One dimensional quantum wire

[lion8172]

I have seen, in several places, derivations of a formula for the current flow in a 1d quantum wire connected to electron reservoirs at its respective terminals. All of these derivations at some point invoke the formula for the number of states per unit volume of a 1d quantum "box,"
n(k) dk = (2 /pi) dk, but note, at the same time, that the conduction electrons are only confined in the directions transverse to the wire's length, and not in the longitudinal direction. My question is quite simple. In view of the fact that the derivation of the density of states expression rests upon the assumption that the system is longitudinally confined (correct me if I'm wrong), why is it valid to invoke the result?

 

[BigJDubs]

Thank you in advance. A:You are right that the expression for the density of states assumes longitudinal confinement. However, this is not a problem since the quantum wire can be treated as a one-dimensional system in the limit that the transverse confinement is much stronger than any longitudinal potential. The resulting cross-sectional area (which determines the density of states) is then determined by the transverse confinement.

 

[charng]

The derivation of the formula for the number of states per unit volume of a 1d quantum box does indeed assume longitudinal confinement. However, in the case of a 1d quantum wire, the electrons are effectively confined in the transverse direction due to the presence of the wire's potential energy barrier. This results in a quantization of the electron energy levels, similar to a 1d quantum box.

Therefore, the formula for the number of states per unit volume can still be applied in this case, as it is based on the principle of quantization in one dimension. It is important to note that this formula is not specific to a particular type of confinement, but rather a general principle that applies to any type of 1d confinement.

In addition, the assumption of longitudinal confinement is not necessary for the derivation of the current flow formula in a 1d quantum wire. This formula takes into account the transverse confinement and the resulting quantization of energy levels in the wire. Therefore, it is valid to use the formula for the number of states per unit volume in this context.

Overall, while the derivation of the density of states expression may rest upon the assumption of longitudinal confinement, it is still applicable in the case of a 1d quantum wire due to the principles of quantization and the transverse confinement of electrons.