The operatorHhas eigenvalues that are bounded below.
The Hamiltonian observableHwill have a physical interpretation in terms of en-
ergy, with the boundedness condition necessary in order to assure the existence
of a stable lowest energy state.
~is a dimensional constant, called Planck’s constant, the value of which
depends on what units one uses for time and for energy. It has the dimensions
[energy]·[time] and its experimental values are
1 .054571726(47)× 10 −^34 Joule·seconds = 6.58211928(15)× 10 −^16 eV·seconds(eV is the unit of “electron-Volt”, the energy acquired by an electron moving
through a one-Volt electric potential). The most natural units to use for quan-
tum mechanical problems would be energy and time units chosen so that~= 1.
For instance one could use seconds for time and measure energies in the very
small units of 6. 6 × 10 −^16 eV, or use eV for energies, and then the very small
units of 6. 6 × 10 −^16 seconds for time. Schr ̈odinger’s equation implies that if one
is looking at a system where the typical energy scale is an eV, one’s state-vector
will be changing on the very short time scale of 6. 6 × 10 −^16 seconds. When we
do computations, usually we will set~= 1, implicitly going to a unit system
natural for quantum mechanics. After calculating a final result, appropriate
factors of~can be inserted to get answers in more conventional unit systems.
It is sometimes convenient however to carry along factors of~, since this
can help make clear which terms correspond to classical physics behavior, and
which ones are purely quantum mechanical in nature. Typically classical physics
comes about in the limit where
(energy scale)(time scale)
~is large. This is true for the energy and time scales encountered in everyday
life, but it can also always be achieved by taking~→0, and this is what will
often be referred to as the “classical limit”. One should keep in mind though
that the manner in which classical behavior emerges out of quantum theory in
such a limit can be a very complicated phenomenon.
1.2.2 Principles of measurement theory
The above axioms characterize the mathematical structure of a quantum theory,
but they don’t address the “measurement problem”. This is the question of
how to apply this structure to a physical system interacting with some sort
of macroscopic, human-scale experimental apparatus that “measures” what is
going on. This is a highly thorny issue, requiring in principle the study of two
interacting quantum systems (the one being measured, and the measurement
apparatus) in an overall state that is not just the product of the two states,
but is highly “entangled” (for the meaning of this term, see chapter 9). Since a
macroscopic apparatus will involve something like 10^23 degrees of freedom, this
question is extremely hard to analyze purely within the quantum mechanical