Wave Function Lab
Motivation
The focus of this lab session is to let you get some experience with wave functions. We will first look at familiar ones for the 1D particle in a box, and do some visualizing and calculating with them. After that we will use numerical methods to solve for the wave functions of the quantum harmonic oscillator (QHO) and find the energies of the states.
Particle in a Box
As we have discussed in class, the wave function for a particle in a box is just a sine wave that satisfies appropriate boundary conditions. The probability density is the square of the wave function. If you integrate it over some range x=a to x=b, you get the probability that the particle will be found within that range of coordinates (between x=a and x=b). In order to integrate a squared sine wave by hand, one must remember or look up the double angle formula. In case you are like me, it is sometimes easier to derive that formula than it is to remember it. Even if you are not like me, we will do this first.
- Euler's formula is used often in the mathematics of complex functions. It is also useful for doing proofs of trigonometric identities. It states Use this identity and the rules of mathematics for exponents like and to derive the double angle formula for which we need to integrate the wave function for the particle in a box.
- Use the relation that you derive to find the average value of the squared sine function over any number of complete half cycles: x=0 to x=nπ.
- Based on this result, come up with a normalization constant for the function so that you can write and have a normalized wave function.
- Open GeoGebra, and define constants L, n, A. Define your wave function (call it psi(x) using the greek font if you want) in terms of these constants, for a 1D particle in an infinite box with walls at x=0 and x=L. Adjust the plot window so that you see the wave function between x=0 and x=L.
- Plot the probability density for the particle. Using the integrate function, verify that the wave function is normalized.
- For each of the first three quantum states, please answer these questions: a. At which location(s) is the particle most likely to be found? b. Least likely to be found? c. In which section of width L/10 of the box is the particle most likely to be found? (Give midpoint of the section and value of probability) d. What width w is the probability unchanging for each quantum state? e. In part (d) what is the probability? f. In which section of width L/10 of the box is the particle least likely to be found? (Give midpoint of the section and value of the probability) g. What is the average location of the particle in the box? Please do this via appropriate calculation even if the answer seems obvious. h. Based on your calculation (integral), is there a meaning for the value of the integrated curve at one specific point, or only the total integral? i. How do your answers for a-g depend on the constants L or m (mass of particle)?
- Find the energy of the first three quantum states, and based on the particle being an electron in a 1.0nm box, what wavelengths of light can be absorbed by the system?
Quantum Harmonic Oscillator
As you saw in the section on the QHO, the vibrational energy of a quantum oscillator is quantized. That means it only takes on specific values, and can not just vibrate with any arbitrary amplitude. In this section we will find those energies and the associated wave functions by using numerical methods.
- Plug in the two linear differential equations mentioned in the chapter on the QHO into GeoGebra. Once they are entered we need to solve them.
- We will need to solve the differential equations both for x>0 and x<0 to get the best results for the wave functions. To solve for the region x>0, we type
NSolveODE[{p', dpdx'},0,{1,0},10]. Note that my initial conditions are P(0)=1, and dPdx(0)=0. These choices will lead to only even functions as solutions to the equations. - To get the solution in the region where x<0, change the last entry in the NSolveODE function to -10 rather than 10. That will solve the equation backwards to lower values of x. Together the two functions should look like one function that smoothly transitions through x=0.
- Let's show the potential energy function on the plot. Type
U(x)=1/2*x^2. Additionally typey=Eto show the energy of our system. You may want to right click on those two functions and display them in different colors or with dashed lines by changing their properties. - Additionally, let's show the value of x where our energy E intercepts the potential energy function and put up dashed lines as walls at those values. This is significant since in the macroscopic world, those would be the absolute allowable limits of the motion of an oscillator. As you'll see in this calculation, the quantum oscillator is allowed to make excursions beyond those values. To show the intercept of E and U(x) use the intercept tool that looks like
. Click it and then click on the two functions.
Points will be created at the intercepts. They will likely be named A and B. If you type x=x(A) and x=x(B) two lines will be added to the graph showing at the x-coordinates of the two points. - Find all the energies up to 10 that satisfy the boundary conditions such that the wave function dies off exponentially in the forbidden region where These energies are in multiples of
- If we want odd functions, we need to choose initial conditions {0,1} instead, and again enter two NSolveODE functions for the x>0 and x<0 regions. Please do this, and once again find all the energies up to that satisfy the equation.
- If you make a list of all the energy values for both the even and the odd solutions, what is the pattern? How does the ground state energy compare with the energy intervals between successive quantum states? The ground state is also called the zero point energy.
- Write the energy as a function of quantum number while allowing N=0, and not N=1 to represent the ground state.