Wells Characteristic Features

You are given four different potentials, each with its wave function. The potential is 0, between $-0.5nm<L<0.5nm$.

Please answer and explain:

(a). What state corresponds to each of the given wave function (N = 1 is the ground state)

(b). Which of the four wave functions has the greatest energy.

(c) Which of the four wave functions corresponds to the smallest energy.

(d) Where the probability of finding the electron at $x=0$ is maximum?

(e) For each potential is the value of the potential elsewhere ($|x| >0.5 nm$) negative, finite of infinite?  

a)

For N=1 (ground state) the wave function has no nodes inside the well (it has a difference of phase between ends of pi). For one node inside the well the energy level is N=2. The upper figures have 2 nodes inside the well, it means N=3 for both upper wave functions). The lower figures have 3 nodes inside the well, it means N=4 for both.

e,f)

For infinite wells the electron is confined just inside the well. At the ends of the infinite well the wave function is exactly zero. By contrary for finite wells the wave functions penetrates into the walls (it is probable to find the electron also outside the well into the walls). Therefore left wells are infinite and right wells are finite (for potential 1, U= infinity; for potential 2, 0<U<infinity)

b.c)

Energy increases with level. For the same level, it is necessary a bigger energy to confine the electron in a smaller space. Since the wave function of finite wells is extended over a larger distance (it enters also the walls and thus the available space is larger) than for infinite wells, it follows that for the same level the energy is higher for infinite wells.

Thus the electron with the smallest energy is on the right upper side, and the electron with the highest energy is on the left bottom side.

d)

The probability to find the electron at a given position x is $|psi(x)|^2$. Since in figure a) (left upper side) the amplitude at $x=0$ (center of well) of the wave function is the biggest, then here it is most likely to find the electron at $x=0$ (center of well).

Both upper figures (a and b) have about the same amplitude but for infinite wells the maximum amplitude ($A=sqrt{2/L}$ is a bit higher since the available space L is a bit smaller.