THE UNCERTAINTY PRINCIPLE. Fredric M. Menger, Emory University, Atlanta, GA, USA
The author was once walking through a city park where an art fair was in progress. Alongside a sidewalk, a man was filling small paper cups with differently colored paint, and then randomly throwing the cups at a large canvas placed on the ground. The man, who perceived that I was a bit startled at his mode of painting, looked at me and said, “This painting reflects the uncertainty principle.” I did not bother to remind him that the uncertainty principle, often called the Heisenberg uncertainty principle, relates to one of the most famous ideas in physics. I am about to discuss below essential aspects of uncertainty.
Werner Heisenberg
To begin with it is necessary to point out that the uncertainty principle is not to be confused with the so-called “observer-effect.” An observer-effect arises when a measurement on a system cannot be made without simultaneously altering the system. For example, to “see” an electron, a photon must first interact with it, but that interaction would then change the path of the electron. In a field other than physics, an example of an observer-effect would be the change in people’s behavior when they know they are being watched. The uncertainty principle, in contrast, relates to a fundamental property of all quantum systems independent of any observer.
The uncertainty principle says that one cannot simultaneously measure precisely both the position x and the momentum p of a small particle. (Momentum is tantamount to energy or velocity). Algebraically, this is expressed in Eq. 1 where = a measure of uncertainty, and h = Planck’s constant. Thus, the more accurately
The uncertainty principle says that one cannot simultaneously measure precisely both the position x and the momentum p of a small particle. (Momentum is tantamount to energy or velocity). Algebraically, this is expressed in Eq. 1 where = a measure of uncertainty, and h = Planck’s constant. Thus, the more accurately
xp ≥ h/4 (1)
we know position x of a proton (i.e. the smaller the x), the less accurately we know its momentum p (i.e. the larger p). Uncertainties in energy E and in time t also have an inverse relationship (Eq. 2).
Et ≥ h/4 (2)
Uncertainties expressed in Eq. 1 and Eq. 2 have nothing to do with instrumental inaccuracies. Uncertainty, being a fundamental property of Nature, is independent of the quality of any experimental method.
It is indeed possible to simultaneously determine the position and velocity of a baseball or an automobile. Does this not conflict with the uncertainty principle? The answer is no because the uncertainty principle applies only to the ultra-small. As particles decrease in size, they become best described by wave behavior, instead of classical Newtonian physics, when atomic dimensions are reached. In other words, the uncertainty principle must be interpreted in terms of wave mechanics, as is attempted in the next paragraph.
Let us assume an electron is represented by a simple sine-wave. Now the momentum of the particle is mathematically related to the amplitude (height) of the sine wave. Since the amplitude of every cycle of the sine wave is identical, there is only one momentum value associated with the sine wave. In other words, we know the momentum p precisely. But the sine wave does not give a clue as to where to find the electron, i.e. its location x. This is because the square of the value at each peak gives the probability of observing the electron at that point, and that probability at each peak is the same. Thus, we have a precise p but uncertain x, and the uncertainty principle is affirmed.
Now in reality the waves associated with electrons are not pure sine waves but, rather, complex waves with a high amplitude at one point and peaks of lower amplitudes on either side of it. Clearly, each of the component peaks has a different momentum, thereby creating uncertainty in the momentum p. On the other hand, the probability that the electron appears at the central maximum of the wave packet is high, therefore allowing greater certainty in position x. Again, the uncertainty principle is affirmed.
The uncertainty principle can be used to explain many aspects of Nature. Everyone knows, for example, that atoms consist of negatively-charged electrons circulating around positively-charged nuclei. Since oppositely charged species attract one another, electrons might have been expected to crash into the nucleus rather than maintain their orbits outside the nucleus. The uncertainty principle helps explain why this does not happen. If an electron gets close to the nucleus, its position in the small volume of confined space would be known with high accuracy. This means that the error in momentum would be enormous. In fact, the electron could even have sufficient energy to escape the atom altogether.
Or take the example of a radioactive atom emitting an alpha particle consisting of two protons and two neutrons. Normally, a great deal of energy would be needed to overcome the forces that hold all the nuclear components in place. But since the alpha particle in a radioactive nucleus has a well-defined velocity, the position of the alpha particle is more uncertain. In fact, there is a small, but non-zero, chance that the alpha particle could find itself outside the nucleus. This escape is known at radioactivity.
