In Chapter 1, we saw that current flow typically involves a channel connected to two contacts that are out of equilibrium with each other, having two distinct electrochemical potentials. One contact keeps filling up the channel while the other keeps emptying it causing a net current to flow from one contact to the other. In the next chapter we will take up a quantum treatment of this problem. My purpose in this chapter is to set the stage by introducing a few key concepts using a simpler example: a channel connected to just one contact as shown in Fig. 8.1.

Since there is only one contact, the channel simply comes to equilibrium with it and there is no current flow under steady-state conditions. As such this problem does not involve the additional complexities associated with multiple contacts and nonequilibrium conditions. This allows us to concentrate on a different physics that arises simply from connecting the channel to a large contact: the set of discrete levels broadens into a continuous density of states as shown on the right-hand side of Fig. 8.1.

In Chapter 1 I introduced this broadening without any formal justification, pointing out the need to include it in order to get the correct value for the conductance. My objective in this chapter is to provide a quantum mechanical treatment whereby the broadening will arise naturally along with the “uncertainty” relation γ = ħ/τ connecting it to the escape rate 1/τ for an electron from the channel into the contact.