FOUNDATIONS OF QUANTUM MECHANICS

IV. 3. DEBATE BETWEEN EINSTEIN EN BOHR 91
Upon closure of the shutter we have, according to Einstein, a choice: either we weigh the box
again and determine how much mass has vanished so that we can, using the relation E = m c 2 ,
retrieve the energy of the escaped photon, or we open the box and read off the clock mechanism to
determine when the shutter has been opened, which enables us to predict the time of exit of the photon
and therefore its time of arrival at a remote detector. We can choose between both options long after
the photon has left.
Bohr’s answer is not entirely clear. It may be assumed that he did not understand Einstein’s
intentions correctly. 1 He explains Einstein’s objection as an attempt to refute the uncertainty relation
between energy and time; he shows that both determinations cannot possibly be made at the same
time.
Bohr reasons as follows. Assume that the box hangs in equilibrium from a spring in a gravitational
field. When in a time interval T a mass δm escapes, it receives an upward impulse F ∆t of magnitude
g δm T. (IV. 9)
We can keep T finite by, at some moment, hanging a small weight to the box to compensate for the
loss of mass. Suppose we want to determine the mass of the photon by measuring this momentum
transfer then, again, the momentum of the box at the start of the experiment must be exactly known,
δp g δm T. (IV. 10)
But now the same argument applies as used in the double slit experiment. This precise determination
of momentum is only possible if the fixation of the position of the box is given up. The box itself
must be considered a quantum mechanical object, and therefore the uncertainty relation δ pδ q h
applies to it. The position of the box is unknown with an uncertainty of magnitude
δq
g δm T
(IV. 11)
from which it follows that the gravitational potential ϕ g to which the clock is exposed is also uncertain,
δϕ g ≃ g δq
. (IV. 12)
δm T
But according to the red shift formula from the general theory of relativity (!) the pace of a clock is
influenced by the gravitational potential,
∆T
T
= δϕ g
, (IV. 13)
c2 therefore, the pace of the clock is also uncertain, and consequently the time of opening of the clock is
unknown. Under the circumstances in which we can determine the energy of the photon, we cannot
retrieve its exit time exactly.
Although Bohr seems to rebuke Einstein with his own theory, Bohr’s answer evokes, among other
things, the question whether it is appropriate that the correctness of quantum mechanics relies on
the correctness of the general theory of relativity, which is a classical theory, and is, strictly spoken,
contradictory to quantum mechanics.
1 That Einstein indeed had the intention to point out the freedom of choice is apparent in a letter to Bohr from Paul
Ehrenfest, who heard the argument from Einstein earlier.