24The size of an atom
1Suppose we have a hydrogen atom, and measure the position of the electron; we2 must not be able to predict exactly where the electron will be, or the momentum
3 spread will then turn out to be infinite. Every time we look at the electron, it is
4 somewhere, but it has an amplitude to be in different places so there is a probability
5 of it being found in different places. These places cannot all be at the nucleus; we6 shall suppose there is a spread in position of order ɑ. That is, the distance of the7 electron from the nucleus is usually about a. We shall determine a by minimizing
8 the total energy of the atom.
The spread in momentum is roughly h/ɑ because of the uncertainty relation, so

11 that if we try to measure the momentum of the electron in some manner, such as by
12 scattering x-rays off it and looking for the Doppler effect from a moving scatterer,
13 we would expect not to get zero every time ― the electron is not standing still ―14 but the momenta must be of the order p ≈ (a15 roughly ½mv 2 = (b). Then the kinetic energy is) = h2/ 2mɑ 2. (In a sense, this is a kind of

16 dimensional analysis to find out in what way the kinetic energy depends upon
17 Planck's constant, upon m, and upon the size of the atom. We need not trust our
18 answer to within factors like 2, π, etc. We have not even defined ɑ very precisely.)
19 Now the potential energy is minus e2 over the distance from the center, say

26(2.10)

We do not know what ɑ is, but we know that the atom is going to arrange itself

27 to make some kind of compromise so that the energy is as little as possible. In order
28 to minimize E, we differentiate with respect to ɑ, set the derivative equal to zero,
29 and solve for ɑ. The derivative of E is30

dE/dɑ = (e),(2.11)

31 and setting dE/da = 0 gives for ɑ the value
ɑ0 = (f)33
34= (g35= (h) meter) angstrom.

(2.12)

36
37 This particular distance is called the Bohr radius, and we have thus learned that38 atomic dimensions are of the order of angstroms, which is right. This is pretty good
39 ― in fact, it is amazing, since until now we have had no basis for understanding
40 the size of atoms! Atoms are completely impossible from the classical point of view,
41 since the electrons would spiral into the nucleus.
42), where e2 is the charge of an electron squared, divided by43

21 4πε0. Now the point is that the potential energy is reduced if ɑ gets smaller, but the44

22 smaller ɑ is, the higher the momentum required, because of the uncertainty

45

23 principle, and therefore the higher the kinetic energy. The total energy is4620 (c).

2532910E = (dNow if we put the value (2.12) for ɑ0 into (2.10) to find the energy, it comes out
E0 = (h) = (i= (j).

)(2.13)
1

47 What does a negative energy mean? It means that the electron has less energy when
48 it is in the atom than when it is free. It means it is bound. It means it takes energy
49 to kick the electron out; it takes energy of the order of 13.6 eV to ionize a hydrogen
50 atom. We have no reason to think that it is not two or three times this ― or half of
51 this ― or (1/π) times this, because we have used such a sloppy argument. However,
52 we have cheated, we have used all the constants in such a way that it happens to53 come out the right number! This number, 13.6 electron volts, is called a Rydberg
54 of energy; it is the ionization energy of hydrogen.
55
56

R.P. Feynman, R.B. Leighton, and M.L. Sands:

57

The Feynman Lectures on Physics (Addison-58

Wesley, 1965) Vol. III, Sec. 2-4 “The size of an59atom”.

2