Consider the hydrogen atom to be a proton embededded in a cavity of radius (Bohr radius) whose charge is neutralised by the addition of an electron to the cavity in a vacuum initially slowly .Estimate the average total energy of an electron in to ground state .Also if the magnitude of the average kinetic energy is half of the magnitude of the average energy ,find the average potential energy

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the System
We have a hydrogen atom represented as a proton in a cavity of radius equal to the Bohr radius (denoted as \( a_0 \)). An electron is added to neutralize the charge of the proton.

### Step 2: Calculate Potential Energy
The potential energy (\( U \)) between the proton and the electron can be calculated using the formula:
\[
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