Quantum Mechanics Solution Manual |
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© Leon van Dommelen |
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4.1.4.2 Solution harmd-b
Question:
Show that the ground state wave function is maximal at the origin and, like all the other energy eigenfunctions, becomes zero at large distances from the origin.
Answer:
According to the answer to the previous question, the ground state is
where is the distance from the origin. Now according to the qualitative properties of exponentials, an exponential is one when its argument is zero, and becomes less than one when its argument becomes negative. So the maximum is at the origin 0.
The other eigenfunctions do not necessarily have their maximum magnitude at the origin: for example, the shown states and are zero at the origin.
For large negative values of its argument, an exponential becomes very small very quickly. So if the distance from the origin is large compared to , the wave function will be negligible, and it will be zero in the limit of infinite distance.
For example, if the distance from the origin is just 10 times , the exponential above is already as small as 0.000 000 000 002 which is clearly negligible.
As far as the value of the other eigenfunctions at large distance from the origin is concerned, note from table 4.1 that all eigenfunctions take the generic form
For the distance from the origin to become large, at least one of , , or must become large, and then the blow up of the corresponding exponential in the bottom makes the eigenfunctions become zero. (Whatever the polynomials in the top do is irrelevant, since an exponential includes, according to its Taylor series, always powers higher than can be found in any given polynomial, hence is much larger than any given polynomial at large values of its argument.)
It may be noted that the eigenfunctions do extend farther from the nominal position when the energy increases. The polynomials get nastier when the energy increases, but far enough away they must eventually always lose from the exponentials.