FOUNDATIONS OF QUANTUM MECHANICS

IV. 5. THE UNCERTAINTY RELATIONS 103
From this inequality it follows that the probability densities of position and momentum cannot
simultaneously be made arbitrarily small, in the sense that a fraction α is concentrated on a arbitrarily
small interval. Finally, 34 years after the birth of the uncertainty principle that of which everyone
thought follows from the standard uncertainty relations was proven.
For the square wave function ψ ss (IV. 43) and its Fourier transform (IV. 44) we find
W α (Q, ψ ss ) ≃ a and W α (P, ψ ss ) ≃ , (IV. 52)
a
so that the product is in the order of magnitude of .
IV. 5. 4
TIME AND ENERGY
In the same article in which Heisenberg (1927) introduces the uncertainty relation for position
and momentum, he also discusses the uncertainty relation between time and energy, starting from the
‘well - known’ equation Et − tE = ih. This equation has caused many problems.
If t is taken to be the universal time parameter, the spectrum of the operator t must be the real axis.
But then the commutation relation can only be satisfied by an energy operator of which the spectrum
is the real axis also. On the other hand, we know that the energy spectrum of quantum mechanical
systems is generally bounded from below and can even be totally or partially discrete. Hence, the
conclusion was soon drawn that there is no time operator in quantum mechanics (Von Neumann 1932,
Pauli 1933). In the light of the existence of a position operator and with the theory of relativity in
mind it was felt that in quantum mechanics something strange was going on with ‘time’. This is
expressed in almost all textbooks and articles concerning this subject. Nevertheless, it has to do with
a conceptual confusion which has not been noticed for a remarkably long time.
As it happens, the comparison between q and t is faulty if t is understood to be a universal time
parameter. After all, q is a dynamic variable of a specific physical system, for example of a particle,
and therefore there are a lot of q’s in a multiple particle system. There is, however, only one time
parameter. This does not belong to a certain physical system but must be put on a par with the
universal position coordinates x, y, z, with which it is linked in the theory of relativity. No more
than these position coordinates, the time coordinate t is an operator in quantum mechanics. Only the
dynamic variables of physical systems can be operators, and the problem outlined above is therefore
a pseudo - problem.
Nevertheless, one can wonder if dynamic variables exist which are just as ‘timelike’, literally
speaking, as q is ‘positionlike’. The answer is affirmative. Such variables exist in systems we call
‘clocks’, think, for example, of the position or the orientation of the hand of a clock. But also very
simple, microscopic systems can have such variables. In quantum mechanics these dynamic time
variables become operators. They occur in specific systems and therefore they are not universal.
And, similar to other dynamic variables, generally the spectrum of such time operators in quantum
mechanics is not the entire real axis (see further J. Hilgevoord 2002).