In classical physics the angular momentum of a particle is
We will use this, together with the momentum operators, to
construct the corresponding quantum mechanical operators for the components
Substituting for the momentum operators
one can show by direct calculation that
As we have seen, the non-commutativity of these operators
means that in general no two components of L can be known
simultaneously with infinite precision. (The only exception is that
they can all be zero simultaneously.)
On the other hand, one shows that
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Thus each component commutes with L2, so the square
of the angular momentum and any component can be simultaneously known.
Momentum in Spherical Coordinates
We have written the operators for the components of L
in terms of cartesian coordinates. But it is much more useful to have them
expressed in spherical coordinates.
To make the direct transformation from (x,y,z) to (r,q,f)
is possible but algebraically tiresome. We will sketch the procedure for
the simplest one, which turns out to be Lz.
First we write the spherical set
as functions of the cartesian set:
Then we use the chain rule. For example
After working all this out, we look at
After a bit of algebra we find a simple
The other components of L have
mor complicated operators. For this reason, one usually chooses to treat
Lz as the one component that is precsely known.
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For L2 the answer is not so
simple, but it is important to us:
Forces and Angular Momentum
The above expression for L2
appeared earlier in the Hamiltonian for central forces.
This is easy to understand from classical
physics. The momentum of the particle can be decomposed into components
along r and perpendicular to r. We call the former the "radial
momentum" and denote it by pr; the latter (denoted by p^)
related to the angular momentum by L = rp^.
The kinetic energy then becomes
Now the TISE for this kind of situation
was found earlier to be
We see that the left side of this is
exactly the quantum mechanical equivalent of the classical energy above.
The first term on the left involves the square of the operator for radial
momentum, which is
(This is interpreted as multiplication
by r, followed by the other operations.) This operator has the same commutation
relation with r as px has with x:
Earlier we had for the angular part of
the separated solutions of the TISE:
Referring to the expression above for
L2 we see that this equation is the same as
Thus the angular part of the separated
solution of the TISE is an eigenfunction of L2. It is important
that we know what those eigenfunctions and eigenvalues are.
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Since L2 and Lz
commute, they can be known simultaneously. This means that the system can
be in a state described by a function that is an eigenfunction of both
operators. We will look for such functions.
We start with requiring the function
to be an eigenfunction of Lz, since that operator is very simple.
Now we bring in a physical requirement:
wavefunction must be single-valued.This
means that if we add 2p
to f (which brings
us to the same point in space) the wavefunction must be the same. But
is the same only if m is an integer.
of Lz are thus restricted to
This is a remarkable
new property of angular momentum in quantum mechanics: it is quantized
in units of h/2p
(as Bohr guessed). Actually, this is the property of a component of
not of the magnitude.
We thus have that the simultaneous eigenfunction of L2
and Lz has the form
We substitute this in the original eigenvalue equation for
L2 and obtain
This must be solved for P(q).
It is one form of Legendre's equation, known from the 18th century.
There are two solutions for any values
of K and m. But most of the solutions become infinite for q
either 0 or p. These
are therefore unacceptable as a part of a wavefunction.
There are well-behaved solutions, however,
for special values of K and m. They take the form of polynomials in sinq
and cosq. The special
conditions on the constants are these:
Since the eigenvalue of L2
we have found the following restrictions on the values of the angular momentum
magnitude and one of its components:
These are general properties of (orbital)
angular momentum, and have nothing to do with what system is being considered.
Note that the magnitude of L is
while the maximum value of Lz
This means that the magnitude of
L is greater than any component can be. The reason is this: If Lz
were the same as the magnitude of L, then both Lx and
Ly would have to be zero. But no two components can be simultaneously
known, so they cannot both be zero. (The only exception is when all components
are zero, i.e., L is zero.)
The eigenfunction we are seeking thus
depends on the two quantum numbers, so we denote it by
The polynomials P(q)
are called "Associated Legendre Polynomials" and are tabulated. Some are
given in the text.
This analysis give a complete account
of (orbital) angular momentum. We can use it wherever it is appropriate,
which is usually in cases where angular momentum is conserved. That is
the case for all central forces.
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We will find later that there is another
kind of angular momentum, called spin, that has nothing to do with the
particle's location or momentum, and is entirely non-classical. The rules
for the eigenvalues, however, are very similar to the ones here.