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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Bohr–Einstein debates | 4/7 | https://en.wikipedia.org/wiki/Bohr–Einstein_debates | reference | science, encyclopedia | 2026-05-05T16:35:15.514856+00:00 | kb-cron |
In many textbook examples and popular discussions of quantum mechanics, the principle of indeterminacy is explained by reference to the pair of variables position and velocity (or momentum). It is important to note that the wave nature of physical processes implies that there must exist another relation of indeterminacy: that between time and energy. In order to comprehend this relation, it is convenient to refer to the experiment illustrated in Figure D, which results in the propagation of a wave which is limited in spatial extension. Assume that, as illustrated in the figure, a ray which is extremely extended longitudinally is propagated toward a screen with a slit furnished with a shutter which remains open only for a very brief interval of time
Δ
t
{\displaystyle \Delta t}
. Beyond the slit, there will be a wave of limited spatial extension which continues to propagate toward the right. A perfectly monochromatic wave (such as a musical note which cannot be divided into harmonics) has infinite spatial extent. In order to have a wave which is limited in spatial extension (which is technically called a wave packet), several waves of different frequencies must be superimposed and distributed continuously within a certain interval of frequencies around an average value, such as
ν
0
{\displaystyle \nu _{0}}
. It then happens that at a certain instant, there exists a spatial region (which moves over time) in which the contributions of the various fields of the superposition add up constructively. Nonetheless, according to a precise mathematical theorem, as we move far away from this region, the phases of the various fields, at any specified point, are distributed causally and destructive interference is produced. The region in which the wave has non-zero amplitude is therefore spatially limited. It is easy to demonstrate that, if the wave has a spatial extension equal to
Δ
x
{\displaystyle \Delta x}
(which means, in our example, that the shutter has remained open for a time
Δ
t
=
Δ
x
/
v
{\displaystyle \Delta t=\Delta x/v}
where v is the velocity of the wave), then the wave contains (or is a superposition of) various monochromatic waves whose frequencies cover an interval
Δ
ν
{\displaystyle \Delta \nu }
which satisfies the relation:
Δ
ν
≥
1
Δ
t
.
{\displaystyle \Delta \nu \geq {\frac {1}{\Delta t}}.}
Remembering that in the Planck relation, frequency and energy are proportional:
E
=
h
ν
{\displaystyle E=h\nu \,}
it follows immediately from the preceding inequality that the particle associated with the wave should possess an energy which is not perfectly defined (since different frequencies are involved in the superposition) and consequently there is indeterminacy in energy:
Δ
E
=
h
Δ
ν
≥
h
Δ
t
.
{\displaystyle \Delta E=h\,\Delta \nu \geq {\frac {h}{\Delta t}}.}
From this it follows immediately that:
Δ
E
Δ
t
≥
h
{\displaystyle \Delta E\,\Delta t\geq h}
which is the relation of indeterminacy between time and energy.
=== Einstein's second criticism ===
At the sixth Congress of Solvay in 1930, the indeterminacy relation just discussed was Einstein's target of criticism. His idea contemplates the existence of an experimental apparatus which was subsequently designed by Bohr in such a way as to emphasize the essential elements and the key points which he would use in his response. Einstein considers a box (called Einstein's box, or Einstein's light box; see figure) containing electromagnetic radiation and a clock which controls the opening of a shutter which covers a hole made in one of the walls of the box. The shutter uncovers the hole for a time
Δ
t
{\displaystyle \Delta t}
which can be chosen arbitrarily. During the opening, we are to suppose that a photon, from among those inside the box, escapes through the hole. In this way a wave of limited spatial extension has been created, following the explanation given above. In order to challenge the indeterminacy relation between time and energy, it is necessary to find a way to determine with adequate precision the energy that the photon has brought with it. At this point, Einstein turns to mass–energy equivalence of special relativity:
E
=
m
c
2
{\displaystyle E=mc^{2}}
. From this it follows that knowledge of the mass of an object provides a precise indication about its energy. The argument is therefore very simple: if one weighs the box before and after the opening of the shutter and if a certain amount of energy has escaped from the box, the box will be lighter. The variation in mass multiplied by
c
2
{\displaystyle c^{2}}
will provide precise knowledge of the energy emitted. Moreover, the clock will indicate the precise time at which the event of the particle's emission took place. Since, in principle, the mass of the box can be determined to an arbitrary degree of accuracy, the energy emitted can be determined with a precision
Δ
E
{\displaystyle \Delta E}
as accurate as one desires. Therefore, the product
Δ
E
Δ
t
{\displaystyle \Delta E\Delta t}
can be rendered less than what is implied by the principle of indeterminacy.
The idea is particularly acute and the argument seemed unassailable. It's important to consider the impact of all of these exchanges on the people involved at the time. Leon Rosenfeld, who had participated in the Congress, described the event several years later:
It was a real shock for Bohr...who, at first, could not think of a solution. For the entire evening he was extremely agitated, and he continued passing from one scientist to another, seeking to persuade them that it could not be the case, that it would have been the end of physics if Einstein were right; but he couldn't come up with any way to resolve the paradox. I will never forget the image of the two antagonists as they left the club: Einstein, with his tall and commanding figure, who walked tranquilly, with a mildly ironic smile, and Bohr who trotted along beside him, full of excitement...The morning after saw the triumph of Bohr.