Introducing two-particle quantum mechanics and entanglement.
We have so far only looked at the quantum mechanics of a single
particle in one dimension. That certainly provided some interesting
results and contrasts with classical mechanics. But the full glory of
quantum mechanics only becomes visible if we allow ourselves to look at
somewhat more complicated systems, with more degrees of freedom. The
system which is closest to what we have analysed so far is that of
two particles in one dimension. The mathematics is not all that
much more complicated than what we have seen so far, but the results we
get out will challenge our intuition, as we touch on the important
concept of entanglement.
14.1
Two-particle wave functions
A system of two (or three, or one hundred) particles only has one wave function.
If we have two particles in one dimension, our system is described by
two positions (and classically, two momenta). So instead of having a
wave function , we will now
need a wave function ,
and a probability density . It
is important to understand what this probability density describes. It
gives us, for any pair of positions of the first and second particle,
the probability density of finding the system in that particular
situation (state). Whereas for a single particle we had we now have a Note that this is probabilitity density is constructed
from one function of two variables. You may have
thought that a system of two particles requires two wave functions, one
for the first and one for the second particle. But that is not how
things work. You have one wave function, which maps any point
in the space of classical configurations (with points labelled by ) to a single complex
number.
If you are only interested in the probability density of one of the
particles, we need to integrate the density over the position of the
other. So we can write and similar for . Intuitively this should make
sense: if you do not care about where particle 2 is located, you need to
‘collect’ all situations which lead to particle 1 being at position
.
By analogy with the one-particle situation, you will not be surprised
to learn that if you measure both the positions of particle 1
and particle 2 to be
and , the wave function
collapses to the product of position eigenstates of the two particles,
What happens if you decide to only
measure the position of, say, particle 1, but not measure particle 2?
Well, in that case the wave function collapses according to That is to say, the wave function now is a
‘slice’ of the original wave function, taken at the position where we
found particle 1.
14.2
Hamiltonian eigenfunctions in a box
The eigenfunctions of a the Hamiltonian of two free particles are simply products of single-particle eigenfunctions; these are separable.
To keep things concrete, let us now assume that our two particles are
put in a box of size , so that the
positions and satisfy and . We will also assume that
the potential vanishes. The Hamiltonain for two free, or
non-interacting, particles is simply the sum of two single-particle
Hamiltonians. If they have equal masses, then we have It
is therefore easy to find eigenfunctions: they are simply products of
single-particle eigenfunctions. So is a unit-normalised
eigenfunction of for
any two integers and . The time-dependence can be found
easily by using our knowledge of stationary states. We simply need to
find the eigenvalue of this wave function, and then the time-dependence
is a simple factor . The Hamiltonian acting
on the wave function above gives and the energy eigenvalue
is simply the sum of the eigenvalues of the individual particle wave
functions.
A wave function of the type is called ‘separable’,
as it separates into a product of an -dependent function and an -dependent function.
14.3
Non-separable wave functions: entanglement
Non-separable wave functions describe entangled particles, for which measurement of the position of one particle influences the subsequent measurement of the position of the other.
Interesting things happen when we add two basis functions together (remember,
Schrödinger’s equation is linear, so we can do that). An example is
This is no
longer a separable wave function; you cannot write it as the product of
one function of only and
another one of only . The
probability density is plotted in the figure below.
A separable wave-function on the left,
versus a non-separable wave function on the right, for the two-particle
system. Lighter colours indicate larger probability density . The horizontal axis
corresponds to , the vertical to
.
The probability density of the position of particle 1 is obtained as
above by integrating over . This
computation gives However, if you first
measure the position of particle to be , the wave function collapses
to . In this case the probability density is The probability density integrates, for every
value of the density along a
vertical line in the plot. The density , on the other hand,
simply takes a horizontal slice through the plot. These clearly do not
have to agree. To make this concrete, the density for our example state
before the
measurement is On the other hand, if we measure, for example, the
position of particle 2 to be , then the density for the other
particle after that measurement will be This clearly is not the same.
Probability density before (solid curve) and after
(dashed curve) the measurement of , for the example wave function
used in this section ( for
convenience).
We thus see that the measurement of one particule influences
the subsequent measurement of the other particle. Quantum states with
that property are called entangled states. Non-separable wave
functions thus describe entangled particles.
Now that you have seen how one-particle wave functions can be used to
build two-particle wave functions, you can of course apply knowledge
from previous chapters to construct more interesting two-particle
states. One useful example is the combination of Gaussian wave
functions. A Gaussian for two particles is given by This is clearly
a separable state. It can be considered an initial wave function, and
its time evolution then follows by using the results computed in the
previous chapter. More complicated wave functions can be obtained by
linear superposition. A separable and a non-separable example are given
in the figure below.
A separable and a non-separable
two-particle wave function, built by multiplying and adding
single-particle Gaussian wave functions.
More complicated things happen when we consider interacting
particles, that is, systems for which . Needless to say,
solving the Schrödinger equation for such systems is even more
complicated than for a single particle with a non-zero potential, and
this almost always requires numerical techniques. This goes beyond the
scope of the current module. We may touch on these briefly in a problem
session later.
For further reading on the topic in this chapter, see Jon J. V. Maestri, Rubin H. Landau, and Manuel J. Páez.
Two-particle schrödinger equation animations of wave packet–wave packet scattering.
American Journal of Physics, 68(12):1113–1119, 2000.
URL: http://dx.doi.org/10.1119/1.1286310, doi:10.1119/1.1286310.,
and also see Schroeder’s book.
14.4Problems
Verify the normalisation constant for the example wave function
used in the notes,
Solution ▶To avoid having to do integrals,
write this function in terms of single-particle wave functions, We can now use orthonormality of the to compute the norm of the wave
function, where the cross-terms drop out because of
orthonormality of the single-particle wave functions. The wave function
is thus correctly normalised.
How does the wave function in the previous problem evolve in
time?
Solution ▶The two terms in the wave function
are each eigenfunctions of the Hamiltonian, so we know their time
evolution is simply multiplication with with the energy eigenvalue for each term. So
we get with the energy eigenvalues