The Schrodinger Equation

The Schrodinger equation is another great discovery that explains how a particle state changes over time. It was formulated in the 1920s by Erwin Schrodinger. It is one of the earliest foundations of quantum physics. I found this equation interesting to me, so I decided to included it in my blogs.

Partial Derivatives

Since Schrodinger’s equation, and many other equations in quantum physics include partial derivatives, I shall explain it here. Derivatives simply tell you the rate of change of something, or how things change over time. However, partial derivatives only differenciate the terms with the variable it’s respected to. The other variables act as constants.

Let’s take the partial derivative of $4x^3 + 3xy^2 - 4y$ with respect to $x$. The first term has $x$, so we differenciate it and get $12x^2$. We differenciate the second term, but remember $y^2$ acts as a constant, so we get $3y^2$. The last term is a constant and the derivative of a constant alone in 0. Therefore, our answer is $12x^2 + 3y^2$. If we differentiate with respect to $y$, we get $6xy - 4$.

Time-Dependent

The time-dependent Schrodinger equation, or the most common one is for describing the rate of change of the state of a particle. The wave function $\psi$ represents the state of particle. The partial derivative (represented with backward 6s) acts on the wave function. It’s with respect to time, so it finds how the state $\psi$ changes over time. The equation is usually written as:

                               $i\hbar \frac{\partial }{\partial t}\Psi (\mathbf{r},t)=\hat{H}\Psi (\mathbf{r},t)$

where $i$ is the imaginary unit, $\hbar$ is the reduced Planck’s constant, and $\hat{H}$ is the hamiltonian operator (potential energy + kinetic energy).

Time-Independent

The time-independent Schrodinger equation represents the energy of a particle, kinetic energy + potential energy. It is an eigenvalue equation, and is represented as $\hat{H} \psi = E \psi$.

Conclusion

As you can see, Schrodinger’s equation is a fascinating equation. This is a useful discovery in quantum mechanics, discribing change. However, this equation doesn’t work in the macroscopic world - Planck’s constant is extremely small, so this is only used for atoms and particles.