Question

a. Sketch graphs of the probability density ${\left|\psi \left(x\right)\right|}^2$for the four states in the finite potential well of Figure $40.14$a. Stack them vertically, similar to the Figure $40.14$a graph of $\psi x$.

b. What is the probability that a particle in the$n=2$ state of the finite potential well will be found at the center of the well? Explain.

c. Is your answer to part b consistent with what you know about standing waves? Explain.

Short answer

a. The graphs of the probability density ${\left|\psi \left(x\right)\right|}^2$is given below in step.

b. since, the wave function ${\psi }_2$us zero at the center of the well, so the probability of finding the particle at the point is zero.

c. Yes, the answer in part (b) is consistent with knowing about standing waves.

Step-by-step solution

Part (a) step 1: Given Information 

We have to given the probability density${\left|\psi \left(x\right)\right|}^2$.

Part (a) step 2: Explanation

The probability density graph will have all of its above the positive horizontal axis, as in the graph below.

Part (b) step 1: Given Information

We need to find the probability that a particle in the$n=2$state of the finite potential well will be found at the center of the well.

Part (b) step 2: Explanation

Hence, the wave function ${\psi }_2$us zero at the center of the well, so the probability of finding the particle at the point is zero.

Part (c) step 1: Given Information 

We need to find know about standing waves.

Part (c) step 2: Explanation

Yes, the answer in part (b) is consistent with knowing about standing waves, there the $\left(n=2\right)$standing wave on a string that is tied at both ends has a node in the middle.