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Fundamental Concepts
We have already seen that the instantaneous
state of a system consisting of a single nonrelativistic particle, whose position coordinate
is , is fully specified by
a complex wavefunction . This wavefunction is interpreted
as follows. The probability of finding the particle between and
at time is given by
. This interpretation only makes sense if the wavefunction is
normalized such that

(417) 
at all times. The physical significance of this normalization requirement is
that the probability of the particle being found anywhere on the axis
must always be unity (which corresponds to certainty).
Consider a system containing nonrelativistic particles, labeled , moving in one dimension. Let and be the position coordinate
and mass, respectively, of the th particle.
By analogy with the singleparticle case, the instantaneous state of a multiparticle system is
specified by a complex wavefunction
.
The probability of finding the first particle between and ,
the second particle between and , etc., at time
is given by
.
It follows that the wavefunction must satisfy the normalization condition

(418) 
at all times, where the integration is taken over all
space.
In a singleparticle system, position is represented by the algebraic operator ,
whereas momentum is represented by the differential operator
(see Sect. 4.6). By analogy,
in a multiparticle system, the position of the th particle
is represented by the algebraic operator , whereas the corresponding
momentum is represented by the differential operator

(419) 
Since the are independent variables (i.e.,
), we conclude that the
various position and momentum operators satisfy the following commutation
relations:
Now, we know, from Sect. 4.10, that two dynamical variables
can only be (exactly) measured simultaneously if the operators which represent
them in quantum mechanics commute with one another. Thus,
it is clear, from the above commutation relations, that the only restriction
on measurement in a onedimensional multiparticle system is that it is impossible to
simultaneously measure the position and momentum of the same
particle. Note, in particular, that a knowledge of the position or momentum of a given
particle does not in any way preclude a similar knowledge for a different
particle. The commutation relations (420)(422) illustrate
an important point in quantum mechanics: namely, that operators corresponding to different degrees
of freedom of a dynamical system tend to commute with one another.
In this case, the different degrees of freedom correspond to the different
motions of the various particles making up the system.
Finally, if
is the Hamiltonian of the system then
the multiparticle wavefunction
satisfies
the usual timedependent Schrödinger equation [see Eq. (199)]

(423) 
Likewise, a multiparticle state of definite energy (i.e., an
eigenstate of the Hamiltonian with eigenvalue ) is written (see Sect. 4.12)

(424) 
where the stationary wavefunction satisfies the timeindependent
Schrödinger equation [see Eq. (296)]

(425) 
Here, is assumed not to be an explicit function of .
Next: NonInteracting Particles
Up: MultiParticle Systems
Previous: Introduction
Richard Fitzpatrick
20100720