \longrightarrow \frac{(\mathbf{P} - q\mathbf{A})^2}{2m} \rm{,}
$$
where $q$ is the charge of the particle (for the electron $q=-|e|$, for the proton $q=+|e|$, etc.).


Then in this case, if your 0-charge particle with a magnetic moment is heavier than the moving charge (such as proton), you can move to the rest-frame of this heavier 0-charge particle, i.e., you put this particle in the center of your coordinate system. Then with respect to this coordinate system you may consider the magnetic field produced by the moving charge as a classical external field (no operators involved).

Yes I want the interaction term associated with the magnetic moment of the neutral particle interacting with the magnetic field produced by the moving charged particle. The neutral particle mass may be much smaller, much larger, or comparable to the moving charge mass, so I'm interested in working in the center of mass frame of the system. Can I still apply this treatment in that case? Effectively what the magnetic moment sees is a magnetic field generated by the charge particle moving with the relative velocity between the magnetic moment and charged particle, so I'd use this field for the interaction term right?

 

[div_grad]

The neutral particle mass may be much smaller, much larger, or comparable to the moving charge mass, so I'm interested in working in the center of mass frame of the system.

If you move to the center-of-mass frame of 2 interacting particles, then your problem changes into a (simpler) problem of a single fictitious particle with some reduced mass moving in some external field (see Kepler problem). Carry out the transformation between the two coordinate systems and try to identify the Hamiltonian terms associated with the magnetic field interactions.

Effectively what the magnetic moment sees is a magnetic field generated by the charge particle moving with the relative velocity between the magnetic moment and charged particle, so I'd use this field for the interaction term right?

The magnetic moment "sees" the magnetic field generated by a moving charged particle from the point of view of its own rest-frame. If you have 2 particles then you can also move to a reference frame in which both of these particles are stationary (think of a diatomic molecule and a reference frame co-rotating with that molecule, with $Z$-axis along the internuclear axis).

Does the Hamiltonian have a definite form? If not, then start with the Hamiltonian for 2 particles written in some "laboratory" frame. Carry out the transformation to the center-of-mass frame that you wish to use in your calculations. Check if you can identify in this Hamiltonian the operators associated with magnetic interactions that you wish to consider.