Page 112 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

3.11 Heisenberg uncertainty principle 103

1.2. Only the spatial dependence of Ø has been derived in equation (3.85). The
state function Ø may also depend on the time through the possible time
dependence of the parameters c, ë, hxi, and hp x i.

Energy±time uncertainty principle
We now wish to derive the energy±time uncertainty principle, which is
discussed in Section 1.5 and expressed in equation (1.45). We show in Section
1.5 that for a wave packet associated with a free particle moving in the x-
direction the product ÄEÄt is equal to the product ÄxÄp x if ÄE and Ät are
de®ned appropriately. However, this derivation does not apply to a particle in a
potential ®eld.
The position, momentum, and energy are all dynamical quantities and
consequently possess quantum-mechanical operators from which expectation
values at any given time may be determined. Time, on the other hand, has a
unique role in non-relativistic quantum theory as an independent variable;
dynamical quantities are functions of time. Thus, the `uncertainty' in time
cannot be related to a range of expectation values.
To obtain the energy-time uncertainty principle for a particle in a time-
^
^
independent potential ®eld, we set A equal to H in equation (3.81)
1 ^ ^
(ÄE)(ÄB) > jh[H, B]ij
2
^
^
where ÄE is the uncertainty in the energy as de®ned by (3.75) with A ˆ H.
Substitution of equation (3.72) into this expression gives

" dhBi
(ÄE)(ÄB) > (3:86)
2 dt
In a short period of time Ät, the change in the expectation value of B is given
by
dhBi
ÄB ˆ Ät
dt
When this expression is combined with equation (3.86), we obtain the desired
result
"
(ÄE)(Ät) > (3:87)
2
We see that the energy and time obey an uncertainty relation when Ät is
de®ned as the period of time required for the expectation value of B to change
by one standard deviation. This de®nition depends on the choice of the
dynamical variable B so that Ät is relatively larger or smaller depending on
that choice. If dhBi=dt is small so that B changes slowly with time, then the
period Ät will be long and the uncertainty in the energy will be small.