Discrete 3-particle system - Condensed Matter Pysics

In summary, the conversation discusses a system of three atoms with valence electrons that can tunnel to the nearest neighbor. The Hamiltonian for this system is given, along with its eigenvalues and eigenstates. The conversation then moves on to finding the momentum values for the simultaneous eigenstates, using the fact that the system has translational invariance and assuming plane wave solutions. It is suggested to think about the relationship between k and n and position and momentum.
  • #1
8
0
1. The problem statement, all variables and givenknown data
We have a system of three atoms arranged in a circular arrangements. They each have a valence electron that can tunnel to the nearest neighbor. For a tunneling rate [itex]-A/\hbar[/itex] we have the Hamiltonian (shifted by an energy [itex]E_a[/itex] on the diagonal)
[tex]
H = \begin{pmatrix}
0 & -A & -A\\
-A & 0 & -A\\
-A & -A & 0
\end{pmatrix}
[/tex]
which eigenvalues [itex]-2|A|[/itex] corresponding to the eigenstate (1,1,1) and degenerated eigenvalues [itex]A[/itex] corresponding to eigenstates (-1,0,1) and (-1,1,0). Since we have translational invariance, [itex]p[/itex] is a good quantum number, hence [itex][H,p]=0[/itex] and [itex]p[/itex] and [itex]H[/itex] can be diagonalized simultaneously. Assuming plane wave solutions [itex]\psi_n \sim \mathrm{e}^{ikn}[/itex] we get from boundary conditions that [itex]k = 2/3 \pi N[/itex] for [itex]N \in \mathbb{Z}[/itex]. Now, the three lowest cases are [itex]k=0[/itex] and [itex]k= \pm 2/3 \pi[/itex]. [itex]k=0[/itex] corresponds to the eigenvalue [itex]-2|A|[/itex] since for [itex]k=0[/itex] we have [itex]\psi_1=\psi_2=\psi_3=1[/itex], i.e. the electrons are evenly distributed over the three atoms. The other two cases give the following linear combinations:
[itex]k=2/3 \pi[/itex]: [tex]|\psi_2 \rangle + (-1/2 -i \sqrt{3}/2)|\psi_3 \rangle[/tex]
[itex]k=-2/3 \pi[/itex]: [tex]|\psi_2 \rangle + (-1/2 +i \sqrt{3}/2)|\psi_3 \rangle[/tex]

My question is now: How do I find the momentum values [itex]p[/itex] for each of the 3 simultaneous eigenstates?

Homework Equations


[itex]\hat{p} | \psi_n \rangle = p | \psi_n \rangle[/itex]


The Attempt at a Solution


I'm really at a loss here. How can we find the momentum values for this discrete system with only assumed plane wave solutions?
 
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  • #2
Hixy said:
Since we have translational invariance, [itex]p[/itex] is a good quantum number, hence [itex][H,p]=0[/itex] and [itex]p[/itex] and [itex]H[/itex] can be diagonalized simultaneously. Assuming plane wave solutions [itex]\psi_n \sim \mathrm{e}^{ikn}[/itex] we get from boundary conditions that [itex]k = 2/3 \pi N[/itex] for [itex]N \in \mathbb{Z}[/itex].

You should think about how we arrive at this form of the plane-wave. What do #k# and #n# have to do with position and momentum?
 

1. What is a discrete 3-particle system in condensed matter physics?

A discrete 3-particle system in condensed matter physics refers to a system consisting of three particles that are separated from each other and interact with each other in discrete steps. This means that the particles can only occupy certain energy levels and their interactions occur in quantized units.

2. How is a discrete 3-particle system different from a continuous system?

A continuous system is one in which the particles can occupy any energy level and their interactions occur in a continuous manner. In contrast, a discrete 3-particle system has a limited number of energy levels and interactions occur in discrete steps.

3. What are the applications of studying discrete 3-particle systems in condensed matter physics?

Studying discrete 3-particle systems can provide insights into the behavior of more complex systems in condensed matter physics. It can also help in understanding the properties of materials and developing new technologies, such as quantum computing.

4. How are discrete 3-particle systems relevant to understanding phase transitions?

Phase transitions, such as the transition from a solid to a liquid, are often studied using discrete 3-particle systems. These systems can exhibit similar behavior to larger systems, making them useful models for understanding phase transitions in condensed matter physics.

5. Are there any real-life examples of discrete 3-particle systems?

Yes, there are many real-life examples of discrete 3-particle systems, such as the three atoms in a water molecule or the three ions in a sodium chloride crystal. Additionally, some materials, such as graphene, can exhibit discrete 3-particle behavior at a nanoscale level.

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