The correct Schrodinger's wave equation for a electron with total energy E and potential energy V is given by:

Options

A. `(del^(2)Psi)/(delx^(2))+(del^(2)Psi)/(dely^(2))+(del^(2)Psi)/(delz^(2))+(8pi^(2))/(mh^(2))(E-V)Psi=0`
B. `(del^(2)Psi)/(delx^(2))+(del^(2)Psi)/(dely^(2))+(del^(2)Psi)/(delz^(2))+(8pim)/(h^(2))(E-V)Psi=0`
C. `(del^(2)Psi)/(delx^(2))+(del^(2)Psi)/(dely^(2))+(del^(2)Psi)/(delz^(2))+(8pi^(2)m)/(h^(2))(E-V)Psi=0`
D. None of these

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 Schrödinger wave equation is a fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes over time. It is expressed in terms of the wave function \( \psi \).

### Step 2: Write the Time-Independent Schrödinger Equation
For a particle in a potential \( V \), the time-independent Schrödinger equation can be written as:

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

where:
- \( \hbar \) is the reduced Planck's constant,
- \( m \) is the mass of the electron,
- \( \nabla^2 \) is the Laplacian operator, which in three dimensions is given by:

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

### Step 3: Rearranging the Equation
Rearranging the equation gives:

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

### Step 4: Multiply by \(-\frac{2m}{\hbar^2}\)
To express the equation in a standard form, we can multiply both sides by \(-\frac{2m}{\hbar^2}\):

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

### Step 5: Substitute \( \hbar \) with \( h \)
Since \( \hbar = \frac{h}{2\pi} \), we can express the equation in terms of \( h \):

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

### Step 6: Final Form of the Schrödinger Wave Equation
Thus, the final form of the Schrödinger wave equation for an electron with total energy \( E \) and potential energy \( V \) is:

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

### Conclusion
The correct Schrödinger wave equation is:

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

The correct option that expresses this equation is option C.

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