1. Angular Momentum
1.1. Angular momentum operators
We will have operators corresponding to angular momentum about different orthogonal axes , , and though they will not commute with each other, in contrast to the linear momentum operator , and which do commute. We will, however, find another useful angular momentum operator which does commute separately with each of , and . The eignfunctions for , , are simple. Those for , the spherical harmonics, are more complicated, but can be understood relatively simply and form the angular shapes of the hydrogen atom orbitals.
The classical angular momentum of a small object of (vector) linear momentum centered at a point given by the vector displacement relative to some origin is .
where , , are unit vectors in , and directions. is perpendicular to the plane of and . is the angle between the vector and . is unit vector in the direction of the vector . Note that, in previous equation, the ordering of the multiplications in the second line is chosen to work also for operators instead of numbers for one or other vector, the sequence of multiplications in each term is always in the sequence of the rows from top to bottom.
With classical angular momentum, we can explicitly write out the various components
Now we can propose a quantum mechanical angular momentum operator based on substituting the position and momentum operators
and similarly write out component operators
which are each Hermitian, and so, correspondingly, is the operator itself.
The operators corresponding to individual coordinate directions obey commutation relations
These individual commutation relations can be written in a more compact form
Unlike operators for position and for linear momentum, the different components of this angular momentum operator do not commute with one another. Though a particle can have simultaneously a well-defined position in both and directions or have simultaneously a well-defined momentum in both the and directions. A particle cannot in general simultaneously have a well-defined angular momentum component in more than one direction.
###1.2. Angular momentum eigenfunctions
The relation between spherical polar and Cartesian coordinates is
is called polar angle and is the azimuthal angle.
In inverse form
With these definitions of spherical polar coordinates and with standard partial derivative relations of the form
for each of the Cartesian coordinate directions, we can rewrite the angular momentum operator components in spherical polar coordinates.
From previous obtained commutators, we obtain
Using , we solve for the eigenfunctions and eigenvalues of . The eigen equation is
where is the eigenvalue to be determined. The solution of this equation is
The requirements that the wavefunction and its derivative are continuous when we return to where we started, mean that must be an integer (positive, negative or zero). Hence we find that the angular momentum around the axis is quantized with units of angular momentum of .
##2. The L Squared Operator
###2.1. Separating the L squared operator
In quantum mechanics, we also consider another operator associated with angular momentum the operator . This should be thought of as the “dot” product of with itself and is defined as
It is straightforward to show when that
when the operator is given by
which is actually and part of the Laplacian () operator in spherical polar coordinator hence the notation.
commutes with each of , , and . Of course, the choice of the direction is arbitrary. We could equally well have chosen the polar axis along the or directions. Then it would similarly be obvious that commutes with or . And the reason why commute with each of , and is that we can choose the eigenfunctions of to be the same as those of any one of , , and .
We want eigenfunctions of or, equivalently, and so the equation we hope to solve is of the form
We anticipate the answer by writing the eigenvalue in the form but it is just any arbitrary number to be determined. The notation also anticipates the final answer but it is arbitrary function to be determined.
We presume that the final eigenfunctions can be separated in the form
where only depends on and only depends on .
Substituting this form in the previous equation gives
Multiplying by and rearranging, gives
The left hand side depends on only whereas the right hand side depends only on so these must both equal a (“separation”) constant. Anticipating the answer, we choose a separation constant of where is still to be determined.
Now for equation, we have in the following form:
The solutions to an equation like this are of the form , or . We choose the exponential form , so is also a solution of the eigen equation
We expect that and its derivative are continuous, so this wavefunction must repeat every of angle , hence, must be an integer.
For equation, we have in the following form (already rearranged)
This is the associated Legendre equation whose solution are the associated Legendre function
The solutions required that and ( is integer).
The associated Legendre functions can conveniently be defined as using Rodrigue’s formula
We see that these functions have following properties:
- The highest power of the argument is always .
- The functions for a given for and are identical other than for numerical perfactors.
- Less obviously, between -1 and +1 and not including the values at those end points the function have zeros.
Putting this all together, the eigen equation is
with spherical harmonics as the eigenfunctions which, after normalization, can be written
where , where is an integer, and the eigenvalues are
As is easily verify these spherical harmonics are also eigenfunctions of the operator. Explicitly, we have the eigen equation
with eigenvalues of being .
