An electron is trapped in an infinitely deep potential well of width 0.500 nm.
1. If the electron is in its ground state, draw a careful diagram to show the probability function for the electron. (Would be great if I could just get a general description of what this function would look like in terms of shape and maybe a few values to plot).
2. Explain clearly, with aid of your diagram, how the probability of finding the electron within 0.125 nm of the left-hand wall can be estimated to be approximately equal to 0.125. Note: Your method of estimation does not need to be accurate, but you must show clearly how you arrive at the final value.
3. Consider now that the electron is in the 99th energy state above the ground state. Explain clearly again, with the aid of a new diagram, how the probability of finding the electron within 0.125 nm of the left-hand wall can be estimated to be approximately equal to 0.25. Again you must show clearly how you arrive at your value.
1. If the electron is in its ground state, draw a careful diagram to show the probability function for the electron. (Would be great if I could just get a general description of what this function would look like in terms of shape and maybe a few values to plot).
2. Explain clearly, with aid of your diagram, how the probability of finding the electron within 0.125 nm of the left-hand wall can be estimated to be approximately equal to 0.125. Note: Your method of estimation does not need to be accurate, but you must show clearly how you arrive at the final value.
3. Consider now that the electron is in the 99th energy state above the ground state. Explain clearly again, with the aid of a new diagram, how the probability of finding the electron within 0.125 nm of the left-hand wall can be estimated to be approximately equal to 0.25. Again you must show clearly how you arrive at your value.
-
1. For the ground state, the wave function is a simple standing wave with the nodes at the ends. The link shows this. The wave function is half a cycle of a sine curve. I'm not clear what you mean by 'probability function', but I will interpret it as the probability density which is the square of the wave function (you may want to check this is what is meant). So it is a 'sin^2' curve. Plot out sin^2(x) for x = 0 degrees to x =180degrees for the shape.
2. 0.125nm is 1/4 of the well width. The probability of finding an electron between a and b is the integral of the probability density between a and b (area under curve between and b). In this example it corresponds to the area under the curve from one side to 0.125nm. If you have sketched the curve, you should see the area under the curve from one side to 0.125nm is about 1/8 of the total area. Since the total area represents the probability of finding the electron in the well, which is 1, 1/8 of the area represents the probability of finding the electron in the region of interest; so the probability = 1/8 = 0.125.
3) You can tell from the link diagram that the n-th state has a wave function made up of n half-waves.
For n = 99, there will be 99 half-waves (about 50 full waves). The probability density (sin^2) will have twice this number of waves (if you plot sin(x) and sin^2(x) over a few cycles you will see this), i.e. 100 cycles. So the area enclosed from one side to 1/4 of the width is simply the area under 25 of the 100 cycles - i.e. 0.25 of the total area, giving a probability of 0.25.
Hope that makes sense.
2. 0.125nm is 1/4 of the well width. The probability of finding an electron between a and b is the integral of the probability density between a and b (area under curve between and b). In this example it corresponds to the area under the curve from one side to 0.125nm. If you have sketched the curve, you should see the area under the curve from one side to 0.125nm is about 1/8 of the total area. Since the total area represents the probability of finding the electron in the well, which is 1, 1/8 of the area represents the probability of finding the electron in the region of interest; so the probability = 1/8 = 0.125.
3) You can tell from the link diagram that the n-th state has a wave function made up of n half-waves.
For n = 99, there will be 99 half-waves (about 50 full waves). The probability density (sin^2) will have twice this number of waves (if you plot sin(x) and sin^2(x) over a few cycles you will see this), i.e. 100 cycles. So the area enclosed from one side to 1/4 of the width is simply the area under 25 of the 100 cycles - i.e. 0.25 of the total area, giving a probability of 0.25.
Hope that makes sense.