Density of States: Varying Boundary Conditions

In summary, when considering waves in a box, the density of states is typically calculated by enforcing either vanishing or periodic boundary conditions. In the case of vanishing boundary conditions, we only consider positive wave numbers because the solutions for +k and -k are identical and counting them twice would be redundant. However, with periodic boundary conditions, we take both positive and negative wave numbers.
  • #1
PineApple2
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Consider waves in a box. It is customary to calculate the density of states either by enforcing vanishing boundary conditions, then the wave numbers are
[itex]k=\frac{n\pi}{L}[/itex] and we take only positive [itex]k[/itex],
or using periodic boundary conditions, in which case [itex]k=\frac{2n\pi}{L}[/itex]
and taking all wave numbers.

My question is - why in the case of vanishing boundary conditions do we take only positive wave numbers? and why in the case of periodic boundary conditions do we take both positive and negative?
 
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  • #2
In the first case, vanishing boundary conditions, the solutions are standing waves, and the solutions for +k and -k are exactly the same. So we use just positive k to avoid counting them twice.
 

1. What is the density of states and why is it important in physics?

The density of states (DOS) is a concept in physics that describes the number of available quantum states per unit energy in a system. It is an important quantity in understanding the behavior of materials and their electronic properties, as it provides information about the energy levels available to particles in a system.

2. How does varying boundary conditions affect the density of states?

Boundary conditions refer to the constraints placed on a system at its boundaries. Varying boundary conditions can affect the density of states by changing the available energy levels for particles in the system. For example, changing the size or shape of a material can alter the boundary conditions and therefore impact the density of states.

3. What are some examples of varying boundary conditions in real-world systems?

Examples of varying boundary conditions in real-world systems include the size and shape of a material, the presence of impurities or defects in a material, and the application of external forces such as pressure or an electric field. These varying conditions can alter the electronic structure and therefore affect the density of states in a material.

4. How is the density of states calculated in a system with varying boundary conditions?

The density of states can be calculated using mathematical models and equations that take into account the specific boundary conditions of the system. These calculations can involve solving for the energy levels and wave functions of particles in the system, and then using this information to determine the density of states at different energy levels.

5. What are the practical applications of studying the density of states with varying boundary conditions?

Studying the density of states with varying boundary conditions has practical applications in various fields such as material science, nanotechnology, and semiconductor technology. It can help researchers understand and predict the electronic properties of materials, as well as guide the development of new materials with desired properties for specific applications.

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