Schrödinger equation for particle on a ring in a magnetic field

In summary, the conversation was about the Schrödinger equation for a particle in a ring under the influence of a magnetic field, taking into consideration the particle's spin. The correct equation involves a gauge-covariant derivative and a Hamiltonian that includes the scalar and vector potentials of the electromagnetic field. To consider spin, the Hamiltonian is modified to include the spin operator and a homogeneous magnetic field can be introduced using the appropriate substitutions. To solve the problem of a particle restricted to a circle, the gradient and Laplacean are substituted in spherical coordinates.
  • #1
235
0
hi

i need the schrödinger equation for a particle(electron) in a ring under the influence of a magnetic field that goes through perpendicular to the plane of the ring and i want to consider the spin too.

Well, the particle in the ring is pretty easy:

[itex]- \frac{ \hbar^2}{2mr^2} \psi''(\phi)=E \psi (\phi) [/itex]

but what is about the magnetic field?

i thought that if i take

[itex]E_{pot}=\mu B=\frac{e m v r}{2m}B=\frac{e l B}{2m} [/itex]

and therefore:
[itex]- \frac{ \hbar^2}{2mr^2} \psi''(\phi)+\frac{i \hbar e}{2m} \psi '(\phi) B=E \psi (\phi) [/itex]

and then i would get an additional term if i want to consider the spin:


[itex]E_{pot}=\mu B=\frac{-g_s \mu_B \sigma }{\hbar }B [/itex]
and therefore:
[itex]- \frac{ \hbar^2}{2mr^2} \psi''(\phi)+\frac{i \hbar e}{2m} \psi '(\phi) B+\frac{-g_s \mu_B \sigma }{\hbar }B\psi(\phi)=E \psi (\phi) [/itex]

so, is this the right equation?
 
Physics news on Phys.org
  • #2
That's not completely correct.

The important point of electromagnetic fields is that it is a gauge field, and thus to get a gauge invariant equation, you have to introduce the covariant derivative

[tex]D_{\mu}=\partial_{\mu}+\mathrm{i} q A_{\mu},[/tex]

where [itex]A_{\mu}[/itex] is the four potential of the electromagnetic field. In the following non-relativistic limit I write [itex]A_0=\Phi[/itex] for the scalar potential and [itex]\vec{A}[/itex] for the vector potential.

For non-relativistic without spin you start with the Schroedinger equation for a free particle

[tex]\mathrm{i} \frac{\partial \psi}{\partial t}=-\frac{1}{2m} \Delta \psi.[/tex]

To couple the electromagnetic field to it, you just substitute all derivatives by their gauge-covariant derivatives, i.e.,

[tex]\partial_t \rightarrow D_0=\partial_t+\mathrm{i} q \Phi.[/tex]

For the gradient one has

[tex]D_i = \frac{\partial}{\partial x^i} + \mathrm{i} q A_i = \frac{\partial}{\partial x^i} - \mathrm{i} q A^i,[/tex]

i.e. the correct substitution for the nabla operator in 3D-vector analysis notation reads

[tex]\vec{\nabla} \rightarrow \vec{D}=\vec{\nabla}-\mathrm{i} q A^i.[/tex]

Plugging this into the Schrödinger equation, one gets

[tex]\mathrm{i} \frac{\partial \psi}{\partial t}-q \Phi \psi=\frac{1}{2m} (-\mathrm{i} \vec{\nabla}-q \vec{A})^2 \psi[/tex]

or to extract the Hamiltonian

[tex]\mathrm{i} \frac{\partial \psi}{\partial t}=\hat{H} \psi=\frac{1}{2m} [-\Delta \psi - \mathrm{i} \vec{\nabla} \cdot (\vec{A} \psi) - \mathrm{i} \vec{A} \cdot \vec{\nabla} \psi - q^2 \vec{A}^2 \psi]+ q \Phi \psi.[/tex]

For a homogeneous magnetic field you set

[tex]\vec{A}=-\frac{1}{2} \vec{x} \times \vec{B}.[/tex]

To consider spin you indeed only need to add

[tex]\hat{H}_{\text{Spin}}=-\frac{q}{2m} g_S \hat{\vec{S}} \cdot \vec{B}.[/tex]

This leads to the Pauli equation.

For the problem of a particle restricted to a circle, just substitute the gradient and Laplacean in spherical coordinates and set [itex]\partial/\partial r=\partial/\partial \vartheta=0[/itex].
 

What is the Schrödinger equation for a particle on a ring in a magnetic field?

The Schrödinger equation for a particle on a ring in a magnetic field is a specific form of the Schrödinger equation that describes the behavior of a quantum particle confined to a ring-shaped region and subject to a magnetic field. It takes into account both the wave-like nature of the particle and the effects of the magnetic field on its motion.

How is the Schrödinger equation for a particle on a ring in a magnetic field derived?

The Schrödinger equation for a particle on a ring in a magnetic field is derived by combining the general Schrödinger equation with the Hamiltonian operator for a particle confined to a ring and the magnetic field operator. This results in a differential equation that describes the time evolution of the particle's wave function.

What are the key features of the Schrödinger equation for a particle on a ring in a magnetic field?

The Schrödinger equation for a particle on a ring in a magnetic field includes terms for the kinetic and potential energy of the particle, as well as the effects of the magnetic field on its motion. It is a time-dependent equation, meaning that the wave function of the particle evolves over time according to the equation.

What is the significance of the Schrödinger equation for a particle on a ring in a magnetic field in quantum mechanics?

The Schrödinger equation for a particle on a ring in a magnetic field is an important tool in quantum mechanics for understanding the behavior of particles in confined spaces and under the influence of external fields. It allows us to make predictions about the behavior of particles at the microscopic level and has wide-ranging applications in fields such as materials science and quantum computing.

Can the Schrödinger equation for a particle on a ring in a magnetic field be solved analytically?

In most cases, the Schrödinger equation for a particle on a ring in a magnetic field cannot be solved analytically, meaning that there is no exact mathematical solution. However, it can be solved numerically using computational methods, and there are some special cases where analytical solutions are possible, such as when the magnetic field is uniform or the particle is in a specific energy state.

Suggested for: Schrödinger equation for particle on a ring in a magnetic field

Back
Top