momentum, and energy. Thus, the classical ideal of precise determination
of the properties of objects gave way to an interpretation in which there
is only a probability of knowing the properties. The correctness of this
interpretation was heavily debated but was accepted eventually.
Heisenberg Uncertainty Principle
Due to the wave nature of all objects, we cannot determine both the
position and momentum of a particle precisely. For example, if we try to
observe where a particle is, we must probe the particle and the probe
will move the particle in an unknown manner. Likewise, if we try to
observe what velocity the particle has, then when we probe it the location
will change. This led to a fundamental concept by Werner Heisenberg in
1927 (Nobel Prize winner in Physics in 1932) that relates the uncertainty
of the parameters according to:
(Δpx)(Δx) ≥Z/2
(ΔE)(Δt) ≥Z/2
(9.55)
where Δpx, Δx, ΔE, and Δtare the uncertainties in momentum, position,
energy, and time, respectively.
How critical is this? For a 1 g bullet moving with a small uncertainty at
a velocity of 10−^6 ms−^1 , the estimated uncertainty in position is 10−^26 m,
and so is undetectable. However, for very small objects, such as electrons,
the uncertainties become important. For example, an electron moving with
an uncertainty at a speed of 5 × 105 ms−^1 has an uncertainty in position
of 1.1 × 10 −^10 m, which is twice the Bohr radius and so is comparable to
the size of an atomic orbital. Experimentally this uncertainty becomes
evident when processes become very fast. A laser that emits light as a
fast time pulse loses its monochromatic precision as the precision in time
causes a corresponding uncertainty in energy:
(9.56)
Using optical spectroscopy, the function of proteins that contain pigments
can be followed in time by using lasers (Chapter 17). For a laboratory
experiment, extremely fast reactions are studied using lasers that can oper-
ate with a time of 0.1 fs, or 10−^16 s, yielding an uncertainty in frequency
of approximately 8 × 1014 s−^1 or equivalently an uncertainty in wavelength
of 380 nm that is comparable to the wavelength itself. By comparison,
the same laser operating in a steady-state mode would have an accuracy
in wavelength that is a small fraction of a per cent. Thus, the decrease
in the time interval results in the laser light changing from being a mono-
chromatic color such as red or green to having many colors, or effectively
becoming white.
ΔΔ
Δ
Δ
Δ
Eh
tt
=≥νν≥
π
Z
2
1
4
or
192 PART 2 QUANTUM MECHANICS AND SPECTROSCOPY
