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The time-energy uncertainty relation

Given a normalized state Ф, that is, one for which (Ф, Ф) = 1, and an observable A, then the expectation value {A) of the observable relative to that state is given by {A) = (Ф, A Ф). This is the quantum equivalent of the mean or average value of the classical variable A corresponding to A, determined by setting up the same state many times in the laboratory and measuring the variable each time.

However, an average is just one particular piece of statistical information about a system, albeit a very useful one. What is often of importance is the spread of the various observations about that expectation value. To obtain this, we define the mean square deviation A2a of the observable A relative to the state Ф as

The square root of A2a is known as the standard deviation in statistics and as the uncertainty in QM and is denoted by Aa .

Consider two observables A, B which need not commute, and a given normalized Ф. It is a standard QM exercise to show that

known as the Kennard-Heisenberg relation [Kennard, 1927]. Here [A, B] is the commutator [A, B] = AB BA of the operators A, B. Consider now the special case when A and B represent conjugate dynamical variables and satisfy the canonical commutation relation [A, B] = ih. Then we deduce AaAb ^ 1 h. In particular, if A is the position operator x for a single particle and p is its conjugate momentum operator, then we arrive at the famous Heisenberg Uncertainty relation [Heisenberg, 1927]

This result concerns the statistics of many repeated measurements, each one done once on one of an ensemble of identically prepared pure states. It says nothing about a one-off act of measurement such as a single particle decay or a single photon impact on a detector. We cannot predict any individual outcome with certainty. This uncertainty is not the result of imprecision in our measuring apparatus; neither does it reflect some unavoidable disturbance to an otherwise well-defined classical value. The uncertainty relation tells us how closely we can ever come to believing in the existence of a classical, absolute reality before any measurement is taken.

We come now to the point. We saw in Chapter 15 that in our extended phase space, the original time parameter t appeared to be conjugate to (minus) the Hamiltonian. Since QM generally quantizes conjugate pairs of classical variables by replacing them with non-commuting operators, we might be tempted to believe that in QM, time should be represented by an operator, t. We might even believe that there were quanta associated with such an operator, thereby justify the term ‘chronon’ for the fundamental quantum of time.

Unfortunately, whilst this is an appealing notion still discussed occasionally by theorists [Muga & Egusquiza, 2008], there are three plausible reasons for not believing this.

  • 1. Pauli’s theorem asserts that the physics of the known universe cannot support a time ‘operator’ conjugate to the ordinary Hamiltonian H [Pauli, 1933]. The argument is straightforward. Suppose there were such a commutation relation of the form [t, H] = —ih. With reference to our discussion of generators in Chapter 14, we interpret H as a generator of translations in t (a reasonable concept) and t as a generator of translations in energy. This latter conclusion would be in conflict with the fact that for normal Hamiltonians, energies are invariably bounded below. If they were not, then the inevitable and irreversible dissipation of energy into the expanding universe would result in the net local disappearance of SUOs with such Hamiltonians, incompatible with the stability of atoms that we observe all around us. Therefore, the proposed commutator seems incompatible with physics as we know it.
  • 2. If there were such a commutator, then we might expect a time-energy uncertainty relation analogous to (24.15), that is, AtAE ^ 1 h. Now such a relationship is admissible if we look at the way experiments are conducted carefully. In (24.14), take the operator B to be the Hamiltonian H. Then we have at initial time t = 0, AaAe ^ 1 h|(^0, [A,Н/]Ф0)|. Now

for time-independent Hamiltonians, we can show that | (Ф0, [A, Н]Ф0)| = d d

(Ф, AФ) = (A) , so assuming this is not zero, we have

dt t=0 dt t=0

The physical dimensions of the factor involving the operator A on the left- hand side of this inequality has the physical dimensions of a time, so if we

d

define At = Aa/{A) , we appear to have a time-energy uncertainty

dt t=0

relation. The interpretation of this requires some care, because ‘time’ is not an observable and is not being measured: the above inequality is telling us something about the relationship of the initial uncertainty in A to the initial rate of change of expectation value of A. In other words, it is not a statement about any time operator.

3. In Dirac’s constraint analysis discussed in Chapter 15, the set of extended phase-space variables includes what looks like the original t parameter and the Hamiltonian as conjugate variables. However, there is a constraint in the system, viz., the extended Hamiltonian Й is zero on the surface of constraints. Not only does this lead to the notorious ‘problem of time’ in quantum cosmology, but it tells us that the extended variables cannot all be observables. The time coordinate in this interpretation is really like a a gauge-dependent function, which means it is not a physical, dynamical variable: but time was never that in the first place.

 
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