Working in a box of volume V=L^3, show that the number of...

Working in a box of volume $V=L^3$, show that the number of allowed states of momentum with $x$ component in the range $p_x$ to $p_x+d p_x$ is $(L / 2 \pi) d p_x$. Convince yourself that you need to impose periodic boundary conditions on the wavefunction and its derivative to ensure no net particle flow out of the volume.

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Key Concepts

Quantization in a Finite Volume

In quantum mechanics, the confinement of a particle to a finite volume, such as a cube of side length L, leads to the discretization of momentum values. The allowed wavefunctions in this finite space must be compatible with the imposed boundary conditions, which restricts the continuous momentum space into a set of discrete states.

Periodic Boundary Conditions

Periodic boundary conditions require that the wavefunction and its derivative are the same at opposite sides of the box. This choice of condition ensures that there is no net particle flow out of the volume, simplifies the mathematical treatment by eliminating edge effects, and maintains translational symmetry within the confined region.

Density of States in Momentum Space

The density of states in momentum space quantifies how many quantum states are available within a given momentum interval. When applying periodic boundary conditions in a finite volume, one finds that the number of allowed states in a small momentum range is proportional to L/dp in one dimension, ultimately leading to an expression that relates the size of the system to the spacing and count of discrete momentum states.

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