5 The Incredible Quantum Mechanics
89
is the ubiquitous Planck’s constant h divided by 2π. Here by “uncertainty”
we mean the standard deviation, 7 as it is called in statistics.
Heisenberg’s Uncertainty Principle is often explained by saying that any
attempt to measure the position of a particle will perturb the particle’s
momentum. The more accurately we try to locate the particle, the greater
is this effect. Conversely, any attempt to measure its momentum, dislodges
the particle from its original position by an unknown amount.
The pair of quantities, “position” and “momentum”, which satisfy Heisenberg’s Uncertainty Principle in this manner, are called canonical conjugates.
They are not the only such pairs that satisfy this relationship. Energy and
Time are another example, as are Angular Position and Angular Momentum. 8
The latter are important in the physics of electrons bound within the confines
of an atom.
In the everyday world where we live, the idea that one cannot know one’s
velocity and position at the same time seems strange indeed. Certainly, the
traffic police have no such qualms when they send you an infringement notice
for violation of the speed limit at a particular place and time. They are only
able to justify this action because of the extremely small value of Planck’s
constant. 9
5.6 Collapse of the Wave Function
We are now at a stage where we can explore some of the consequences of
the probabilistic nature of QM, and provide an insight into the “incredible”
in the title of this Chapter. It is time to put away the last remnants of our
common sense, and follow our fancy and mathematics, wherever they may
lead us.
Let us begin with a simple thought experiment, where we imagine shooting
a beam of particles, e.g. electrons, through a small slit. There are various ways
to construct such a slit, which need not concern us here. The important thing
is that suitable slits can be constructed without much difficulty in the laboratory. The wave-like nature of electrons means that we expect diffraction to
occur as the waves pass through the slit.
7 For a bell-shaped (or “normal”) distribution of data around a central point x, the standard deviation
σ is defined such that 68% of the data will lie between x - σ and x + σ.
8 According to the story at the start of this Chapter, in Neutrinia “money” and “time” are also a pair
of canonical conjugates.
9 In international units (m, Kg and s) the value of is approximately 10 –34 .
89
is the ubiquitous Planck’s constant h divided by 2π. Here by “uncertainty”
we mean the standard deviation, 7 as it is called in statistics.
Heisenberg’s Uncertainty Principle is often explained by saying that any
attempt to measure the position of a particle will perturb the particle’s
momentum. The more accurately we try to locate the particle, the greater
is this effect. Conversely, any attempt to measure its momentum, dislodges
the particle from its original position by an unknown amount.
The pair of quantities, “position” and “momentum”, which satisfy Heisenberg’s Uncertainty Principle in this manner, are called canonical conjugates.
They are not the only such pairs that satisfy this relationship. Energy and
Time are another example, as are Angular Position and Angular Momentum. 8
The latter are important in the physics of electrons bound within the confines
of an atom.
In the everyday world where we live, the idea that one cannot know one’s
velocity and position at the same time seems strange indeed. Certainly, the
traffic police have no such qualms when they send you an infringement notice
for violation of the speed limit at a particular place and time. They are only
able to justify this action because of the extremely small value of Planck’s
constant. 9
5.6 Collapse of the Wave Function
We are now at a stage where we can explore some of the consequences of
the probabilistic nature of QM, and provide an insight into the “incredible”
in the title of this Chapter. It is time to put away the last remnants of our
common sense, and follow our fancy and mathematics, wherever they may
lead us.
Let us begin with a simple thought experiment, where we imagine shooting
a beam of particles, e.g. electrons, through a small slit. There are various ways
to construct such a slit, which need not concern us here. The important thing
is that suitable slits can be constructed without much difficulty in the laboratory. The wave-like nature of electrons means that we expect diffraction to
occur as the waves pass through the slit.
7 For a bell-shaped (or “normal”) distribution of data around a central point x, the standard deviation
σ is defined such that 68% of the data will lie between x - σ and x + σ.
8 According to the story at the start of this Chapter, in Neutrinia “money” and “time” are also a pair
of canonical conjugates.
9 In international units (m, Kg and s) the value of is approximately 10 –34 .