Time independent perturbation theory

[ice109]

Homework Statement

Assume that H0 describes a paramagnetic system that couples to a magnetic field via
the Zeeman effect, i.e. V = −μB, where μ is the magnetic moment. Note that for the
unperturbed paramagnetic system the probability of having an up-spin is equal to that for
a down-spin. Show that the magnetic susceptibility, defined as E = EGS −$\frac{1}{2}\chi B^2$, where
EGS is the ground state energy in the absence of field, is always non-negative. Write down
the expression for the susceptibility.
Note: A similar expression for a susceptibility is valid for an arbitrary perturbation. This
is known as the Lehmann representation applied to the linear response of the system to
the perturbation.

Homework Equations

The Attempt at a Solution

i have no idea what's being asked here? is the perturbation −μB to a free particle hamiltonian?

 

[Jhero]

Hi there, thank you for your post. It seems that the forum post is discussing the concept of magnetic susceptibility in a paramagnetic system. The perturbation mentioned is the Zeeman effect, which describes the coupling between the magnetic field and the system's magnetic moment. The expression for the magnetic susceptibility is E = EGS - (1/2)χB^2, where EGS is the ground state energy in the absence of field and χ is the magnetic susceptibility. This expression shows that the energy of the system is dependent on the magnetic field, with a higher susceptibility leading to a larger energy change in response to the field. The Lehmann representation is a general mathematical tool used to analyze the linear response of a system to a perturbation, in this case the magnetic field. I hope this helps clarify the forum post for you. Let me know if you have any further questions.