Section 6–5 The uncertainty principle

(Probability density / Einstein’s worry / Electron cloud)

The uncertainty principle is discussed in chapter 2, chapter 5, chapter 6, chapter 37, and chapter 38 of The Feynman Lectures. In this section, the three interesting points discussed are a probability density, Einstein’s worry, and electron cloud.

1. Probability density:

“…We can give a probability density p₁(x), such that p₁(x)Δx is the probability that the particle will be found between x and x+Δx (Feynman et al., 1963, section 6.5 The uncertainty principle).”

Dr. Sands explains the uncertainty principle from a perspective of probability. He elaborates that the concepts of probability are useful in describing the behavior of 10²² or more molecules in a gas because it is clearly impractical to write down the position and velocity of all these molecules. Furthermore, the use of probability is not just a matter of convenience for very complex situations, but it is essential to a description of atomic phenomena. Based on Heisenberg uncertainty principle, there are always uncertainties in the specifications of positions and velocities. Thus, physicists specify the probability of a particle as p₁(x)Δx will have a position between x and x+Δx in which p₁(x) is the probability density that the particle will be found.  

In the previous chapter, Dr. Sands gives a slightly different explanation of the uncertainty principle: the uncertainty in position is related to the error in our knowledge of the momentum of the object whose position we are measuring. In addition, he mentions that the uncertainty in position measurements is related to the wave nature of particles. Nevertheless, Heisenberg distinguishes two kinds of uncertainty: objective and subjective. In his own words, “[t]hese uncertainties may be called objective in so far as they are simply a consequence of the description in the terms of classical physics and do not depend on any observer. They may be called subjective in so far as they refer to our incomplete knowledge of the world (Heisenberg, 1958, p. 28).”

2. Einstein’s worry:

“…Einstein was quite worried about this problem. He used to shake his head and say, ‘But, surely God does not throw dice in determining how electrons should go!’ (Feynman et al., 1963, section 6.5 The uncertainty principle).”

Dr. Sands emphasizes that the most precise description of nature must be stated in terms of probabilities. On the contrary, some physicists have the opinion that they could know the speed and position of a particle simultaneously. For example, Einstein argues that God does not throw dice in determining how electrons should go. Moreover, Einstein, Podolsky, and Rosen (1935) pose the question whether a quantum mechanical description of physical reality can be considered complete. Interestingly, Einstein’s worry is related to a “spooky action at a distance” and it has led to the concept of quantum entanglement.

Dr. Sands mentions that only one or two physicists were working on the problem of describing the physical world in a different way such that uncertainties can be removed. Historically, many notable physicists attempted this problem which results in important contributions in quantum mechanics. For example, Bohm and Aharonov (1957) publish a new version of the Einstein–Podolsky–Rosen (EPR) paradox by reformulating the original argument in terms of spin. Subsequently, Bell (1964) proposes a theorem that could be used to test the EPR paradox. Most important, Einstein’s worry has led to active research by the physics community on quantum entanglement and useful applications such as quantum cryptography.

Note: In a speech titled Simulating physics with computers, Feynman (1982) discusses EPR paradox without mentioning Bell’s theorem.

3. Electron cloud:

“…our best “picture” of a hydrogen atom is a nucleus surrounded by an “electron cloud” (although we really mean a “probability cloud”) (Feynman et al., 1963, section 6.5 The uncertainty principle).”

Dr. Sands elaborates that the uncertainty in the position of an electron in a hydrogen atom is as large as the atom itself. Based on quantum mechanics, physicists do not describe the electron as moving in an “orbit” around the hydrogen atom or a proton. That is, we speak of a probability, p(r)ΔV, of observing the electron in a volume ΔV at a distance r from the proton. We may visualize the hydrogen atom as having an “electron cloud” that is surrounding the proton. The density of the electron cloud is proportional to the probability density of the electron where it will be found. Essentially, we can describe the electron as having a probability somewhere at a location because nature permits us to know only the chance of locating it.

In the earlier editions of The Feynman Lectures, the probability density of a hydrogen atom is described by the expression p(r) = A exp(−r²/a²) in which the constant a is the “typical” radius of the atom. Thus, there is a small chance of finding the electron at distances from the nucleus significantly greater than a which is about 10⁻¹⁰ meter. In the New Millennium edition of The Feynman Lectures, the probability density for an undisturbed hydrogen atom is revised as p(r) = A exp(−2r/a). This expression can be derived by using a three dimensional (time-independent) Schrödinger equation in spherical coordinates. The mistake of Dr. Sands in using p(r) = A exp(−r²/a²) could be due to his guess that the probability density of the hydrogen atom has a normal distribution.

Questions for discussion:

1. Should a precise description of an atom be described only in terms of probabilities?

2. Is Einstein’s worry about the quantum mechanical description of nature justified?  

3. How is the probability density of an undisturbed hydrogen atom derived?

The moral of the lesson: the uncertainty principle describes an inherent fuzziness that exists in any attempt to describe nature.

References:

1. Bell, J. S. (1964). On the Einstein Podolsky Rosen Paradox. Physics, 1(3), 195-200.

2. Bohm, D., & Aharonov, Y. (1957). Discussion of experimental proof for the paradox of Einstein, Rosen, and Podolsky. Physical Review, 108(4), 1070-1076.

3. Einstein, A. Podolsky, B. & Rosen, N. (1935). Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? Physical Review, 47(10), 777-780.

4. Feynman, R. P. (1982). Simulating physics with computers. International journal of theoretical physics, 21(6), 467-488.

5. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

6. Heisenberg, W. (2007/1958). Physics and Philosophy: The Revolution in Modern Science. New York: HarperCollins.