3.1. Quantum mechanical description of particle motion
137
One would naturally ask how the particle comes to exist in one
particular state, out of a multiplicity of possible states. The answer is determined by the initial experimental condition by which
the state of the particle is prepared.
For example, we might have a case where the particle has passed
through an energy filter, so that the magnitude of the momentum
is selected to have one particular known value. Downstream from
this, the particle might be normally incident on a pair of slits
in an otherwise opaque screen, such that it must pass through
one of the slits. However it is not known which slit the particle
passes through. This initial condition defines the state function on
the front side of the screen as a constant which is independent of
transverse position. Furthermore, the state vector roughly has one
particular coefficient a j equal to one, with all other coefficients
equal to zero. We will see in the following sections that this is
a monoenergetic plane wave, and the experiment is the familiar
two-slit experiment. This represents a particular preparation of
the state, with this preparation being achieved in a controllable
manner experimentally. With this as an initial condition, the state
function and state vector then propagate through space and time,
consistent with the dynamical equation of motion (3.32).
In this example, one might allow the particle to impinge on a
phosphor screen placed further downstream. A flash of light is
emitted at the landing position of the particle. The transverse position of the particle is thus measured with high precision. The
position of a single particle at a single time does not represent a
great deal of useful information, however. It is much more useful to measure the macroscopic properties of a beam of particles.
These properties include the current, the distribution of intensity
as a function of transverse position, the distribution of intensity
as a function of angle, and the distribution of kinetic energies. In
the present example the first two properties are measured. One
can envision an alternative measurement in which the phosphor
screen is replaced by a movable detector or a spectrometer. These
would allow measurement of the intensity as a function of angle
137
One would naturally ask how the particle comes to exist in one
particular state, out of a multiplicity of possible states. The answer is determined by the initial experimental condition by which
the state of the particle is prepared.
For example, we might have a case where the particle has passed
through an energy filter, so that the magnitude of the momentum
is selected to have one particular known value. Downstream from
this, the particle might be normally incident on a pair of slits
in an otherwise opaque screen, such that it must pass through
one of the slits. However it is not known which slit the particle
passes through. This initial condition defines the state function on
the front side of the screen as a constant which is independent of
transverse position. Furthermore, the state vector roughly has one
particular coefficient a j equal to one, with all other coefficients
equal to zero. We will see in the following sections that this is
a monoenergetic plane wave, and the experiment is the familiar
two-slit experiment. This represents a particular preparation of
the state, with this preparation being achieved in a controllable
manner experimentally. With this as an initial condition, the state
function and state vector then propagate through space and time,
consistent with the dynamical equation of motion (3.32).
In this example, one might allow the particle to impinge on a
phosphor screen placed further downstream. A flash of light is
emitted at the landing position of the particle. The transverse position of the particle is thus measured with high precision. The
position of a single particle at a single time does not represent a
great deal of useful information, however. It is much more useful to measure the macroscopic properties of a beam of particles.
These properties include the current, the distribution of intensity
as a function of transverse position, the distribution of intensity
as a function of angle, and the distribution of kinetic energies. In
the present example the first two properties are measured. One
can envision an alternative measurement in which the phosphor
screen is replaced by a movable detector or a spectrometer. These
would allow measurement of the intensity as a function of angle