The Schrodinger equation for a free electron of mass m and energy E written in terms of the wave function `Psi` is `(d^(2)Psi)/(dx^(2))+(8pi^(2)mE)/(h^(2))Psi=0`. The dimensions of the coefficient of `Psi` in the second term must be

Options

A. `[M^(1)L^(1)]`
B. `[L^(2)]`
C. `[L^(-2)]`
D. `[M^(1)L^(-1)T^(1)]`

Explanation

To find the dimensions of the coefficient of \( \Psi \) in the Schrödinger equation, we start with the equation given:

\[ \frac{d^2 \Psi}{dx^2} + \frac{8\pi^2 m E}{h^2} \Psi = 0 \]

We need to focus on the second term, which is:

\[ \frac{8\pi^2 m E}{h^2} \Psi \]

The coefficient of \( \Psi \) is:

\[ \frac{8\pi^2 m E}{h^2} \]

### Step 1: Identify the dimensions of each component

1. **Mass \( m \)**: The dimension of mass is given by:
\[ [m] = M \]

2. **Energy \( E \)**: The dimension of energy can be expressed as:
\[ [E] = M L^2 T^{-2} \]

3. **Planck's constant \( h \)**: The dimension of Planck's constant can be derived from the relation \( E = h \nu \), where \( \nu \) is frequency. The dimension of frequency is:
\[ [\nu] = T^{-1} \]
Thus, the dimension of \( h \) is:
\[ [h] = [E][\nu]^{-1} = (M L^2 T^{-2})(T^{-1}) = M L^2 T^{-3} \]

### Step 2: Substitute the dimensions into the coefficient

Now we substitute the dimensions into the coefficient:

\[ \frac{8\pi^2 m E}{h^2} \]

The dimension of \( h^2 \) is:
\[ [h^2] = (M L^2 T^{-3})^2 = M^2 L^4 T^{-6} \]

Now, substituting the dimensions into the coefficient:

\[ \text{Dimension of } \frac{m E}{h^2} = \frac{[m][E]}{[h^2]} = \frac{M \cdot (M L^2 T^{-2})}{M^2 L^4 T^{-6}} \]

### Step 3: Simplify the expression

Now, simplifying this expression:

\[ = \frac{M^2 L^2 T^{-2}}{M^2 L^4 T^{-6}} = \frac{L^2 T^{-2}}{L^4 T^{-6}} = L^{-2} T^{4} \]

### Final Result

Thus, the dimensions of the coefficient of \( \Psi \) in the second term of the Schrödinger equation is:

\[ [M^0 L^{-2} T^0] \quad \text{or simply} \quad L^{-2} T^{4} \]