S3: Hydrogen

The Hydrogen atom consists of a proton at the nucleus and an electron in a cloud surrounding it. Because the proton is so much heavier than the electron, we think of the electron as sitting in a potential well due to the proton: if the electron is a distance r from the proton, there is a Coulomb energy

holding the electron to the proton.

How can we solve this problem? To solve it, we need to solve for a three-dimensional wave function ; we only know how to solve Schrödinger's equation in one dimension! It turns out that one can break up the problem, solving the radial part of the equation separately from the angular dependence. It so happens that a slightly different function can be defined,

which satisfies exactly our one-dimensional Schrödinger equation:

.
The boundary conditions also remain the same:

. How convenient!

We won't go into the angular parts of Hydrogen's wave equation in this course. For the chemists among us, the angular parts give the differences between the s, p, and d states. Our wavefunctions , when divided by r, will give exactly the 1s, 2s, 3s, 4s, 5s, and 6s states. It happens (for truly obscure reasons) that the energies of the p and d states are the same as the energies for the s states, so we can compare the energies we numerically calculate in this lab directly to those we will measure in the experimental Spectrum of Hydrogen lab. We'll label our states, instead of 1s, by their energies E1, E2, ....

(S3a) The Ground State

Find the energy of the ground state of hydrogen. Change the potential type to Hydrogen. Note that the black line now shows the hydrogenic potential energy, and that the range has shrunk to 0.5nm. You should start in the region between -13eV and -14eV, and then zero in on a good value. Note that the energies in this section are negative! This is just because of the convention for the Coulomb potential: V(x) can't be set to zero at zero (why?) so we set it to zero at infinity. (You can see the hints for the Energy slider for tips on zooming in efficiently.) You should be able to get the energy to five significant figures (-13.XXX). (The last decimal place isn't correct, actually: you can hit Configure and make delta_x half the size to get a better number if you're ambitious.) Include the ground-state energy in your writeup, and sketch for the ground state. The ground state is also called the 1s state, and we'll call its energy E1.

(S3b) The Excited States

Find the energies of the first three excited states (four states in all). As you sweep the energy, every time the tail whips from up to down or down to up you've passed an eigenstate. You should be able to tell if you've missed a state by counting the nodes in the wave-function. Be careful

to expand the Distance to Shoot each time you raise the energy. You want to make sure at least the right-most third of the graph is showing the exponential growth or decay, so that you can kill it carefully,
in counting nodes, not to count the one which - for the exact solution - would be at infinity: if the tail of the wave-function diverges in the wrong direction your approximate wave-function might show that node in the tail,
to get at least two significant figures after the decimal point.

Include these energies in your writeup, and label them E2, E3, as appropriate. Sketch in your lab notebook the function

for the energy E3 (the second excited state), over the range 0 < X < 2.5nm, or (if your TA prefers) use the ``Copy Graph'' button to save the PostScript file, print it, and include that in your writeup.

(S3c) The Energy Level Diagram

As in part 1, draw an energy level diagram in your writeup for the first four energy states of the hydrogen atom, leaving room at the top for states between the highest energy you found and E=0. Plot the energies to scale! (This should occupy the entire length of a sheet of paper.) You should find that they form a simple pattern, bunching up as they approach the top of the Hydrogen potential well at E=0. Label the states with their energies in eV, and with their numbers E1, E2, ...

(S3d) Comparing to the Answer

It's rare in quantum mechanics that one can find solutions to the equations. The hydrogen atom is one of those rare cases: the energies for the states (and the wave-functions) can be found exactly (ignoring relativistic effects). In your writeup, compare the first four energies you found to the exact formula

How close did we come? Use the formula and your ground-state energy to predict where the next two energy states should lie. Type in these energies, and verify that your predictions are close to the next two energy states in the ladder. Add them to your energy level diagram.

(S3e) The Visible Spectrum of Hydrogen

Look back to the end of part 1 to review how to get the frequencies of the emitted light, given the energy differences between the initial state of the electron and the final state. Is the light emitted from the transition from the first excited state to the ground state in the visible range? How about the transition from En to E1, for n> 2? In the experimental Spectrum of Hydrogen lab, we'll measure several visible lines, which will turn out to be transitions into the first excited state E2, from the levels E3, ... E6. (These are the Balmer lines. The inventors of quantum mechanics used these lines to guess the equations: it's like listening to a violin string and deducing the wave equation.) Using your measured values, predict the frequencies (in cycles/sec) and the wavelengths for these four transitions. What colors should the light be?

We will measure precisely these four lines (if your eyes are good) in the experimental Spectrum of Hydrogen lab. Include the predicted colors and frequencies in your writeup, and also jot them down on a separate piece of paper to bring to the experimental lab session!

Links Back

Statistical Mechanics: Entropy, Order Parameters, and Complexity, now available at Oxford University Press (USA, Europe).