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I am having trouble to think how to solve the following problem:

The plane $x=0$ separates two parts of space: when $x>0$ there is homogeneous magnetic field, which induction vector $B_x=B_y=0$, $B_z =B_0$, but when $x<0$ there is no magnetic field. From $x=-\infty$ to the separating plane along $x-$ axis the neutron is released. Find the probability that neutron will be reflected from $x=0$ plane. Make an assumption that the neutron only has a spin magnetic moment ${\mu}_N$, while the energy of the neutron is $E={2{\mu}_NB_0}$ and its spin is directed downwards the $z$ axis.

As far as I understand, this should be one of the "Potential barrier/well" problems in Quantum physics. However, how to write the Hamiltonian for it? And what type of wave functions are the eigenfunctions for Schrodinger equation? Should I solve stationary Schrodinger equation?

Would be very thankful, if someone could help!

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The Hamiltonian describes the energy. Your neutron has kinetic energy $H_{\rm{kin}} = p^2/2m$ and since it couples to the magnetic field it also has potential energy $H_{\rm{pot}} = 2\mu_N B_0 \theta(x)$ which is given to you in the problem. Note that $\theta(x)$ makes sure that this magnetic field only acts in $x>0$. Now you can combine it to a Hamiltonian $$H = \frac{p^2}{2m}+2\mu_N B_0 \theta(x)$$ and you can solve the stationary Schrödinger equation, finding the wave functions of the eigenstates and their scattering as they impinge upon the potential at $x=0$. The boundary condition you are looking for is that of incoming waves from the left.

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  • $\begingroup$ Thank you very much for the answer! However, I was wondering, shouldn't the magnetic field component be just ${\mu}_NB_0{\sigma}_z$ as magnetic field is acting on z axis and the energy should only come in comparision when solving the differential equation as $b^2-4ac>0$ or $b^2-4ac<0$ determines the solutions of differential equation? $\endgroup$
    – aerospace
    Commented Mar 9, 2020 at 10:55
  • $\begingroup$ As you are explicitly given that the spin of the neutron is pointing down in the $z$ direction, and the energy of it, I don't think there is need to enter $\sigma_z$ into the Hamiltonian. The spin degree of freedom doesn't play here any role as it is fixed and therefore inert. I'm not sure that are the $a, b$ and $c$ in the expression you wrote afterward. What is the relation between these coefficients and the differential equation? $\endgroup$
    – user245141
    Commented Mar 9, 2020 at 11:03

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