Since the problem is rotationally symmetric rather than square-symmetric, it makes more sense to use a coordinate system that reflects the symmetry in the problem. Hence, I'd like to solve this problem is spherical coordinates (r'--).
In spherical coordinates Schrödinger's equation reads:
Inside the well, where the potential is zero, U_{0} would be set to zero.
The stuff in the square brackets (with the overall minus sign) is the (dimensionless) angular momentum squared operator: L'^{2}. The eigenfunctions of this operator are degenerate, so there is a choice to make in defining the basis states. In physics folks conventionally use the Y_{lm}, which are simultaneously eigenfunctions of both L'^{2} and L'_{z}:
L'^{2} Y_{lm} = l(l+1) Y_{lm}
L'_{z} Y_{lm} = m Y_{lm}
I direct your attention to the H-atom and SHM, where we have discussed a bit about these Y_{lm}.
Taking a as our length scale and ^{2}/(2ma^{2}) as our energy scale, we gain the dimensionless form
Seeking a solution where the dependence on r' factors from the dependence on angles ("separation of variables", =R(r') Y_{lm}(,)), we find:
We will find it convenient to divide the above equation by E'-U_{0}', but we will need to pay attention as to the sign of (E'-U_{0}'). When (E'-U_{0}')>0 we are in a classically allowed region, and we expect the wavefunction to oscillate with a wavelength that depends on the particle's momentum... sin(kr) like solutions are to be expected. When (E'-U_{0}')<0 we are in a classically disallowed region, and we expect the wavefunction to exponentially decay...exp(-r) like solutions are to be expected. We'll handle both cases at the same time, sometimes by using ± signs where the top case (+ here) refers to a classically allowed region and the bottom case (- here) refers to a classically disallowed region. We substitute:
in our r' differential equation, producing a differential equation which I'll write in three equivalent forms:
As usual we'll have to work to solve the radial differential equation! Start by seeing how the the equation must work for large . If in the last equation we ignore the (very small) ^{-2} term, we get the familiar oscillator differential equation. For E>U_{0} we find:
That is we see the expected oscillations as a function of r', but for large r' with ever smaller amplitude, because of the 1/r factor.
For E<U_{0} we find:
That is we see the exponential behavior as a function of r' expected in a classically disallowed region.
For small , the ^{-2} term dominates and we can ignore the right hand side in comparison. If we try a power-law solution ^{n}
we find n=l.
Factoring out all the required behavior for small , we hope to find a simple power series for R:
Plugging this in to the first R equation:
Or:
The result is a two term recursion relation (i.e., the result has just two as so, for example, given a_{0} we can calculate a_{2}, from which we can calculate a_{4}, etc. until we're done.) Note that the the recursion relation connects even k to even k and since a_{0} may not be zero (as the leading term ^{l} was needed to make the small differential equation work). Thus the series will have only even terms and we write our even k=2i, where i is an integer.
The series does not terminate; the result (for E>U_{0}) can be recognized as the spherical Bessel function j_{l} from the hypergeometric formula (_{0}F_{1}).
As shown above, spherical Bessel functions are related to normal Bessel functions with half-integral index.
As we expect that a second order linear differential equation should have two independent solutions, we're only half way there. j_{l} is regular at =0, the other spherical Bessel function: y_{l} (also known as n_{l}) explodes at =0, and hence is not part of normalizable wavefunctions. Here are the basic large and small argument behaviors of j_{l} and y_{l}.
As we showed above, for large we expect oscillations like exp[±i]/. For small we expect ^{l} or ^{-l-1}, exactly as shown above. Here are some plots:
y and j are real functions and hence oscillate like sin and cos. If we want, we can put y and j together to produce something that for large oscillates like exp[±i]/. The results are related to Hankel functions and are also called spherical Bessel function of the third kind:
For E<U_{0} we've found the exponentially rising modified spherical Bessel function related to I_{l+½}:
As above there is a second solution that is exponentially falling related to K_{l+½}:
The above expression may make spherical Bessel functions look as complex as the generic Bessel functions, but in fact, the spherical Bessel functions can be expressed in terms of well-known functions (whereas a generic Bessel function can not be so easily expressed):
j_{l}(k'r')=0 for r'=1
That is k' must be a zero of j_{l}. The energy of the wavefunction can then be calculated from
E'=k'^{2}
Here is the overall solution:
Because the angular dependence is "simple" we can usefully plot the wavefunction just as a function of r'. Here are "stacked wavefunction" plots for l=0,1,2:
The red line is the classical "effective potential": l(l+1)/r'^{2} (which is the "centrifugal potential") plus the infinite square well. Note that the centrifugal barrier excludes non-zero l wavefunctions from the origin.
l,n_{r} | root | E' |
---|---|---|
0,0 | 3.1 | 9.87 |
1,0 | 4.5 | 20.19 |
2,0 | 5.8 | 33.22 |
0,1 | 6.3 | 39.48 |
3,0 | 7.0 | 48.83 |
1,1 | 7.7 | 59.68 |
4,0 | 8.2 | 66.95 |
2,1 | 9.1 | 82.72 |
5,0 | 9.4 | 87.53 |
0,2 | 9.4 | 88.83 |
You should check each of the above roots on the above plots of j_{j}.
Here is what the set of energy levels looks like: