Positronium is just like a H-atom with the proton replaced by the positively charged anti-particle of the electron (called the positron which is as massive as the electron). What would be the ground state energy of positronium ?

Options

A. `3.4eV`
B. `-5.2eV`
C. `-6.8eV`
D. `-10.2eV`

Explanation

To find the ground state energy of positronium, we can follow these steps:

### Step 1: Understand the System
Positronium is similar to a hydrogen atom, but instead of a proton, it has a positron (the antiparticle of the electron) which has the same mass as the electron.

### Step 2: Recall the Formula for Energy Levels
The energy of an electron in the nth orbit of a hydrogen-like atom is given by the formula:
\[ E_n = -\frac{m e^4}{8 \epsilon_0^2 n^2 h^2} \]
where:
- \( m \) is the reduced mass of the system,
- \( e \) is the charge of the electron,
- \( \epsilon_0 \) is the permittivity of free space,
- \( h \) is Planck's constant,
- \( n \) is the principal quantum number (for ground state, \( n = 1 \)).

### Step 3: Calculate the Reduced Mass
For positronium, both the electron and positron have the same mass \( m_e \). The reduced mass \( \mu \) is given by:
\[ \mu = \frac{m_e \cdot m_e}{m_e + m_e} = \frac{m_e^2}{2m_e} = \frac{m_e}{2} \]

### Step 4: Substitute the Reduced Mass into the Energy Formula
Now, substitute the reduced mass into the energy formula. For the ground state (\( n = 1 \)):
\[ E_1 = -\frac{\left(\frac{m_e}{2}\right) e^4}{8 \epsilon_0^2 (h^2)} \]

### Step 5: Relate to Hydrogen Atom Energy
The ground state energy of hydrogen is:
\[ E_{H} = -\frac{m_e e^4}{8 \epsilon_0^2 h^2} = -13.6 \text{ eV} \]
Thus, for positronium, since the reduced mass is half of the electron's mass:
\[ E_{Ps} = \frac{1}{2} E_{H} = \frac{1}{2} (-13.6 \text{ eV}) = -6.8 \text{ eV} \]

### Final Answer
The ground state energy of positronium is:
\[ \boxed{-6.8 \text{ eV}} \]

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