**Problem 4:** Find the most probable radial position of an electron in the 3d state of the hydrogen atom.---*Educational Note:*Finding the most probable radial position involves determining where the electron is most likely to be found in the 3d orbital of a hydrogen atom. This involves using the radial probability density function, which is derived from the square of the radial wave function associated with the specific quantum state. Solving this involves calculus and key concepts from quantum mechanics.

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Answered: 4. Find the most probable radial…

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Normalize the wavefunction for the 2s orbital of a hydrogen electron. 3 Rn. 1 (r) = = (²/7) ² (2 − p) e - ² - a 2Zr na
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3. Suppose an electron in a hydrogen atom is in a 2p state, and the radial wavefunction is (2a.)3/2 /3a. 2ao , where a, is the Bohr radius. (a) 2-axis? What possible angles might the angular momentum vector L make with the (b) What is the most probable radius (in terms of a,) at which the electron is found? (c) What is the expectation value of r in this state? Note: xe-"dx = 120. (p) S° x*e-dx = 23.91. What is the probability of finding such an electron between a, and oo? Note:
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3. Suppose an electron in a hydrogen atom is in a 2p state, and the radial wavefunction e 2ao, where a, is the Bohr radius. 1 is (2ао)3/2 VЗа. (а) What possible angles might the angular momentum vector L make with the Z-axis? (b) What is the most probable radius (in terms of a.) at which the electron is found? (c) What is the expectation value of r in this state? Note: S xe-"dx 120. (d) What is the probability of finding such an electron between a, and ∞? Note: ° x*e-"dx = 23.91.
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4. Find the most probable radial position of an electron in the 3d state of the hydrogen atom.