In all the examples below, we will inspect electrons interacting with various potentials in one dimension (i.e., along x axis). Think carefully about how each potential looks. Do not engage in overcomplicated numerical calculations, rather make a sketch and see the least complicated way to provide an answer. (A) Assume that a quantum particle (electron) is trapped in a finite potential well (i.e., it is localized in the well). At t=0, the corresponding wavefunction y(x) is purely real and that is properly normalized. Show that the average momentum in this state must be zero. (Hint: use integration by parts and think carefully about how the graph of the wave- function will look.) (B) Consider the interaction of a moving electron with a square barrier. The quantum particle has energy Evo, where vo is the potential height of the barrier. Make a hand drawn figure which depicts the wavefuntion V (c) which is transmitted across the barrier and comment on the important features of the wavefuntion in different areas of the space (i.e., in various regions along x). Make another figure capturing the fundamentals of scanning tunneling microscopy (STM), where vacuum between the STM tip and the sample represents the barrier. Is the energy of the electron in the tip below or above the height of the potential barrier? In all the examples below, we will inspect electrons interacting with various potentials in one dimension (i.e., along x axis). Think carefully about how each potential looks. Do not engage in overcomplicated numerical calculations, rather make a sketch and see the least complicated way to provide an answer. (A) Assume that a quantum particle (electron) is trapped in a finite potential well (i.e., it is localized in the well). At t=0, the corresponding wavefunction y(x) is purely real and that is properly normalized. Show that the average momentum in this state must be zero. (Hint: use integration by parts and think carefully about how the graph of the wave- function will look.) (B) Consider the interaction of a moving electron with a square barrier. The quantum particle has energy Evo, where vo is the potential height of the barrier. Make a hand drawn figure which depicts the wavefuntion V (c) which is transmitted across the barrier and comment on the important features of the wavefuntion in different areas of the space (i.e., in various regions along x). Make another figure capturing the fundamentals of scanning tunneling microscopy (STM), where vacuum between the STM tip and the sample represents the barrier. Is the energy of the electron in the tip below or above the height of the potential barrier
