# Visual Examples: Bound States of the Hydrogen Atom

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Three different quantum states of the hydrogen atom, and the corresponding Bohmian motion for 40 representative points in each case. The motion is rather boring, though, in cases (a) and (b): since the wave function has constant phase, the particle is at rest. Not so in case (c): the motion is periodic, the actual period length being 4.06 10^{-16} seconds.

These are plots of the radial distribution density function *4pi r^2 |psi(r)|^2* varying with time. For stationary states such as (a) and (b), the distribution is a constant of time, i.e. it does not change. The upward axis is the density, the right axis the radius in multiples of the Bohr atom radius. Observe that the electron tends to be closer to the nucleus in the ground state (a) than in the excited state (b). In their superposition (c), one can observe that the value at *r = 2* is time-independent, which comes from the fact that the |2s> state (b) has a zero there.

Another look at the density function: the southwest axis is radius, the southeast axis is time. The unit of time is Planck's constant over the Rydberg (= ground state) energy, which is the natural unit of time for the hydrogen atom; the period of wave function (c) is 4/3 units because |1s> rotates in Hilbert space with frequency 1 while |2s> rotates with 1/4 (remember that the energy levels are 1/n), so that the relative rotation has frequency 3/4. In the picture one sees almost 2 periods. The color does not carry any information.

For those interested in pictures of wave functions, take a look at the pictures of Hydrogen eigenstates on the webpage Visual Quantum Mechanics by Bernd Thaller.

By Roderich Tumulka.

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