PHYS 27/193: Homework 3

1. The Stefan-Boltzmann law states that an object at temperature T (in Kelvin), radiates energy at a rate that depends on the 4th power of the temperature. Thus if you double the temperature of an object, it radiates 16 times more energy. This energy is what we "feel" as heat or see if the object is hot enough to glow.

   Specifically the Stefan-Boltzmann law is

   

   Here, flux is the power radiated (in Watts) per square meter of surface of the object.

   Use gnuplot as a calculator to verify that the coefficient in from of the T⁴ is 5.67 x 10^-8 W/m²/K ^-4

   To enter numbers in scientific notation to gnuplot, use the following transcription:

   6.626 x 10 ^-34 is entered to gnuplot as 6.626e-34

   The e above stands for the "'exponent' on the 10". Note that the sign is really really important!!

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2. The temperature, T, of the surface of the Sun is about 5.5 x 10³ K. The radius, R, of the Sun is 7 x 10⁸ m.

   The above formula says that the total thermal power emmitted by an object is the thermal power flux x surface area.

   Use gnuplot as a calculator to find the total power output of the sun in Watts

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3. According to quantum mechanics, when a particle is trapped in a particular region of space (either stuck in a box, or trapped in an atom), it can only have specific energies, ie. it can only exist in certain stated--for an electron in an atom, these are the orbitals or shells you will remember from chemistry.

   A standard quantum mechanics problem is to find the energy levels of a particle trapped in a well (or box). The easy version is when the walls are infinitely high and the probability that the particle can be found outside the box is zero.

   The harder (and real) version is when the well has a finite depth. This problem applies to many structure in modern electronics, particularly nano devices such as quantum dots.

   Below are a couple of links about the quantum mechanics of such structures:

   • Lecture (PDF) by Dr. Leonard Gamberg, Penn State University, on the quantum mechanics of confined particles.
   • Java Applets showing energy levels of trapped particles (scroll down to "Electrons in Quantum Structures".
   • And one of my favorites: The Brittany Spears Guide to Semiconductor Physics

   For a particle of mass m in a 1-dimensional box of width L and "potential depth" V_o, the energy levels are given by solutions of the following equation:

   

   We will measure lengths, masses, and energies in units so that m=1, h=1, and L=1 in the equation above. In this case, the equation becomes:

   

   Given V_o we can solve this equation for the allowed E values. Thus, your third homework problem of this set:

   For a well of depth V_o = 50, plot the two sides of the equation above, as two separate functions. Since you want to solve for E, you will let that be represented by the variable x in your plot.

   How many solutions are there? (ie. places where the two curves meet). This is the number of different states that the particle can be in this "Quantum Dot". Note that the energy must be positive!

   What are the energies, E, of each of the states?

   When plotting these, you may see a spurious "vertical line" in the tan(x) plot. As you zoom in, this will disappear.