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S4f. How real is the wave function?
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In thought experiments one often assigns a state to a single particle.
How defendable is this, and what is the meaning of the state?
In a statistical interpretation - see the section on measurements -,
this would make no sense, since there the state is a property
of the ensemble of particles generated by a given source. But then
it is difficult to visualize what happens in each single case.
Thus many people prefer the 'realistic' language of particles having
definite states. So let us discuss some of its implications.
Suppose that the particle is in the pure state represented by the wave
function psi. It is possible to give the wave function, or rather its
absolute valued squared, a geometric interpretation:
m(x)=m|psi(x)|^2
is the mass density and
e(x)=e|psi(x)|^2
the charge density.
Thus while the wave function itself has no tangible interpretation,
certain fields computable from it have.
This extends - but not quite in the obvious way - to multiparticle
systems:
For a system of several, say n particles, the wave function is
3n-dimensional psi(x_1,...,x_n), each x_i being an ordinary
3-dimensional position vector, but the correct densities are
still 3-dimensional, obtained by integration:
m(x) = sum_a m_a integral dx_1...dx_n delta(x-x_a)|psi(x_1:n)|^2,
e(x) = sum_a e_a integral dx_1...dx_n delta(x-x_a)|psi(x_1:n)|^2.
This reduces for n=1 to the above, and is consistent with the
definition of mass and charge density in quantum field theory as
m(x) = ,
e(x) = ,
where Psi_0(x) is the time component of the relevant matter field.
These formulas are the common starting point for the derivation from
first principles of the semiconductor equations in solid state physics.
It is also what chemists draw as molecular shapes, using a cutoff where
m(x) and e(x) are negligible to delineate the boundary. Indeed,
chemists use such an interpretation all the time when visualizing
molecules in terms of orbitals, and with great success. The charge
distribution of the electron cloud of a molecule is one of the
important outputs of quantum chemistry packages such as
GAUSSIAN (commercial)
http://www.gaussian.com/
MOLPRO (commercial)
http://www.molpro.net/
GAMESS (free after registration)
http://www.msg.ameslab.gov/GAMESS/pcgamess.shtml
In the ground state (but also in definite excited states),
the mass or charge distribution is spread out over an infinite region,
although it becomes negligibly small outside a tiny core region
(or, sometimes, such as in Stern-Gerlach experiments, where the
wave function is multimodal, outside a few disconnected core regions).
The infinite extension invites apparent paradox in that upon
collapse (e.g., due to hitting a detector screen), the particle
contracts from its infinite extension to a single spot. This seems
to violate the central tenet of relativity that information cannot
flow faster than the speed of light.
However, special relativity only restricts the observational
consequences of theory. Since most of the wave function of an
individual particle is unobservable, there is no contradiction.
(It is like the nonlocality in tests of Bell's inequalities.
Nonlocality is unavoidable in QM, but the observable consequences
respect the bound relativity puts on the speed of information flow.)
For example, on a TV set, one observes just 3 position degrees of
freedom of each electron reaching the screen, while - in contrast
to the case of a classical particle - the wave function
characterizing a pure state of the electron sits
in a space of functions of 3 variables, which has infinitely many
degrees of freedom. Thus one observes only a tiny little bit about
the electron's state. It is like knowing the velocity of the wind
(a 3-dimensional vector field) in the earth's atmosphere
at a single point (giving a velocity vector with 3 coordinates)!
This unobservability of most of the state causes a problem for
those who require that everything a theory is talking about is
observable. But this requirement is not satisfied anyway in current
microphysics - no one ever observed a quark, but it is generally
believed that they make up most of the matter in our universe.
Thus, while it is reasonable to require that theory has observable
consequences in agreement with Nature, it is not reasonable to
require that everything the theory talks about is observable.
Then the unobservability of most of the state of a single particle
is harmless.
On the other hand, one can probe the state of particles in detail
if one has a large ensemble of identically prepared particles
(to make sure that they have the same state). These are usually created
by a carefully calibrated source, such as a laser. Then one can
subject them to different kinds of measurements from which one can
reconstruct a reasonable approximation of the state by quantum
tomography. In theory, one can make the approximation arbitrarily good.
Similarly a particle bound to a surface in a stationary state will
be measurable repeatedly if after the measurement the particle returns
to its state (which is natural if the bound system is in equilibrium).
Therefore one can measure equilibrium properties quite accurately.
In this sense one can say that the state of a single particle is
indeed real, and objective.
Note that single particles can nowadays be routinely prepared and
studied; see, e.g.,
D. Leibfried, R. Blatt, C. Monroe, and D. Wineland,
Quantum dynamics of single trapped ions,
Reviews of Modern Physics 75 (2003), 281-324.
S.M. Reimann and M. Manninen,
Electronic structure of quantum dots,
Reviews of Modern Physics 74 (2002), 1283-1342.