88
R. Barrett and P. P. Delsanto
a gambler winning on their various poker machines. They plan their budgets
based on this knowledge. When a large number of gamblers use the machines,
the reality reflects these predictions accurately. Actuaries in insurance companies examine past records to determine the risk associated with the various
behaviours and lifestyles of their clients, and adjust the premiums they must
pay accordingly. Some of the classical laws of physics, e.g. those dealing with
the properties of gases, are statistically based. They achieve their accuracy
because of the enormous numbers of molecules involved, which make the
statistical predictions very precise indeed.
However, when dealing with individual atoms, QM does not allow an
accurate prediction of events. In a sample of radium, one cannot tell in
advance which will be the next atom to undergo radioactive decay, only how
many are expected to decay in a given time.
5.5 Heisenberg’s Uncertainty Principle
As we have just seen, if we wish to know the location of a particle, we need
to obtain its wave function by solving Schrödinger’s equation. We will not
obtain a precise location, but only a probability distribution for the particle.
If the location is known fairly accurately, this distribution will be narrow;
contrarily, if the distribution is broad, it means that our uncertainty of the
particle’s location is great.
We could apply the same methodology to determining the particle’s
velocity (or more strictly, its momentum, i.e. its mass multiplied by its speed).
Again, we would obtain a probability distribution, and a narrow one would
mean we have a fairly precise knowledge of the momentum, and a broad one
the opposite.
Heisenberg in 1927 realised that these two probability distributions, one
for the particle’s location, and the other for its momentum, are not independent of each other, but closely related. If we can locate a particle accurately, we
can measure its momentum only roughly, and vice versa. In the extreme case,
where we are able to locate the particle exactly, we can have no knowledge at
all of its momentum.
Of course, in physics we do not deal with “hand-waving” statements, such
as “accurately” or “roughly”. Heisenberg presented his principle in a mathematical form, stating that the product of the uncertainty in the particle’s
location and the uncertainty in its momentum cannot be less than /2, where
R. Barrett and P. P. Delsanto
a gambler winning on their various poker machines. They plan their budgets
based on this knowledge. When a large number of gamblers use the machines,
the reality reflects these predictions accurately. Actuaries in insurance companies examine past records to determine the risk associated with the various
behaviours and lifestyles of their clients, and adjust the premiums they must
pay accordingly. Some of the classical laws of physics, e.g. those dealing with
the properties of gases, are statistically based. They achieve their accuracy
because of the enormous numbers of molecules involved, which make the
statistical predictions very precise indeed.
However, when dealing with individual atoms, QM does not allow an
accurate prediction of events. In a sample of radium, one cannot tell in
advance which will be the next atom to undergo radioactive decay, only how
many are expected to decay in a given time.
5.5 Heisenberg’s Uncertainty Principle
As we have just seen, if we wish to know the location of a particle, we need
to obtain its wave function by solving Schrödinger’s equation. We will not
obtain a precise location, but only a probability distribution for the particle.
If the location is known fairly accurately, this distribution will be narrow;
contrarily, if the distribution is broad, it means that our uncertainty of the
particle’s location is great.
We could apply the same methodology to determining the particle’s
velocity (or more strictly, its momentum, i.e. its mass multiplied by its speed).
Again, we would obtain a probability distribution, and a narrow one would
mean we have a fairly precise knowledge of the momentum, and a broad one
the opposite.
Heisenberg in 1927 realised that these two probability distributions, one
for the particle’s location, and the other for its momentum, are not independent of each other, but closely related. If we can locate a particle accurately, we
can measure its momentum only roughly, and vice versa. In the extreme case,
where we are able to locate the particle exactly, we can have no knowledge at
all of its momentum.
Of course, in physics we do not deal with “hand-waving” statements, such
as “accurately” or “roughly”. Heisenberg presented his principle in a mathematical form, stating that the product of the uncertainty in the particle’s
location and the uncertainty in its momentum cannot be less than /2, where