The correct Schrodingers's wave equation for an electron with e as tatal energy and V as potential energy is :

Options

A. `(del^(2)psi)/(del x^(2))+(del^(2)psi)/(dely^(2))+(del^(2) psi)/(delz^(2))+(8pi^(2))/(mh^(2))(E-V)psi=0`
B. `(del^(2)psi)/(del x^(2))+(del^(2)psi)/(dely^(2))+(del^(2) psi)/(delz^(2))+(8pim)/(h^(2))(E-V)psi=0`
C. `(del^(2)psi)/(del x^(2))+(del^(2)psi)/(dely^(2))+(del^(2) psi)/(delz^(2))+(8pi^(2)m)/(h^(2))(E-V)psi=0`
D. `(del^(2)psi)/(del x^(2))+(del^(2)psi)/(dely^(2))+(del^(2) psi)/(delz^(2))+(8pim^(2))/(h)(E-V)psi=0`

Answer

To derive the correct Schrödinger wave equation for an electron with total energy \( E \) and potential energy \( V \), we can follow these steps:

### Step 1: Understand the Schrödinger Wave Equation
The time-independent Schrödinger wave equation describes how the quantum state of a physical system changes in space. The general form of the equation is:

\[ -\frac{\hbar^2}{2m} \nabla^2 \psi + V \psi = E \psi \]

Where:
- \( \hbar \) is the reduced Planck's constant,
- \( m \) is the mass of the electron,
- \( \nabla^2 \) is the Laplacian operator,
- \( \psi \) is the wave function,
- \( V \) is the potential energy,
- \( E \) is the total energy.

### Step 2: Rearranging the Equation
We can rearrange the equation to isolate the Laplacian term:

\[ \nabla^2 \psi = \frac{2m}{\hbar^2} (V - E) \psi \]

### Step 3: Expressing in Terms of the Laplacian
The Laplacian operator in three dimensions can be expressed as:

\[ \nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} \]

Substituting this into our rearranged equation gives:

\[ \frac{\partial^2 \psi}{\partial x^2} + \frac{\partial^2 \psi}{\partial y^2} + \frac{\partial^2 \psi}{\partial z^2} = \frac{2m}{\hbar^2} (V - E) \psi \]

### Step 4: Final Form of the Schrödinger Wave Equation
Multiplying through by \(-1\) gives us the correct Schrödinger wave equation:

\[ \nabla^2 \psi + \frac{2m}{\hbar^2} (E - V) \psi = 0 \]

### Step 5: Identifying the Correct Form
In the context of the problem, we can express \( \frac{2m}{\hbar^2} \) in terms of \( 8\pi^2 \) by recognizing that \( \hbar = \frac{h}{2\pi} \) and adjusting the constants accordingly. This leads to the final form:

\[ \nabla^2 \psi + \frac{8\pi^2 m}{h^2} (E - V) \psi = 0 \]

### Conclusion
Thus, the correct Schrödinger wave equation for an electron with total energy \( E \) and potential energy \( V \) is:

\[ \nabla^2 \psi + \frac{8\pi^2 m}{h^2} (E - V) \psi = 0 \]

The correct option is part C.

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