Quantum Mechanics Solution Manual |
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© Leon van Dommelen |
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4.1.4.1 Solution harmd-a
Question:
Write out the ground state wave function and show that it is indeed spherically symmetric.
Answer:
Repeating an earlier exercise, taking the generic expression
and substituting 0, you get the ground state eigenfunction
Now substitute for from table 4.1:
where the constant is as given in table 4.1. You can multiply out the exponentials:
According to the Pythagorean theorem, is the distance from the origin , so
It follows that the wave function only depends on the distance from the origin, not on the angular orientation compared to it. That is the definition of spherically symmetric: it looks the same from any angle.