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Schrödinger equation for particle on a ring in a magnetic field

  1. Oct 28, 2011 #1
    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?
     
  2. jcsd
  3. Oct 28, 2011 #2

    vanhees71

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    2016 Award

    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].
     
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