Shankar, R. (1994), Principles of Quantum Mechanics, Plenum Press. Chapter 12, Exercise 12.6.10.
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The radial function for a free particle can be either a spherical Bessel function or a spherical Neumann function . If the solution space includes the origin, then only is acceptable since the functions diverge as .
In rectangular coordinates, a free particle wave function has the form
where the energy is
For a free particle travelling in the direction, this becomes
Since the solutions of the free-particle Schrödinger equation in spherical coordinations form a complete set, we must be able to express this wave function as a linear combination of these solutions, so that
where the are constants. Because we’re looking at motion in the direction, there is no angular momentum about the axis, which is reflected in the fact that does not depend on . Thus and . We therefore have
The problem, of course, is to find these constants. We can do this using the identities given by Shankar in his problem 12.6.10, which are
The last line follows because is an even function and is therefore symmetric about .
We can use the standard procedure for isolating by multiplying both sides by and using 9.
This relation must be true for all values of , so we can look at the limit of small (but not zero, since both sides are then zero) . We have the asymptotic relation for the spherical Bessel functions
We thus have
We can then look at the integral on the LHS and hope that, when we expand the exponential, that the terms in for vanish. We can then match the coefficients of on both sides to find .
We can see that this will work because the Legendre polynomials are a complete set of functions, and the polynomial has degree . This means that any polynomial of degree can be written as a linear combination of the , where . Because of 9, this means that
Therefore, when we expand in a power series, we have
In the limit of small , higher order terms in the sum on the RHS can be ignored, so we get
Now consider the integral in the last line. Using 11 we have
We can integrate by parts repeatedly until the derivative in the integrand disappears. Note that the th derivative of will always contain a factor of to some power for any , and thus is zero at both limits of integration. Since the integrated term in the integration by parts always contains such a derivative, all integrated terms are zero at both limits. We therefore integrate ( times) and differentiate ( times) and keep only the residual integral after each iteration. The differentiation of ( times) introduces a factor of . Since the sign of the residual integral alternates as we perform each integration by parts, the final result is
where we used 13 in the last line. The double factorial in the numerator can be written as
We therefore have
Plugging this back into 22 we have
The wave function for a free particle moving in the direction is therefore