Density Matrix for Spin 1/2 particle in a magnetic field

In summary, the conversation discusses the process of creating a density matrix for a spin-1/2 particle in thermal equilibrium at temperature T and a constant magnetic field in the x-direction. The equations for finding the energy eigenvalues are given, but it is noted that there is an error which leads to two solutions. The question is raised as to why a magnetic field in the x-direction does not break the degeneracy of the energy eigenstates, while in the z-direction it does. Ultimately, it is clarified that the error in the equations was the cause of this confusion.
  • #1
khfrekek92
88
0
Hi everyone!

I am trying to create the density matrix for a spin-1/2 particle that is in thermal equilibrium at temperature T, and in a constant magnetic field oriented in the x-direction. This is a fairly straightforward process, but I'm getting stuck on one little part.

Before starting I need to find the energy eigenvalues (In order to find the partition function):

H=-μS⋅B=-μBσ_x

But since σ_x is an off-diagonal matrix (unlike σ_z), plugging this Hamiltonian into the Schrodinger Equation yields two equations (By letting |ψ>=(ψ1,ψ2))

Eψ1=-μBψ2
Eψ2=-μBψ1

And then solving these like normal for E gives us only one energy eigenstate for this system (with degeneracy 2):

E=μB

However, when the magnetic field was in the z direction, the z pauli spin matrix was diagonal and didn't switch the positions of ψ1 and ψ2, which gave me two energy eigenstates (±μB).

So my question is, why would a magnetic field in the x-direction NOT break the degeneracy of the energy eigenstates, while in the z-direction it does? These directions are completely arbitrary and should yield the same results, right?

Thanks
 
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  • #2
You have made a mistake. Your equations lead to ##E^2=(\mu B)^2##, which has two solutions.
 
  • #3
Ah man such a small algebra error! That makes everything work out perfectly, duh! Thank you so much!
 

FAQ: Density Matrix for Spin 1/2 particle in a magnetic field

1. What is the density matrix for a spin 1/2 particle in a magnetic field?

The density matrix for a spin 1/2 particle in a magnetic field is a mathematical representation of the quantum state of the particle. It is a 2x2 matrix that contains information about the probability amplitudes for the particle to be in a certain spin state.

2. How is the density matrix calculated for a spin 1/2 particle in a magnetic field?

The density matrix is calculated by taking the outer product of the particle's state vector with its conjugate transpose. This involves multiplying the state vector by its complex conjugate and then transposing the result to create a 2x2 matrix.

3. What does the density matrix tell us about the spin state of a particle in a magnetic field?

The density matrix provides information about the probabilities of the particle being in a certain spin state. The diagonal elements of the matrix represent the probability of the particle being in either the spin up or spin down state, while the off-diagonal elements represent the probability of the particle being in a superposition of these states.

4. How does the density matrix change when the magnetic field is varied?

The density matrix changes when the magnetic field is varied because the state of the particle is affected by the magnetic field. As the magnetic field is increased or decreased, the probabilities of the particle being in different spin states will also change, resulting in a different density matrix.

5. What is the significance of the density matrix in quantum mechanics?

In quantum mechanics, the density matrix is a useful tool for describing the state of a system. It allows us to calculate the expected values of observables and make predictions about the behavior of the system. It is also used to study the dynamics of a system, such as how the state of the particle changes over time in the presence of a magnetic field.

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