If the wave function $\Psi$ is spherically symmetric then the radial probability density is given by: A) 4 $\pi$r²$\Psi$ B) $\mid$$\Psi$$\mid$² C) 4 $\pi$r²$\mid$$\Psi$$\mid$² D) 4 $\pi$$\mid$$\Psi$$\mid$² E) 4 $\pi$r$\mid$$\Psi$$\mid$²

Question

Quizplus · Accepted Answer

The radial probability density is given by the equation 4πr2|R(r)|2, where R(r) is the radial part of the wave function. In this case, the wave function is spherically symmetric, which means it depends only on the magnitude of the position vector r. Therefore, the radial part of the wave function is given by R(r) = |ψ(r)|, where ψ(r) is the wave function. Substituting R(r) = |ψ(r)| into the equation for the radial probability density gives: 4πr2|R(r)|2 = 4πr2|ψ(r)|2 Therefore, the correct expression for the radial probability density is: 4πr2|ψ(r)|2 = 4πr2|F1F1F1S1F1F1F10F1F1F1S1F1F1F10F1F1F1S1F1F1F10²|2 = 4πr2|(1/sqrt(2))(²)(F1F1F1S1F1F1F10)|2 Expanding the expression inside the absolute value signs and simplifying, we get: 4πr2(1/2)|F1F1F1S1F1F1F10|2 = 4πr2(1/2)|F1F1F1S1F1F1F10F1F1F1S1F1F1F10F1F1F1S1F1F1F10²|2 The final expression matches choice C.

If the Wave Function Ψ\PsiΨ Is Spherically Symmetric Then the Radial Probability Density Is Given

If the Wave Function $\Psi$ Is Spherically Symmetric Then the Radial Probability Density Is Given