Heisenberg’s Microscope
Lecture time with Werner Heisenberg.
Introduction:
Heisenberg’s Uncertainty Principle is one of the more difficult concepts to grasp in the study of quantum mechanics. When measuring the position or momentum of an object, there will always be a degree inaccuracy in either measurement. This arises not from technological underdevelopment, but from a fundamental property of all particles. Heisenberg’s Uncertainty Principle directly relates the accuracy of the position and momentum of an object in an inverse relationship.
∆x ∆p = ħ/2 (1)
Where ∆x is the range of error for the position, ∆p is the range of error for the momentum and ħ is the reduced Planck’s constant (h / 2).Therefore, as the accuracy of position increases, it decreases for momentum and vice versa. This concept however is not found in classical mechanics, where both the position and momentum of any object can be accurately measured with certainty. In order to better demonstrate this novel concept of uncertainty between position and momentum, Heisenberg developed a thought experiment, which we refer to as the Heisenberg Microscope experiment.
The Original Experiment:
Figure 1 Figure 2
Figure 1 : The electron being detected (blue) is hit by a high-energy gamma ray (green). The scattered gamma ray (red) is what is observed after the collision and the photons that reach the lens are in the range of .
Figure 2: When the aperture of the microscope is small, the smaller angle decreases the range of error for the momentum measurement and thus increases accuracy. However a microscope with a small aperture also has very poor resolving power, so increased information about the momentum is obtained by compromising information about the position.
In Heisenberg’s original experiment, he considers a theoretical microscope that can observe the exact location of an electron through the use of gamma rays (Figure 1). Gamma rays have an extremely small wavelength and thus would provide a very accurate position measurement of the electron. At the same time, gamma rays also have very high energy that would drastically alter the momentum of the electron it strikes. In order to minimize the change in the electron’s momentum, a photon with a large wavelength and low energy would have to be used. However, this limits the resolution of the image provided by the microscope.
Bohr’s Criticism of the Heisenberg Microscope:
When Heisenberg presented his idea, it was not without criticisms. The most important of these comes from Niels Bohr. Heisenberg attributed uncertainty to the “kick” the electron receives after the collision with the photon, which would cause what he referred to as a “discontinuous” change in the particle’s momentum. For this to be true, the change in momentum of the particle must be absolutely indeterminate, a point Heisenberg never directly states. Bohr argued that this is not the cause of uncertainty. While it is indeed true that the momentum the particle receives after the collision can be uncertain, it may well be that its behavior well defined and measurable. Instead, Bohr relates the indeterminacy to instrumentation and the properties of the microscope itself.
A microscope focuses light with its resolving powergiven by the following equation:
Resolving Power= / (2 sin (½ )) (2)
Where is the wavelength of light and is the angle of the lens of the microscope. For a microscope to distinguish the small details of an object well, it must have a low numerical value for resolving power. Although confusing, the term “high resolving power” is used to describe a resolving power with a low value. Light enters the microscope within its given angle of focus. A small microscope aperture would have a very small angle of focus, thus giving us a very small range within which the exact momentum of the particle lies (Figure 2). At the same time, such a small aperture would have very poor resolving power (Figure 2), limiting the accuracy of the measurement of position. The reverse is true for a large aperture. Therefore, while the uncertainty principle still stands, the uncertainty itself is a result of the inaccuracies presented by instrumentation that is used to observe the behavior of the electron.
Revised Experiment:
As the photon collides with the electron, both wave particles change their momentum. This means that it is possible that the microscope may not detect the photon. If the photon does not hit the aperture of the microscope, the viewer will not see the photon and therefore will not know that the interaction occurred. However, if the photon is visualized on the microscope, it means that there is a finite range of momentum that this photon could have (Figure 2). Therefore, the smaller the aperture is, the smaller the range of the photon’s momentum. Consequently, if an infinitely small lens were to detect a photon, we would know the exact momentum of the detected particle. However, the location of the electron still has to be considered.
In order to be able to see where the electron is located, the microscope needs to have high resolving power to be able to distinguish between the electron and its surroundings. Evaluating the equation for resolving power (Equation 2) using the wavelength of a gamma ray and an infinitely small value for the angle (which is needed to pinpoint the momentum of the photon) yields an infinitely high number. This means that the microscope is unable to resolve anything, no less an extremely small electron. As a result, we know exactly the momentum of the electron but have idea no information about its position since we are unable to differentiate it from its surroundings.
Similarly, if we make the aperture of the microscope infinitely large it makes the angle of the lens infinitely large and the resolving power extremely small. This way we can see the exact position of the electron in comparison with its surroundings. However, this also means that we have a larger range of momentum changes that we can perceive. Therefore, one can never accurately measure both position and momentum simultaneously but instead there is a tradeoff between the two as described by the uncertainty principle (Equation 1).
Discussion:
Heisenberg’s microscope uses classical concepts to describe quantum phenomena. Therefore, one must be careful when considering the Heisenberg microscope. There is no way to accurately describe the quantum nature of the uncertainty principle using a classical system such as described by the experiment. However, it is important to note that the uncertainty principle rises from a fundamental aspect of nature and not from a technicality or limited technological advancement. Therefore the Heisenberg microscope’s function is primarily as a teaching tool to illustrate the uncertainty principle. If one were to want a more thorough understanding of the uncertainty principle, one must approach the topic with more complex mathematics and quantum theory.
Luckily, in the macroscopic world, one does not have to constantly consider the effects of the uncertainty principle of every object he encounters. This task would be daunting and would severely alter the way humans understand nature. Since macroscopic objects are much more massive than electrons, the degree of indeterminacy of velocity becomes so small that it is negligible.
Resources:
Resolving Power
Cassidy, David. “Heisenberg - Quantum Mechanics 1925-1927: The Gamma-Ray Microscope.” The American Institute of Physics. 1998-2009. Web. 13 September 2009. <www.aip.org/history/heisenberg/p08b.htm>.
Hey, Tony; Walters, Patrick. The New Quantum Universe. Cambridge: Cambridge University Press, 2003.
Hughes, R.I.G. The Structure and Interpretation of Quantum Mechanics. USA, Harvard University Press, 1992, p.167-170.
Tanona, S. Studies in History and Philosophy of Modern Physics. 2004, 35, 483–507.
References:
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