Consider a fictitious one-dimensional system with one electron. The wave function for the electron, drawn below, is c1x2 = sin x from x = 0 to x = 2p. (b) At what value or values of x will there be the greatest probability of finding the electron?

Video transcript

Welcome back everyone in this example, we have the following diagram which shows the hypothetical function of a electron in a one dimensional space. So the below diagram is generated by taking the way function of X, which is set equal to the sine of three halves times X. And it generates X values from zero to X equals two pi. We need to use this info to determine at which values of X is the probability of finding an electron going to be the highest. So we should recall that in order to determine this value. We want to calculate probability density of X and recall that probability density is represented by the symbol for wave function being squared. So this is going to represent the probability that we should recall for that. An electron will be found at a given point in space. And so scrolling down below for more room. We want to plot the wave function squared of our X values in our calculators. And in doing so we should generate the diagram below which I will paste here in our space beneath here. So we would generate the below diagram. And according to this diagram that we've generated, we have three peaks which correspond to the probability being a value of one. So these are the highest points on our diagram or the highest values of probability. And this is also known as the term maxima. So we can say that this first peak is our first maxima corresponding to the probability of one and the X value at one third pi. Moving on, we have a second maxima at our second peak corresponding to the probability of one and the X value being just pot here. And then we have our third peak representing our and I'll use a different color here. Third maxima which corresponds again to the probability being greatest at the value one. And the this corresponds to the X value being five thirds pi. And so based on this diagram that we've generated in our calculators, we can say that therefore the greatest probability of finding an electron corresponds to our plot where we took the square of our wave function of X. And we generated X values for our first maxima where we have X equal to three over pi, which is just one third pi simplified for our second X value. We generated X is equal to pi for our second maxima. And then for our third maxima or third highest probability of finding an electron that corresponded to an X value equal to five thirds times pi. And so these X values here would be our final answers to complete this example as the points at which we have the highest probability of finding an electron. So what's highlighted in yellow is our final answer. And this will correspond to choice C. In the multiple choice. So I hope that everything that I reviewed was clear. If you have any questions, please leave them down below. And I will see everyone in the next practice video