Consider the electron wave function

Question

ψ (x) = {\begin{matrix} c \sqrt{1 - x^{2}} & ∣ x ∣ \leq 1 cm \\ 0 & ∣ x ∣ \geq 1 cm \end{matrix}

where x is in cm. Draw a graph of |ψ(x)|² over the interval −2 cm ≤ x ≤ 2 cm. Provide numerical scales.

Accepted Answer

Hello, let's go through with this practice problem. The motion of a particle follows si of Y equals a multiplied by the square root of eight minus two Y squared. The way function exists in the range in negative two nanometers to positive two nanometers and decays to zero for other values of Y. The unit of Y is nanometers plot the square of psy of X for the interval negative 2.5 nanometers to positive 2.5 nanometers giving appropriate numerical labels on the axes. OK. So in order to plot the square of the wave function, we first need to determine what the wave function actually is. It is given to us. But we first need to find the value of a, the normalization constant. So we're first going to need to go through the normalization condition for the wave function recall that the normalization condition of the wave function is the integral over all bounds. So from negative infinity to positive infinity, the integral over infinity of the square or of the probability density to the square of si of why? With respect to why must be equal to one? So we're going to take the integral of the function that's given to us set that integral equal to one and solve for a and that's how we'll get the normalization constant. All right. So let's begin taking this integral and we don't actually have to integrate across infinities because the problem specifically tells us that the function only exists from negative two to positive two nanometers. So that makes things a little easier on us. But we're integrating a multiplied by the square root of eight minus two Y squared, all of it is squared and we're integrating with respects to Y. So let's first apply the square. So this A becomes squared and then the square root goes away. And since the A is a constant, we can just pull it out, you can pull the A squared out of the integral entirely. So integrating from negative two to positive two and now we're integrating just eight minus two Y squared. Now, from here on out, this integral is fairly simple to evaluate it's just a simple integral power rule. So the eight integrated with respect to Y becomes an eight Y and then the two Y squared again, the integral power rule is we raise the exponent by one and then divide by the new exponent. So right now it's two Y squared, so we raise that exponent by one. So it's a cube and divide the term by three. Next, we'll apply the bounds using the fundamental theorem of calculus. So first, we apply the upper bound. So that is eight multiplied by two minus two, multiplied by two cubed divided by three, then minus, then we apply the negative bound. So minus eight time or multiplied by negative two minus two, multiplied by the cube of negative two divided by three. Which if you simplify that or put it into a calculator, it's a cubed multiplied by 64 divided by three. And remember this is all equal to one as per the normalization condition. So to find a, we divide both sides of the equation by that, so three divided by 64 and then take the square root of both sides. And so we find that a has a value of 0.22 nanometers raise the power of negative one half. So our wave function then is equal to 0.22 with the units of nanometers rates the power of negative one half multiplied by the square root of eight minus two Y cubed or squared. And since the problem is asking for us to graph the square of it, let's take the square. So squaring 0.22 this means that the coefficient is 0.048 nanometers inverted multiplied by eight minus two Y squared. So this is now the formula that we are now ready to graph. If you want, you can put it into a graphing calculator to see what the graph looks like. Or you can just experiment with it by plugging in different numbers into why and seeing what happens to the wave function. But what we discover is we end up with a graph that look like this or here's our Y axis, here is our probability density axis. The units of nanometers inverted in the horizontal axis. The y axis is units of nanometers. What we find with a bit of experimentation is that the peak. So when Y is equal to zero as a value of 0.384 we also find that the curve is symmetrical over the Y ax over the vertical axis and reaches zero and Y is equal to two nanometers. So let's also set up our axis, our axis points, our ticks based on the fact that we're supposed to be sketching the interval from negative 2.5 to positive 2.5. So negative 2.5 positive 2.5 here's 12345. So here's where one is and here's where two is we do the same thing on the left side. So negative one, negative 12 four and a negative two. So the graph gradually decreases from the peak down to the point where Y is equal to two or negative two. And then from there, it just remains at zero all the way to the 2.5 bounds that were given to us in the problem. So your graph should look something like this and that is it for this problem. I hope this video helped you out. If it did, please consider checking out our other videos, which will give you more experience with these types of problems. That's all for now and I hope you have a nice day. Bye bye.

Consider the electron wave function $ψ (x) = {\begin{matrix} c \sqrt{1 - x^{2}} & ∣ x ∣ \leq 1 cm \\ 0 & ∣ x ∣ \geq 1 cm \end{matrix}$ where x is in cm. Draw a graph of |ψ(x)|² over the interval −2 cm ≤ x ≤ 2 cm. Provide numerical scales.

Key Concepts

Wave Function

Probability Density

Graphing Functions

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