Question

2. Consider a hydrogen-like atom (atomic number Z) in the ground state (n = 1, l = 0). a. (5 points) Calculate $\sigma_r = \sqrt{\langle r^2 \rangle - \langle r \rangle^2}$. b. (5 points) Calculate $\sigma_p$ [Hint: $\langle p \rangle = 0$. Determine $\langle p^2 \rangle = 2m(\langle E \rangle - \langle U \rangle)$ where U is potential energy and E is total energy]. c. (5 points) Show that this system satisfies the Heisenberg Uncertainty Equation.

          2. Consider a hydrogen-like atom (atomic number Z) in the ground state (n = 1, l = 0).
a. (5 points) Calculate $\sigma_r = \sqrt{\langle r^2 \rangle - \langle r \rangle^2}$.
b. (5 points) Calculate $\sigma_p$ [Hint: $\langle p \rangle = 0$. Determine $\langle p^2 \rangle = 2m(\langle E \rangle - \langle U \rangle)$ where U is potential energy and E is total energy].
c. (5 points) Show that this system satisfies the Heisenberg Uncertainty Equation.

2. Consider a hydrogen-like atom (atomic number Z) in the ground state (n = 1, l = 0).
a. (5 points) Calculate = √(⟨ r^2 ⟩ - ⟨ r ⟩^2).
b. (5 points) Calculate [Hint: ⟨ p ⟩ = 0. Determine ⟨ p^2 ⟩ = 2m(⟨ E ⟩ - ⟨ U ⟩) where U is potential energy and E is total energy].
c. (5 points) Show that this system satisfies the Heisenberg Uncertainty Equation.

Added by Julio N.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Jake Knight

Step 1

To calculate the value of r2 - r, we need to find the values of r and r2. In a hydrogen-like atom, the radial wave function for the ground state (n=1, l=0) is given by: R(r) = (2/Z^3)^(1/2) * exp(-r/Z) where Z is the atomic number and r is the radial distance Show more…

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Consider a hydrogen-like atom (atomic number Z) in the ground state (n=1, l=0). a. Calculate the value of r2 - r. b. Calculate the value of Op. (Hint: p=0. Determine p^2 = 2m(E-U), where U is the potential energy and E is the total energy.) c. Show that this system satisfies the Heisenberg Uncertainty Equation.