Albert Einstein, who did not like the uncertainty principle, proposed the following “thought experiment” which has become known as “Einstein’s box.” Consider, he wrote, an ideal box lined with mirrors that contains a photon of light indefinitely. The box is weighed and, at a chosen instant, a panel in the box is opened to allow the photon to escape at a precisely known time. The box is then weighed again. The decrease in mass discloses the energy of the emitted light (using the famous Einstein equation, E = mc2 where E = energy, m = mass, and c = speed of light). One can then in principle measure, concluded Einstein, energy and time with any desired precision, thereby violating the uncertainty principle (Eq. 2).
Niels Bohr, another Nobel Prize winner and a proponent of the uncertainty principle, reportedly spent a sleepless night worrying about Einstein’s thought experiment. He finally concluded that Einstein was wrong because there will be an uncertainty in the weight measurement in which a pan on a weighing balance moves in response to gravitational effects. The exact arguments are beyond this essay, but suffice it to say that the uncertainty principle is now an accepted and important notion in modern physics.
It is indeed possible to simultaneously determine the position and velocity of a baseball or an automobile. Does this not conflict with the uncertainty principle? The answer is no because the uncertainty principle applies only to the ultra-small. As particles decrease in size, they become best described by wave behavior, instead of classical Newtonian physics, when atomic dimensions are reached. In other words, the uncertainty principle must be interpreted in terms of wave mechanics, as is attempted in the next paragraph.
Let us assume an electron is represented by a simple sine-wave. Now the momentum of the particle is mathematically related to the amplitude (height) of the sine wave. Since the amplitude of every cycle of the sine wave is identical, there is only one momentum value associated with the sine wave. In other words, we know the momentum p precisely. But the sine wave does not give a clue as to where to find the electron, i.e. its location x. This is because the square of the value at each peak gives the probability of observing the electron at that point, and that probability at each peak is the same. Thus, we have a precise p but uncertain x, and the uncertainty principle is affirmed.
Now in reality the waves associated with electrons are not pure sine waves but, rather, complex waves with a high amplitude at one point and peaks of lower amplitudes on either side of it. Clearly, each of the component peaks has a different momentum, thereby creating uncertainty in the momentum p. On the other hand, the probability that the electron appears at the central maximum of the wave packet is high, therefore allowing greater certainty in position x. Again, the uncertainty principle is affirmed.
The uncertainty principle can be used to explain many aspects of Nature. Everyone knows, for example, that atoms consist of negatively-charged electrons circulating around positively-charged nuclei. Since oppositely charged species attract one another, electrons might have been expected to crash into the nucleus rather than maintain their orbits outside the nucleus. The uncertainty principle helps explain why this does not happen. If an electron gets close to the nucleus, its position in the small volume of confined space would be known with high accuracy. This means that the error in momentum would be enormous. In fact, the electron could even have sufficient energy to escape the atom altogether.
Or take the example of a radioactive atom emitting an alpha particle consisting of two protons and two neutrons. Normally, a great deal of energy would be needed to overcome the forces that hold all the nuclear components in place. But since the alpha particle in a radioactive nucleus has a well-defined velocity, the position of the alpha particle is more uncertain. In fact, there is a small, but non-zero, chance that the alpha particle could find itself outside the nucleus. This escape is known at radioactivity.
Albert Einstein, who did not like the uncertainty principle, proposed the following “thought experiment” which has become known as “Einstein’s box.” Consider, he wrote, an ideal box lined with mirrors that contains a photon of light indefinitely. The box is weighed and, at a chosen instant, a panel in the box is opened to allow the photon to escape at a precisely known time. The box is then weighed again. The decrease in mass discloses the energy of the emitted light (using the famous Einstein equation, E = mc2 where E = energy, m = mass, and c = speed of light). One can then in principle measure, concluded Einstein, energy and time with any desired precision, thereby violating the uncertainty principle (Eq. 2).
Niels Bohr, another Nobel Prize winner and a proponent of the uncertainty principle, reportedly spent a sleepless night worrying about Einstein’s thought experiment. He finally concluded that Einstein was wrong because there will be an uncertainty in the weight measurement in which a pan on a weighing balance moves in response to gravitational effects. The exact arguments are beyond this essay, but suffice it to say that the uncertainty principle is now an accepted and important notion in modern physics.
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