It makes no difference to the eigenfunctions if we multiply them by a function of .
###2.2. Visualizing spherical harmonics
The lowest solution , is called “breathing” mode. The spherical shell expands and contracts periodically. For all other solutions there are one or more nodal circles on the sphere. A nodal circle is one that is unchanged in that particular oscillating mode.
Note the following rules for the spherical shell modes
- the surfaces on opposite sides of a nodal circle oscillate in opposite directions.
- the total number of nodal circles is equal to .
- the number of total nodal circles passing through the poles in , and they divide the sphere equally in the azimuthal angle .
- the remaining nodal circles are either equatorial or parallel to the equator symmetrically distributed between top and bottom halves of the sphere.
###2.3. Notations for spherical harmonics
We often use Dirac notation in writing equations associated with angular momentum. It is common to write
The spherical harmonics arises in the solution of the hydrogen atom problem. Different value of give rise to different sets of spectral lines from hydrogen identified empirically in the 19th century. Spectroscopists identified groups of lines called
- “spectral” (s)
- “principal” (p)
- “diffuse” (d) and
- “fundamental” (f)
Each of these is now identified with the specific values of . Now we also alphabetically extend to higher values: s: , p: , d: , f: , g: , h: and so on. It is convenient that the “s” wavefunctions are all spherically symmetric even though the “s” of the notation originally had nothing to do with the spherical symmetry.
##3. The Hydrogen Atom
###3.1. Multiple particle wavefunctions
We start by generalizing the Schroedinger equation, writing generally for time-dependent problems
where now we mean that the Hamiltonian is the operator representing the energy of the entire system. And is the wavefunction representing the state of the entire system.
For a hydrogen atom, there are two particles: the electron and the proton. Each of these has a set of coordinates associated with it: , and for the electron and , and for the proton. The wavefunction will therefore in general be a function of all six of these coordinates.
###3.2. Solving the hydrogen atom problem
The electron and proton each have a mass and respectively. We expected kinetic energy operators associated with each of theses masses and potential energy from the electrostatic attraction of electron and proton.
Hence, the Hamiltonian becomes
where we mean and similarly for and is the position vector of the electron coordinates and similarly for .
The Coulomb potential energy
depends on the distance between the electron and proton coordinates which is important in simplifying the solution.
The potential here is only a function of . We could choose a new set of six coordinates in which three are the relative positions , , from which we obtain
The position of the center of mass of two masses is the same as the balance point of a light-weight beam with the two masses at possible ends and so is the weighted average of the positions of the two individual masses
where is the total mass .
Now we construct the differential operators we need in terms of these coordinates with
then for the new coordinates in the direction, we have
and similarly for the and directions.
Using the standard method of changing partial derivatives to new coordinates and fully notating the variables held constant. The first derivatives in the direction become
The second derivatives become
So dropping the explicit statement of variables held constant
where is the so-called reduced mass .
The same kind of relations can be written for each of the other Cartesian directions, so if we define
We can write the Hamiltonian in a new form with center of mass coordinates
which now allows us to separate the problem.
Presume the wavefunction can be written
Substituting this form in the Schroedinger equation with the Hamiltonian above. We obtain
The left hand side depends only on and the right hand side depends only on , so both must equal a “separation” constant which we call .
Hence, we have two separated equations
The center of mass motion is the Schroedinger equation for free particle of mass with wavefunction solutions
This is the motion of the entire hydrogen atom as a particle of mass .
The relative motion equation corresponds to the “internal” relative motion of the electron and proton and will give us the internal states of the hydrogen atom.
###3.3. Informal solution for the relative motion
We presume that the hydrogen atom will have some characteristic size which is called the Bohr radius . We expect that the “average” potential energy strictly, its expectation value will therefore be
For a reasonable smooth wavefunction of size , the second spatial derivative will be
Remembering that for a mass , the kinetic energy operator is . The “average” kinetic energy will therefore be
Now, in the spirit of a “variational” calculation. We adjust the parameter to get the lowest value of the total energy. Some variational approaches can be justified rigorously as approximation for the lowest energy.
With our very simple model, the total energy is
The total energy is balance between the potential energy and the kinetic energy. For this simple model, differentiation shows that the chose of that minimizes the energy overall is
which is the standard definition of the Bohr radius. With this choice of , the corresponding total energy of the state is
We can usually define a “Rydberg” energy unit.