Analyzing the Hydrogenic Wavefunction

The given wavefunction is $ψ = N r(6 – Zr)e^{-Zr/3} cos θ$. We need to determine which statement about this wavefunction is incorrect.

Evaluating Option 1: $ψ = 0$ in the xy-plane

• The xy-plane is defined by the angle $θ = \frac{π}{2}$.
• In this plane, $cos θ = cos(\frac{π}{2}) = 0$.
• Substituting this into the wavefunction: $ψ = N r(6 – Zr)e^{-Zr/3} \times 0 = 0$.
• Therefore, the wavefunction $ψ$ is zero in the xy-plane. This statement is correct.

Evaluating Option 2: Two radial nodes are present

• Radial nodes occur when the radial part of the wavefunction is zero, excluding $r=0$.
• The radial part of the given wavefunction is proportional to $R(r) = r(6 – Zr)e^{-Zr/3}$.
• Setting the non-exponential part to zero for $r > 0$: $r(6 – Zr) = 0$.
• This equation yields $6 – Zr = 0$, which means $r = \frac{6}{Z}$.
• There is only one value of $r$ ($r = \frac{6}{Z}$) where a radial node exists.
• Alternatively, comparing the wavefunction to the standard hydrogenic form $ψ_{n,l,m}(r, θ, φ) = R_{n,l}(r) Y_{l,m}(θ, φ)$, we identify $l=1$ (due to $cos θ$) and the radial part $r(6-Zr)e^{-Zr/3}$ corresponds to $n=3$. The number of radial nodes is $n_r = n - l - 1 = 3 - 1 - 1 = 1$.
• Therefore, the statement that two radial nodes are present is incorrect.

Evaluating Option 3: One angular node is present

• Angular nodes occur when the angular part of the wavefunction is zero.
• The angular part of the given wavefunction is proportional to $cos θ$.
• $cos θ = 0$ when $θ = \frac{π}{2}$. This corresponds to the xy-plane.
• The number of angular nodes is equal to the angular momentum quantum number, $l$.
• From the $cos θ$ term, we identify $l=1$.
• Thus, there is one angular node. This statement is correct.

Evaluating Option 4: Orbital size decreases with atomic number

• The term $e^{-Zr/3}$ in the wavefunction shows exponential decay. As the atomic number ($Z$) increases, this term decays faster with increasing $r$.
• The radial node occurs at $r = \frac{6}{Z}$. As $Z$ increases, this node shifts closer to the nucleus (smaller $r$).
• Both factors indicate that the orbital contracts, meaning its effective size decreases as the atomic number ($Z$) increases. This statement is correct.

Conclusion

Based on the analysis, the incorrect statement is that two radial nodes are present. The wavefunction has only one radial node.