1.10. Classical Limit of Hydrogen Atom As we observed in our study of the correspondence principle, a basic question in quantum theory is the way that classical mechanics can be viewed as a limit of quantum theory. Classically the Kepler problem can be formulated as a motion in a plane containing the Runge-Lenz vector and is perpendicular to the angular momentum vector of the particle, and in this plane we have a well-defined elliptic orbit. Since in the classical limit we have a two-dimensional motion, it is convenient to start with the problem of hydrogen atom in two dimensions, the one that we solved in the preceding section.
The uncertainty obtained from Eq. (9.254) and the commutation relation is given by
To get the minimum uncertainty, i.e. satisfying the equality sign in we find the solution of the eigenvalue equation (see Eq. (4.76)) [20]
where is a real parameter and is an eigenvalue of the non-Hermitian operator . Since both and commute with , therefore we can find the eigenfunctions which diagonalize and at the same time satisfies (9.270). The solution of the eigenvalue equation (9.270) gives us the minimum uncertainty product and , which according to (4.82) and (4.83) are:
and these uncertainties satisfy (9.269) with the equality sign. To generate other eigenvalues we introduce the raising and lowering operators by
for . By applying these operators times we find
where is an integer satisfying the condition
The eigenvalue is real and is related to the mean value of the eccentricity of the elliptic orbit (see also Sec. 12.8)
We are particularly interested in the eigenstate which corresponds to the maximum eigenvalue . This eigenstate satisfies the condition
Now for large quantum numbers from equation (9.276) it follows that the eccentricity depends only on and is independent of . In fact in this limit Eq. (9.276) shows that
We can compare this result with the classical expression for the eccentricity of the orbit given in terms of the semi-major and semi-minor axes, and ;
or
in the units that we have chosen.
Figure 9.1: The time-dependent wave packet obtained from Eq. (9.284) is shown for two different times (a) for and (b) for where is the Kepler period for this orbit . The classical orbit for this motion is also shown in both (a) and (b). For this calculation the eccentricity is chosen to be and the average angular momentum is assumed to be . Other parameters used in the calculation are and [20].
Returning to the wave function we note that a general linear superposition of these states for large quantum numbers also minimizes the uncertainty (9.269) and therefore has minimal fluctuations in and . In order to solve Eqs. (9.270) and (9.277) we expand in terms of the eigenfunctions of the Coulomb Hamiltonian and the angular momentum ;
where the coefficients are given by
When is large, say around 40 , the coefficient given by (9.282) can be approximated by a Gaussian function of
The wave function (9.281) which minimizes has a spatial probability distribution which is peaked about the Kepler orbit having the eccentricity
To find the time evolution of this wave packet we superimpose the energy eigenstates with time-dependent factor , i.e.
where the coefficient is sharply peaked about a fixed principal quantum numbers . For instance we can choose
and calculate from (9.284) and then observe the wave packet at different times.
In Fig. 9.1 this wave packet is plotted at two different times. The original wave packet is shown in (a) and the wave packet after a time equal to half of the Kepler period is displayed in (b). The wave packet moves around the elliptic orbit with a period of . Writing this period in units that we are using and noting that , the period becomes . Thus the wave packet starts its motion at the perihelion, Fig. 9.1 (a), and then it slows down, contracts, and becomes steeper as it reaches aphelion Fig. 9.1 (b). As it returns to perihelion it speed up and spreads faster [20]. This motion of the wave packet is counterclockwise. For the motion in the opposite direction we can either choose to be a negative quantity or keep positive but choose the condition
instead of (9.277).
The spreading of the wave packet as it moves around the orbit is due to the initial uncertainty in the position and momentum as is required by the Heisenberg principle (Sec. 4.5). But there is an additional and important quantum interference effect which happens when the head of the wave packet catches up with its tail. This causes a nonuniform varying amplitude of the wave packet along the ellipse [20].
