Homework B on Quantum Wells (solutions)

Feel free to work on the homework in groups. The work you hand in, however, should reflect your understanding of the material and be in your own words. Students who turn in identical (or close to identical) homework assignments will be asked to explain their answers orally to the TA or prof. A student who cannot explain how he or she arrived at a given answer will be charged with academic dishonesty.

You should justify all of your answers for full credit.

Answer the following questions about a proton is in an infinite well of width a.

1. Write down the expression for the spacial wavefunction, c(x), for the states n=2, n=3, and n=5.
In general, the wave function of a particle in an infinite well can be written as:

where n=1,2,3,…

One arrives at the answer by substituting in the appropriate values of n, obtaining

c(x) = B sin (2px/a),   (n=2)

c(x) = B sin (3px/a),   (n=3)

c(x) = B sin (5px/a),   (n=5)

2. Sketch c(x) for those three states. What is the wavelength of each?
Sketch c(x) for those three states. What is the wavelength of each?
Since one labels the left side of the graph x=0 and the right side x=a, the plots of the wave functions are 0 at the edges (which makes sense because there can be no probability of the particle being outside the box). The quantum number, n, is the number of maxima and minima, or "humps" in the plot. i.e. n=2 will have one maxima and one minima, and so forth.
In each of these graphs, the x axis extends from 0 to a.

The wavelength of the wave function is the distance from one point on the graph to the corresponding point (e.g. maxima to maxima), or the distance required for the wave to complete one cycle. In this way one could simply determine the wavelength from the plot of the function. Alternately, using the expression

it is possible to numerically solve for the wavelength. Thus, l₂ = a, l₃= 2a/3, and l₅ = 2a/5.

3. Sketch the probability density |c(x)|² for those three states.

In each of these graphs, the x axis extends from 0 to a.
Here the wavefunctions which were initially both positive and negative going have the negative parts of the graph "flipped up" above c = 0. The peaks will appear steeper also, due to the values being squared.

4. List the values of x at which a particle in each of the states will never be found. (i.e., for n=1, the particle will never be found at x = 0 or at x = a.)
List the values of x at which a particle in each of the states will never be found. (i.e., for n=1, the particle will never be found at x = 0 or at x = a.)
The probability density tells us about the probability of finding the particle in a particular place or region. When the probability density is 0 in a given region there is no chance of finding the particle there. The values where the probability density goes to 0 can be found either by looking at the graph or looking at the mathematical form of the wave function to see where it will equal 0. The wave function can only be equal to 0 when the sine expression is 0. Sine is 0 where its argument is a multiple of p radians (180 degrees). So then we can write:

where m=0,1,…,n
For n=2, the probability is zero at 0, a, and a/2.
For n=3, the probability is zero at 0, a, a/3, and 2a/3.
For n=5, the probability is zero at 0, a, a/5, 2a/5, 3a/5, and 4a/5.

5. If the width of the well is a = 2.0 nm (a nm is 10^-9 meters), calculate the energy of the proton in each of the three listed states. You can list it in either Joules or electronVolts, but just be careful with your units.
If the width of the well is a = 2.0 nm (a nm is 10^-9 meters), calculate the energy of the proton in each of the three listed states. You can list it in either Joules or electronVolts, but just be careful with your units.
In the reading one found that only certain discrete energy levels are allowed in the infinite well. These energies were related to the lowest energy (the "ground energy") by the expression

where the ground energy is given by the expression

Using the values for the constants given, the energies can be computed in Joules. They can be then converted to eV using the fact that 1 eV is equal to 1.6x10^-19 Joules. For a proton in the 2.0 nm well specified, E₀ = 8.22 x 10^-24 J = 5.14 x 10^-5 eV. Thus,
E₂ = 4(5.14 x 10^-5 eV) = 2.06 x 10^-4 eV = 3.29 x 10^-23 J
E₃ = 9(5.14 x 10^-5 eV) = 4.62 x 10^-4 eV= 7.40 x 10^-23 J E₅ = 25(5.14 x 10^-5 eV) = 1.29 x 10^-3 eV = 2.06 x 10^-22 J

6. Which of the following energies IS allowed in the quantum well? To which value of n does it belong?
Ea = 0 eV,   Eb = 3.29 x 10^-3 eV, Ec = 4.63 x 10^-3 eV
Which of the following energies IS allowed in the quantum well? To which value of n does it belong?
Ea = 0 eV,   Eb = 3.29x10^-3 eV, Ec = 4.63x10^-3 eV

After doing the calculations, one finds that Eb is 64 times E₀, so n=8. Ec is 90 times E₀. 90 is not a perfect square, so Ec is not an allowed energy. 0 is also not a valid energy, since the wavefunction for E=0 is zero. Thus it never occurs.

