Q12Engineering Physics

The time-independent Schrödinger equation is the absolute fundamental governing equation of steady-state quantum mechanics, rigidly describing the spatial behavior of matter waves in a stationary potential field.

Step 1: The Classical Wave Equation

We strictly begin with the general, three-dimensional classical differential equation for a standing wave, where $\psi (x,y,z)$ represents the time-independent spatial amplitude of the wave:

Where ${\nabla }^2=\frac{{\partial }^2}{\partial x^2}+\frac{{\partial }^2}{\partial y^2}+\frac{{\partial }^2}{\partial z^2}$ is the standard Laplacian operator, and $\lambda$ is the absolute wavelength.

Step 2: Incorporating Quantum Mechanics (de Broglie Hypothesis)

To specifically adapt this strictly classical equation for a quantum particle, we systematically substitute the fundamental de Broglie wavelength relation, $\lambda =\frac{h}{p}=\frac{h}{mv}$, directly into the classical wave equation:

By rigorously introducing the reduced Planck constant $\hslash =\frac{h}{2\pi }$, the equation simplifies strictly to:

Step 3: Applying Conservation of Energy

The absolute total energy ($E$) of the particle is rigidly defined as the exact sum of its physical kinetic energy ($K$) and potential energy ($V$):

We systematically solve this energy equation strictly for the squared momentum, $p^2$:

Step 4: The Final Equation

Finally, we rigorously substitute this exact expression for $p^2$ completely back into our modified wave equation from Step 2, instantly yielding the final, definitive Time-Independent Schrödinger Wave Equation: