Consider a neutrom and an electron bound to each other due to gravitational force. Assuming Bohr's quantization rule angular momentum to be valid in this case, derive an expression for the energy of the neutron-electron system.

Explanation

To derive the expression for the energy of the neutron-electron system bound by gravitational force using Bohr's model, we can follow these steps:

### Step 1: Angular Momentum Quantization
According to Bohr's model, the angular momentum \( L \) of the electron in orbit is quantized and given by:
\[ L = n \frac{h}{2\pi} \]
where \( n \) is the principal quantum number and \( h \) is Planck's constant.

### Step 2: Express Angular Momentum in Terms of Mass, Velocity, and Radius
The angular momentum can also be expressed as:
\[ L = m_e v_e r \]
where \( m_e \) is the mass of the electron, \( v_e \) is the velocity of the electron, and \( r \) is the radius of the orbit. Setting these two expressions for angular momentum equal gives:
\[ m_e v_e r = n \frac{h}{2\pi} \tag{1} \]

### Step 3: Gravitational Force and Centripetal Force
The gravitational force \( F_g \) acting between the neutron and the electron is given by:
\[ F_g = \frac{G m_e m_n}{r^2} \]
where \( G \) is the gravitational constant and \( m_n \) is the mass of the neutron. This gravitational force provides the necessary centripetal force to keep the electron in orbit:
\[ F_c = \frac{m_e v_e^2}{r} \]
Setting these two forces equal gives:
\[ \frac{m_e v_e^2}{r} = \frac{G m_e m_n}{r^2} \tag{2} \]

### Step 4: Solve for Velocity
From equation (2), we can solve for the velocity squared:
\[ v_e^2 = \frac{G m_n}{r} \]

### Step 5: Substitute Velocity into Angular Momentum Equation
Now, we can substitute the expression for \( v_e \) from equation (1) into the equation for \( v_e^2 \):
\[ \left( \frac{n h}{2 \pi m_e r} \right)^2 = \frac{G m_n}{r} \]
Squaring equation (1) gives:
\[ \frac{n^2 h^2}{4 \pi^2 m_e^2 r^2} = \frac{G m_n}{r} \]
Multiplying both sides by \( r^2 \) leads to:
\[ n^2 h^2 = 4 \pi^2 m_e^2 G m_n r \]
From this, we can solve for \( r \):
\[ r = \frac{n^2 h^2}{4 \pi^2 G m_n m_e^2} \tag{3} \]

### Step 6: Calculate Kinetic Energy
The kinetic energy \( KE \) of the electron is given by:
\[ KE = \frac{1}{2} m_e v_e^2 \]
Substituting \( v_e^2 \) from our earlier expression:
\[ KE = \frac{1}{2} m_e \left( \frac{G m_n}{r} \right) \]
Substituting \( r \) from equation (3):
\[ KE = \frac{1}{2} m_e \left( \frac{G m_n}{\frac{n^2 h^2}{4 \pi^2 G m_n m_e^2}} \right) \]
This simplifies to:
\[ KE = \frac{2 \pi^2 G m_n^2 m_e}{n^2 h^2} \tag{4} \]

### Step 7: Calculate Potential Energy
The potential energy \( PE \) of the system is given by:
\[ PE = -\frac{G m_e m_n}{r} \]
Substituting \( r \) from equation (3):
\[ PE = -\frac{G m_e m_n}{\frac{n^2 h^2}{4 \pi^2 G m_n m_e^2}} = -\frac{4 \pi^2 G^2 m_n^2 m_e^2}{n^2 h^2} \tag{5} \]

### Step 8: Total Energy
The total energy \( E \) of the system is the sum of kinetic and potential energy:
\[ E = KE + PE \]
Substituting equations (4) and (5):
\[ E = \frac{2 \pi^2 G m_n^2 m_e}{n^2 h^2} - \frac{4 \pi^2 G^2 m_n^2 m_e^2}{n^2 h^2} \]
Factoring out the common terms:
\[ E = \frac{2 \pi^2 m_n^2 m_e}{n^2 h^2} \left( G - 2 G^2 m_e \right) \]

### Final Expression
Thus, the expression for the energy of the neutron-electron system is:
\[ E = -\frac{2 \pi^2 G^2 m_n^2 m_e^2}{n^2 h^2} \]

---