Our goal is to obtain an overview of the essential
ingredients in QM. Here are some salient aspects:
Observations or measurements will record many discrete quantities. For
example, a detector will detect an
electron or detect no electron
but it will not detect 0.5 electrons.
The quantum nature of particles allows for a probability of
detection that can span continuously the values from 0-> 1 but
independent of the probability an actual measurement finds that an entire
electron either present or not present.
To account for the wave nature of QM amplitudes are used to describe “how
much”. For example, for a two slit experiment, one needs to know the
amplitude for the particle to pass through slit 1 and arrive at location y
on the screen. Amplitudes are used
contribution of various sequences of events or paths to a final result,
- the content of a state.
To determine the probability of an
event one squares the amplitude.
spaces: In order to incorporate the nature of QM correctly the states are
viewed as constituting a vector space.
One might notice that classically light exhibits wave
behavior. A very interesting
example of how light behaves is the impact of polarizing filters. An
interesting result is the transmission of light through two perpendicular
filters (zero transmission) when a third filter is inserted in between the
two perpendicular ones. In this case
the transmission need not be zero. Although the result might be surprising
it is understandable because of the vector nature of the electric
field. Quantum states are vectors
in Hilbert space. Some states are
fundamentally different from other states (orthogonal) a particle at
location x1 is not at all the same as a particle at location x2. However a
particle at location x1 does and a particle of definite momentum do share
or overlap properties. Both of these states may result in a particle being
detected at x1. The behavior of QS is encapsulated in the mathematics of
addition: The sum of two QS with weight factors (amplitudes) is a valid QS. You can any state A to any state B and the
result is a valid QS.
There exists sets of QS that are linearly
independent and from which any conceivable QS can be built using the
above addition rule.
product: An inner product can be introduced by adding a dual space and a
Familiar vector space of 3-d does
not typically use the notion of a dual space when introducing the inner or dot
product. However 4-d spacetime can be conveniently
cast into the above form and the minus sign for the time component, when
calculating length, can be introduced by allowing the space and its dual to have opposite signs for the t-component.
- Orthonormal: using the inner product one can impose
the condition of normality and orthogonality
for a basis.
These are somewhat formal mathematical rules but they are
essential. If one can learn to formulate
QM as vector space with amplitudes and bases, then the behavior of QS can be
extracted and a certain intuition can be built.
There are two types of idealized problems that are usually
introduced to provide examples of quantum behavior.
slit experiment: Feynman uses this example
to introduce QM. There are several great websites that discuss the ideas.
Again Feynman devotes considerable time to the topic and several
interesting web sites can be found under Stern Gerlach
Spin filters e. g.
light filters are an example of these types of filters.
Typically these two idealized systems give the students a
good introduction to the properties of quantum systems.