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Physics LibreTexts

11.4: The hydrogen atom

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For the hydrogen atom we have a Coulomb force exerted by the proton forcing the electron to orbit around it. Since the proton is 1837 heavier than the electron, we can ignore the reverse action. The potential is thus

V(r)=e24πϵ0r

If we substitute this in the Schrödinger equation for u(r), we find

22m2r2u(r)e24πϵ0ru(r)=Eu(r).

The way to attack this problem is once again to combine physical quantities to set the scale of length, and see what emerges. From a dimensional analysis we find that the length scale is set by the Bohr radius a0,

a0=4πϵ02me2=0.53×1010 m

The scale of energy is set by these same parameters to be

e24πϵ0a0=2Ry

and one Ry (Rydberg) is 13.6 eV . Solutions can be found by a complicated argument similar to the one for the Harmonic oscillator, but (without proof) we have

En=12e24πϵ0a01n2=13.61n2eV

and

Rn=er(na0)(c0+c1r++cn1rn1)

The explicit, and normalised, forms of a few of these states are

R1(r)=14π2a3/20er/a0,R2(r)=14π2(2a0)32[1r2a0er(2a0).

Remember these are normalised to

0Rn(r)Rm(r)dr=δnm

Notice that there are solution that do depend on θ and φ as well, and that we have not looked at such solutions here!


This page titled 11.4: The hydrogen atom is shared under a CC BY-NC-SA 2.0 license and was authored, remixed, and/or curated by Niels Walet via source content that was edited to the style and standards of the LibreTexts platform.

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