Quantum Mechanics for Engineers |
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© Leon van Dommelen |
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Subsections
A.6 Constant spherical potentials
This addendum describes the solutions of the Hamiltonian eigenvalue
problem in spherical coordinates if the potential is constant.
These solutions are important for many reasons. For example, you
might want to create a simplified model for the hydrogen atom that
way. To do so, you could, for example, assume that the potential
energy has a constant negative value up to say the Bohr radius and is
zero beyond it. That is not really a very good model for the hydrogen
atom. However, it works much better for nucleons in atomic nuclei,
chapter 14.12.
The solutions in this note are also important for describing
experiments in which particles are scattered from some target,
{A.30}. And more fundamentally, they give the energy
states of definite angular momentum for particles in empty space.
A.6.1 The eigenvalue problem
The Hamiltonian eigenvalue problem is
In this note it is assumed that the potential is a constant in the
radial region of interest.
To clean the problem up a bit, take the potential energy term to the
other side, and also the coefficient of the Laplacian. That produces
where
The constant is what classical physics would take to be
the linear momentum of a particle with total energy and potential
energy .
A.6.2 The eigenfunctions
Because the potential is spherically symmetric like for the hydrogen
atom, the eigenfunctions are of similar form:
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(A.17) |
Here the are again the spherical harmonics. These
eigenfunctions have definite square angular momentum
where is an nonnegative integer. They also have definite angular
momentum in the chosen -direction equal to , where
is an integer that satisfies .
The radial functions in the eigenfunctions are
different from those of the hydrogen atom. Depending on whatever is
easiest in a given application, they can be written in two ways,
{D.16}.
The first way is as
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(A.18) |
Here and are arbitrary constants. The
functions and are called the “spherical Bessel functions” of the first and second kinds. The
are also called the “Neumann functions” and might instead be indicated by or
.
Expressions for these Bessel functions can be found in advanced
mathematical handbooks, [1]:
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(A.19) |
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(A.20) |
The spherical Bessel functions are often convenient in a region of
constant potential that includes the origin, because the Bessel
functions of the first kind give the solutions that are finite
at the origin. (To see that, note that the Taylor series of
divided by is a power series in , and that
.) In particular for small :
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(A.21) |
Here !! is one of these unfortunate notations. The second ! means
that all even factors are dropped from the factorial.
The Bessel functions of the second kind are singular at the origin and
normally do not appear if the origin is part of the considered region.
Also, the spherical Bessel functions are real for real .
However, in a region where the potential is larger than the energy
of the particles, the argument of the Bessel functions in
(A.18) will be imaginary.
The other way to write the radial functions is as
where and are called the “spherical Hankel functions.”
The spherical Hankel functions can again be found in advanced
mathematical handbooks, [1]:
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(A.22) |
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(A.23) |
The given expressions in terms of the spherical Bessel functions are
readily inverted to give the Bessel functions in terms of the Hankel
functions,
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(A.24) |
For large the spherical Hankel functions can be approximated as
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(A.25) |
This asymptotic behavior tends to make the Hankel functions more
convenient far from the origin. Exponentials are mathematically
simpler and more fundamental than the sines and cosines in the
asymptotic behavior of the Bessel functions.
A.6.3 About free space solutions
The most important case for which the energy eigenfunctions of the
previous subsection apply is for particles in empty space. They
describe energy states with definite square and angular momentum.
However, sometimes particles in empty space are better described by
states of definite linear momentum. And in some cases, like in
scattering problems, you need both types of solution. Then you also
need to convert between them.
The energy states in empty space with definite square and angular
momentum are
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(A.26) |
These states have square angular momentum and angular
momentum in the chosen -direction . They are
nonsingular at the origin.
A state that has definite linear momentum purely in
the -direction has an energy eigenfunction
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(A.27) |
This eigenfunction is not normalized, and cannot be normalized.
However, neither can the eigenfunction above be. It is
the curse of eigenfunctions in infinite empty space. An introduction
to the adjustments that must be made to deal with this is given in
chapter 7.9.
It is sometimes necessary to write a linear momentum eigenfunction of
the form (A.27) in terms of angular momentum ones of the
form (A.26). Rayleigh worked out the correct combination a
very long time ago, {D.16}:
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(A.28) |
Note that only eigenfunctions with 0 are needed.