SCHRÖDINGER'S EQUATION IN THREE DIMENSIONS
(This is a two-page lab: be sure to turn this page over!)

Quantum systems are quantized because of boundary conditions. These boundary conditions are what give rise to discrete energy levels. We will look at two such systems -- the hydrogen atom and the quark-antiquark pair () known as a meson.

A hydrogen atom consists of two charged particles, with the potential energy of the system inversely proportional to the distance between the two particles. If we know the value of the wave function and its derivative with respect to radius at the origin, we can use Schrödinger's equation to crank out the wavefunction throughout the rest of space, provided we know the energy, E, of the electron. The computer program QM3D does just that. The only problem is that, for an arbitrarily-chosen energy, the wavefunction will probably blow up as r → ∞. Only certain values of energy will work: the eigenenvalues for the system. This is where the spectrum of the hydrogen atom comes from -- it comes from insisting that .
1. The Hydrogen Atom: Download QM3D_2007.xls. Run QM3D, choosing the 'Coulomb' potential to simulate a hydrogen atom. Verify that:
  • For a given angular momentum quantum number ℓ, the eigenenergies, in the units of this program, are given by En =-1/n2, where n is an integer larger than ℓ, and En is independent of ℓ. (For E = -1/22, type in "-0.25", not "-1/22".) The number of antinodes in the wavefunction is equal to n - ℓ. (Notice that this definition for n is different from the one you used last week.) NOTE: You may have to change the graph limits for large values of n. If so, double click on the chart axis, and treat it like any Excel chart.
  • Only angular momentum quantum numbers, ℓ, between 0 and n-1 give valid wave functions. Verify this claim for all energy quantum numbers, n, less than or equal to 5.

2. Mesons (The Airy potential): A meson consists of a quark and an antiquark bound by a force which is believed -- to lowest-order approximation -- to be independent of distance, so that the potential energy increases linearly with distance. (Like the triangular potential from last week, only three-dimensional) How do we know that the force behaves this way if we have found it impossible so far to isolate individual quarks? We know this only indirectly, by comparing the energy spectrum of observed mesons with the hypothetical energy spectrum of a system for which potential energy varies linearly with distance -- exactly what you are about to test next using QM3D. (a) Run QM3D, choosing the 'Airy' potential to simulate a pair. Calculate the energy levels, εnl , for (n,l) = (1,0), (2,0), (3,0), (2, 1), (3,2). Do any of the eigenvalues equal those you got for the triangular 1D potential last week? Fill these eigenvalues into the table below.

(b) The energy, ET)nl, of an actual physical meson is related to the corresponding eigenvalue, εnl, by the following expression:
(ET)nl   =   [(ħ k) 2/mq]1/3εnl   +   2(mqc2),

where k is the force of attraction between two quarks, and mq is the mass of the quark and antiquark involved. This is a linear relation between x=εnl and y=(ET)nl, where x is the set of energies you get from QM3D, and y are the energies listed below. By fitting this linear relation, using the program SLUDGE, you can determine the rest energy, mqc2, of a quark, and the force constant, k.) Fit the data in the chart below, and calculate the mass of the charm and the beauty quarks. Save your graphs of ET vs εnlto show whether the Airy potential is a good model for the attraction between quark and antiquark.

(c) Time permitting, calculate the force constant, k, for both mesons. Is it the same? DISCUSSION: A. Compare your calcuated quark masses to the accepted values. (HINT: on a poster in the department) What kind of quarks are J/Ψ and Υ (upsilon) made of?
Be sure to include the uncertainties of your calculated values in the discussion.
B. Decide whether the force constant, k is the same for both types of particles.

Masses of ground and excited states of J/Ψ and Υ particles, in GeV
State (n,l)
εnl from QM3D
ET: J/Ψ
ET: Υ
1S (1,0)
  
3.097
9.458
2P (2,1)
  
3.511
-----
2S (2,0)
  
3.686
10.016
3S (3,0)
  
4.730
10.323
3D (3,2)
  
3.770
-----
4S (4,0)
  
4.414
10.547


Quick re-explanation of what you need to do: 1. Calculate the dimenionless energy eigenfunctions, εnl, using the program QM3D. Record these in the blank column above, making absolutely certain that each entry corresponds to the correct values of n and ℓ. NOTE: n - ℓ is equal to the number of antinodes in the wavefunction.
2. Use SLUDGE to fit a straight line to two sets of data. The εnl data will be the "x" data for each. The "y" data will consist of each of the other two columns. Record the slope and intercept -- and their uncertainties!! -- and set them equal to the appropriate terms in the equation above. Making sense of it all: Why did we do this "compuational experiment with the mesons? We started by saying that we think that the force between quarks acts in a certain way. If this is true, then the excited energies of a two-quark system should be calculatable with this software. The proof of the pudding is in the tasting: do these excited energies behave as the model predicts? For both two-quark systems? What do we conclude about the fitness of the model for describing quark-antiquark pairs? (This exercise is adapted from an example in Das & Melissinos' Quantum Mechanics: A Modern Introduction.)
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