How to Solve for Eigenstates: The Time-Independent Schrödinger Equation

How to Solve for Eigenstates: The Time-Independent Schrödinger Equation

If you’ve ever ventured anywhere near the field of quantum mechanics, you've heard of the “mystical” Schrödinger Equation. Today, we'll use Python to solve it!

If you’ve ever ventured anywhere near the field of quantum mechanics, you’ve almost certainly heard of the “mystical” Schrödinger Equation.

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Much like Newton’s Second Law, which describes the propagation of particles in a classical system, where every particle has definite properties like position and momentum, the Schrödinger equation allows us to understand how a quantum system evolves, where each particle is described by a single, collective wavefunction.

What’s the difference between the Time-Dependent and Time-Independent Schrodinger Equation?

Since the Time-Dependent Schrodinger Equation (TDSE) allows for standing waves, we can solve for those specifically by simplifying the TDSE into the Time-Independent Schrodinger Equation (TISE).

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If you look at the TISE, you’ll see a two main stark differences:

Firstly, there is no time-dependence (though that should have been implied by the name of the equation). Standing waves remain constant over time (except in phase), so the probability density of the particle remains constant. Thus, time at which the wavefunction is captured is irrelevant.

Secondly, instead of the partial derivative with respect to time on the left-hand side (which represents our energy operator), we have a constant energy (E), which represents the energy of the state. Therefore, the solutions to the TISE are energy eigenstates.

If you want to learn more about Quantum Mechanics before we dive in, I highly recommend checking out MIT OpenCourseWare at https://ocw.mit.edu, which has some excellent lectures on Quantum Mechanics.

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