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|1-4: Heisenberg's Way of Thinking|
At almost the same time
as the above-mentioned
Schroedinger's wave mechanics
W. K. Heisenberg
(Germany, 1901 - 76)
published a new theory
which has later been called
the matrix mechanics.
Let us consider a particle of mass m whose momentum and coordinate (position) are represented by p and q , respectively. In the classical mechanics, the coordinate q and the velocity v or the momentum p = mv are determined as a function of time t by Newtonian equation of motion. In other words, the particle moves along an orbit described by a function q (t ), and the momentum p (t ), is definite at any point on the orbit. Accordingly, the orbit and the state of motion of a particle are completely described by the momentum p and the coordinate q in the classical mechanics.
[Heisenberg's Matrix Mechanics]
The above-mentioned variables p and q in classical mechanics are ordinary variables. When they are multiplied by each other, the order of multiplication can be exchanged; i.e.,
However, in Heisenberg's theory, the result of multiplication changes depending on its order. This means that the physical quantities in Heisenberg's theory are not ordinary variables; they are matrices, well-known in mathematics. Let us denote these matrix-variables by boldface letters as p or q .
A matrix consists of many elements as
As with ordinary variables the four basic operations of arithmetic (addition, subtraction, multiplication and division) can be defined for matrices. However, the results of multiplication of p and q differ depending on the order as
Heisenberg assumed their difference to be
where i is the imaginary number unit and I is a unit matrix.
Heisenberg replaced the ordinary variables in classical mechanics with the matrix variables mentioned above and assumed the condition (4). Thereby, he obtained the correct values of frequencies and strengths of the spectra of hydrogen. These results are exactly the same as those obtained by Schroedinger's wave mechanics.
This was surprising because it appeared that there might be nothing common between these two theories, the matrix mechanics and the wave mechanics. Schroedinger has however shown that these two theories are equivalent to each other and one can be derived from the other (1926).
Nowadays, these Heisenberg's matrix mechanics and Schroedinger's wave mechanics are generically named quantum mechanics.
[Stop Drawing the Orbit
-- Abandon Common Sense !]
Looking back at what we have studied so far, we are somewhat skeptical whether the concept of the orbit of a particle may be valid or acceptable in the microscopic world.
In our common sense, a particle is considered to be a point, and the motion of the particle is represented by the position (coordinate) and the velocity (or momentum). Namely, we assume that the moving point particle draws an orbit, and the momentum of that point particle is specified at each point or position on the orbit. This is the common sense in the macroscopic world.
Whereas, in the Schroedinger wave mechanics, a particle does not only move as a particle but accompanies a wave. In the Heisenberg matrix mechanics, the variables of position and momentum are no longer ordinary variables but matrices.
Consequently, it seems that position and momentum might not necessarily be variables of unique values as our common sense suggests. Quantum mechanics seems to say, "Abandon the common sense !"
Is it allowed in the microscopic world to abandon the common understandings which have long been accepted since ancient time?
[Heisenberg's Uncertainty Principle]
As seen above, Heisenberg asserted that "the common sense might be abandoned" (1927). The basis is the following uncertainty principle. In order to explain this, let us borrow an interesting illustration used by G. Gamow (Russia, USA, 1904 - 68). (See the following figure.)
This pleasant illustration was represented in Gamow's enlightening book, "Thirty Years That Shock Physics" (Anchor/Doubleday, New York, 1966), to easily explain the uncertainty relation. As for Gamow, we will see him in the next section.
Now let us consider the thought (Gedanken) experiment shown in the figure below. ("Gedanken" is a German word.)
Gamow's Gedanken experiment
to explain the uncertainty principle
In a completely vacuous room, an electron ejected horizontally from a gun is going down vertically, being acted on by gravity. When we detect the trajectory by shining illuminating light on the electron, what happens?
Let us consider
in the above figure.
In a completely vacuous room, an electron ejected horizontally from a gun is going down vertically, being acted on by gravity.
First, let us analyze this situation with the classical theory. According to Newtonian mechanics, the electron must fall down along a parabolic orbit . To detect this orbit, we turn on the lamp sometimes and illuminate the electron. Since light applies pressure to the electron and distorts the orbit, we should weaken the strength of the light very much to minimize its impact. It is possible to weaken the strength without bound, because there is no lower limit of the strength of light in the classical theory. As a consequence, the orbit measured in this way would approach a parabola ultimately. This is just our common sense that must be proved by the measurement.
However, if we take into account that illuminating light is emitted as photons the above discussion would be incorrect. Let the frequency and the wavelength of the light be and , respectively. The light collides with the electron as a particle with a given energy and momentum . The electron is given a momentum at most. As a result, the orbit of the electron is distorted. In order to reduce this distortion, we have to make the momentum transfer as small as possible, i.e., to make the wavelength extremely large. But we cannot identify the position of the electron, if the wavelength is larger than the size of the telescope being used for observation. Namely, the uncertainty (or error) of the position of the electron is at least about the wavelength of the light, . Accordingly, the uncertainty (or error) of the momentum of the electron becomes
Consequently, we have a relation
This relation (6) is called Heisenberg's uncertainty principle.
This principle means that, if we intend to reduce the uncertainty of position, then the uncertainty of momentum becomes large, and vice versa. In other words, it is impossible to simultaneously specify precisely the values of both the coordinate position and corresponding momentum of a particle in an experiment because of the particle-wave duality of light.
We referred to uncertainty (or error) in the above discussion, but this does not imply the measurement errors like misreading of apparatuses or errors caused from poor experimental settings. Even if the apparatuses have no defect and the experiment is carried out without any error, we would have essential errors (or uncertainties) due to the particle-wave duality of light.
[The Assertion of Heisenberg and Bohr]
After doing various hypothetical (Gedanken) experiments, Heisenberg and Bohr came to the conclusion that no experiment can specify the values of the coordinate position and momentum of a particle more precisely than the limits given by the uncertainty principle (6). Hence, they asserted that we may abandon the classical common sense that the coordinate and the momentum of a particle are ordinary variables possessing precisely specified values and that a particle moves along a trajectory drawn in a curved line.
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