1.	Write down the expression for the spacial wavefunction, c(x), for the states n=2, n=3, and n=5. In general, the wave function of a particle in an infinite well can be written as: where n=1,2,3,… One arrives at the answer by substituting in the appropriate values of n, obtaining c(x) = B sin (2px/a), (n=2) c(x) = B sin (3px/a), (n=3) c(x) = B sin (5px/a), (n=5)
2.	Sketch c(x) for those three states. What is the wavelength of each? Sketch c(x) for those three states. What is the wavelength of each? Since one labels the left side of the graph x=0 and the right side x=a, the plots of the wave functions are 0 at the edges (which makes sense because there can be no probability of the particle being outside the box). The quantum number, n, is the number of maxima and minima, or "humps" in the plot. i.e. n=2 will have one maxima and one minima, and so forth. In each of these graphs, the x axis extends from 0 to a. The wavelength of the wave function is the distance from one point on the graph to the corresponding point (e.g. maxima to maxima), or the distance required for the wave to complete one cycle. In this way one could simply determine the wavelength from the plot of the function. Alternately, using the expression it is possible to numerically solve for the wavelength. Thus, l₂ = a, l₃= 2a/3, and l₅ = 2a/5.
3.	Sketch the probability density \|c(x)\|² for those three states. In each of these graphs, the x axis extends from 0 to a. Here the wavefunctions which were initially both positive and negative going have the negative parts of the graph "flipped up" above c = 0. The peaks will appear steeper also, due to the values being squared.
4.	List the values of x at which a particle in each of the states will never be found. (i.e., for n=1, the particle will never be found at x = 0 or at x = a.) List the values of x at which a particle in each of the states will never be found. (i.e., for n=1, the particle will never be found at x = 0 or at x = a.) The probability density tells us about the probability of finding the particle in a particular place or region. When the probability density is 0 in a given region there is no chance of finding the particle there. The values where the probability density goes to 0 can be found either by looking at the graph or looking at the mathematical form of the wave function to see where it will equal 0. The wave function can only be equal to 0 when the sine expression is 0. Sine is 0 where its argument is a multiple of p radians (180 degrees). So then we can write: where m=0,1,…,n For n=2, the probability is zero at 0, a, and a/2. For n=3, the probability is zero at 0, a, a/3, and 2a/3. For n=5, the probability is zero at 0, a, a/5, 2a/5, 3a/5, and 4a/5.
5.	If the width of the well is a = 2.0 nm (a nm is 10^-9 meters), calculate the energy of the proton in each of the three listed states. You can list it in either Joules or electronVolts, but just be careful with your units. If the width of the well is a = 2.0 nm (a nm is 10^-9 meters), calculate the energy of the proton in each of the three listed states. You can list it in either Joules or electronVolts, but just be careful with your units. In the reading one found that only certain discrete energy levels are allowed in the infinite well. These energies were related to the lowest energy (the "ground energy") by the expression where the ground energy is given by the expression Using the values for the constants given, the energies can be computed in Joules. They can be then converted to eV using the fact that 1 eV is equal to 1.6x10^-19 Joules. For a proton in the 2.0 nm well specified, E₀ = 8.22 x 10^-24 J = 5.14 x 10^-5 eV. Thus, E₂ = 4(5.14 x 10^-5 eV) = 2.06 x 10^-4 eV = 3.29 x 10^-23 J E₃ = 9(5.14 x 10^-5 eV) = 4.62 x 10^-4 eV= 7.40 x 10^-23 J E₅ = 25(5.14 x 10^-5 eV) = 1.29 x 10^-3 eV = 2.06 x 10^-22 J
6.	Which of the following energies IS allowed in the quantum well? To which value of n does it belong? Ea = 0 eV, Eb = 3.29 x 10^-3 eV, Ec = 4.63 x 10^-3 eV Which of the following energies IS allowed in the quantum well? To which value of n does it belong? Ea = 0 eV, Eb = 3.29x10^-3 eV, Ec = 4.63x10^-3 eV After doing the calculations, one finds that Eb is 64 times E₀, so n=8. Ec is 90 times E₀. 90 is not a perfect square, so Ec is not an allowed energy. 0 is also not a valid energy, since the wavefunction for E=0 is zero. Thus it never occurs.