Chapter 9 The Two-Body Problem

1.9. Calculation of the Energy Eigenvalues Using the Runge-Lenz Vector
This method is a version of the elegant solution of the hydrogen atom problem found by Pauli within the framework of the matrix mechanics before the discovery of wave mechanics [16], [17].
In the classical Kepler problem the orbit of a particle attracted by a potential to the center of force close on themselves, i.e. the orbit does not precess. The reason for this is the fact that the period associated with the angular motion, i.e. the time that the polar angle changes from zero to is the same (or in general an integral multiple) of the period that the radial distance goes from its minimum value to the maximum value and back. The non-precessing orbit is also related to the conservation of the Runge-Lenz vector (see Sec. 1.10)[18]. Let which is defined by

denote the Runge-Lenz vector. This vector, , lies in the plane of motion of the particle and is directed along the semi-major axis of the ellipse. Since is constant, the major axis of the ellipse is fixed in space and cannot precess. The quantum mechanical operator form of the Runge-Lenz vector can be found by writing for and their operator forms and symmetrizing the result to get a Hermitian

Note that this is the only Hermitian operator for that we can construct. This operator commutes with either of the Hamiltonians (9.185) or (9.186) and therefore is a constant of motion,

In addition to this property the operator is (a) - perpendicular to ,

and (b) - the square of its magnitude can be expressed in terms of the Hamiltonian

For bound states of the hydrogen atom, the eigenvalues of are negative numbers. When this is the case it is more convenient to work with the vector operator rather than where

This operator, , is Hermitian provided that it acts on the bound state eigenstates. The vector does not commute with itself and we have the following commutators for its components

where is the totally antisymmetric matrix introduced earlier, with its nonzero elements given by (1.32) and (1.33). When we replace by , Eq. (9.229) becomes

In addition to (9.230) we also have the commutator of with , viz,

Equations (9.229) and (9.231) can be derived by writing for and in terms of and and use the commutators of position and momentum to simplify the result, exactly as was done for the commutators of the components of .
By solving Eq. (9.227) for , noting that and are constants of motion we obtain

Since and do not commute with each other we introduce two new operators and by

and

The commutation relations for and found from (9.35), (9.230) and (9.231) are

and

These relations show that and commute with each other and that they satisfy the commutation relations for the components of angular momentum. Using the operators and , we can diagonalize the Hamiltonian together with and since they all commute with each other. Let us denote the eigenstates of these four commuting operators by , then

and

The eigenvalues and are the same as those of discussed earlier with no restriction for the eigenvalues to be integers

and and have the same eigenvalues as those of ;

Now from Eq. (9.226) and (9.228) we find that

Using these relations we have

Therefore for the eigenstate we get

or

This result shows that the eigenvalues corresponding to the eigenstate is given by

where

Let us examine the state . This is an eigenstate of with the eigenvalue which is times an integer. However this state is not an eigenstate of because it is a linear combination of the hydrogen atom states (denoted by ) with fixed and but different values. Now we want to know about the degeneracy of the system, i.e. the number of states with different and values having the same energy. We note that the energy eigenvalue depends only on , Eq. (9.249). Thus for a fixed there are values and for fixed there are values. Altogether there are

different levels corresponding to the same energy eigenvalue

Two-Dimensional Kepler Problem - In Sec. 1.9 we considered the classical formulation of this problem. The Hamiltonian of the system has the simple form of

where . There we found that three conserved quantities are associated with this Hamiltonian, the Runge-Lenz vector and the component of angular momentum which we denoted by . Following Pauli's method we can solve the quantum mechanical version of this motion for the bound states. For the sake of simplicity we use units where . We first construct Hermitian operators for

by symmetrization method and then replace the Poisson brackets (1.206)-(1.208) by the commutators

and

Now we construct a three-dimensional vector operator ;

where . The components of satisfy the commutation relations of the angular momentum

We observe that , therefore

Next we find the square of the Runge-Lenz vector

Expanding the first term on the left-hand side of (9.261) and simplifying the result we get

Similarly we expand other terms in Eq. (9.261)

By substituting (9.262) and (9.263) in and factoring we find

or

Since , we can diagonalize and simultaneously, knowing that the eigenvalues of are where is an integer. Thus from (9.265) we obtain

Now by identifying with the principal quantum number we find the eigenvalues to be [19]

The complete normalized wave function obtained either from the factorization method or directly by solving the Schrödinger equation with the Coulomb potential in polar coordinates is given by

where and are the Laguerre polynomials.