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Chapter 38: Problem 17

The hydrogen atom wave function \(\psi_{200}\) is zero when \(r=2 a_{0} .\) Does this mean that the electron in that state can never be observed at a distance of \(2 a_{0}\) from the nucleus or that the electron can never be observed passing through the spherical surface defined by \(r=2 a_{0}\) ? Is there a difference between those two descriptions?

Short Answer

Expert verified
Answer: The electron cannot be observed at a distance of \(2a_0\) from the nucleus in the state \(\psi_{200}\) as the probability density at that specific position is zero. However, this does not imply that the electron can never be observed passing through the spherical surface defined by \(r=2a_0\). The electron can still transition from one region to another via different paths, which might include passing through regions very close to, but not exactly at, the spherical shell \(r=2a_0\).

Step by step solution

01

Understanding the wave function

The wave function of an electron, in this case, \(\psi_{200}\), represents the probability amplitude for an electron to be found at a certain point in space. The square of the magnitude of the wave function, \(|\psi_{200}|^2\), gives the probability density of finding the electron at that position.
02

Analyzing the given wave function at position \(r=2a_0\)

We are given that the wave function at \(r=2a_0\) is zero, i.e., \(\psi_{200}(2a_0)=0\). Since \(|\psi_{200}(2a_0)|^2=0\), it implies that the probability density of finding the electron at exactly \(r=2a_0\) is zero.
03

Differentiating the two descriptions

The first description, stating that "the electron in that state can never be observed at a distance of \(2a_0\) from the nucleus", means that the electron cannot be found at the exact position \(r=2a_0\) in space. The second description, stating that "the electron can never be observed passing through the spherical surface defined by \(r=2a_0\)", means that the electron cannot be found in a region surrounding the point \(r=2a_0\) while transitioning from one region to another.
04

Deducing the correct description

Based on our analysis, we can conclude that the first description is accurate in this case. Since the probability density of finding the electron at exactly \(r=2a_0\) is zero, the electron cannot be observed at a distance of \(2a_0\) from the nucleus in state \(\psi_{200}\). As for the second description, it is not true that the electron can never be observed passing through the spherical surface defined by \(r=2a_0\). Even though it cannot be found exactly at \(r=2a_0\), it can still transition from one region to another via different spatial paths, which might include passing through regions very close to, but not exactly at, the spherical shell \(r=2a_0\).

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