SOLVING the SCHRODINGER EQUATION | Quantum Physics by Parth G

How to solve the Schrodinger Equation... but what does it even mean to "solve" this equation?

In this video, I wanted to take you through the steps for solving the simplest version of the Schrodinger Equation. As we may know from my old video on this topic (https://www.youtube.com/watch?v=BFTxP03H13k&t=234s) the equation deals with a quantity known as the "wave function" of the quantum system we happen to be studying. More about the wave function in this video of mine: https://www.youtube.com/watch?v=w9Kyz5y_TPw

Solving the Schrodinger Equation just means finding the wave function of a system, given the energy of the system as well as other factors in its environment. Now the equation is the governing equation of quantum mechanics, and it basically deals with the kinetic and potential energies of the system. And the right hand side can be thought of as the total energy of the system, meaning we're basically working with "KE + PE = E_total".

The important thing here is that we are working with the "Time-Independent" Schrodinger equation. This means that the total energy of the system does NOT change with time. We're doing this because the time-independent equation is much easier to solve than the time-dependent one.

The system in question is known as a one-dimensional particle in a box. Essentially, there is a particle that can move freely in one dimension (e.g. left and right, but not up and down or in and out of the screen). However, this free movement can only happen in a particular region (say between x=0 and x=a). This is because we have placed two barriers at these x positions. The particle cannot be found at x=0 or to the left of this position, and it cannot be found at x=a or to the right of this position. It can only be found between x=0 and x=a. In other words, the potential to the left of x=0 is INFINITE! The same is true for the potential to the right of x=a. What this means is that the particle would need to have infinite potential energy in order to be found either within, or beyond, the walls.

What we are looking to do, is to find out what our wave function looks like in the nice, friendly region of space where V=0, between x=0 and x=a. This tells us something about how likely we are to find the particle at various points within this range. To do this, we substitute V=0 into the Schrodinger equation, and then rearrange it so that we've got a second order differential equation that we can solve.

Solving a differential equation is tricky, but luckily the equation we're working with has a simple solution. The wave function, psi, is a sinusoid. It must be either a sine curve or a cosine curve. Because these functions, when differentiated twice, are equal to just a constant multiplied by the original functions themselves.

Additionally, we look at the boundary conditions that the wave function must be equal to zero at both walls. This is because we should not be able to find our particle at the walls. This ends up meaning that the wave function can only look like a half sine wave, or a full sine wave, or 3/2 sine waves, and so on. Integer multiples of half sine waves, basically. Because if this were not true, then the value of the wave function at the walls would be something other than zero - which is not allowed.

When we plug in the condition that the wave function must be zero at the wall where x=a, we get an interesting constraint on the ENERGY of the particle. We find that because the wave function can only be an integer multiple of half a sine wave, the energy can also only take particular values. This is a phenomenon known as quantisation (quantization in the US). In our setup, the particle cannot have any arbitrary value of energy - it can only have specific values! And even more intriguingly, the particle must have a minimum amount of energy in order to exist in the box! This minimum energy is larger than zero, and is often known as zero point energy.

Check this out if you're unfamiliar with radians btw: https://en.wikipedia.org/wiki/Radian

There is one further thing we need to consider, which is known as normalisation (normalization in the US)... but that's for a separate video!

Timestamps:
0:00 - Introduction!
0:18 - The Schrodinger Equation - Wave Functions and Energy Terms
2:15 - Time-Independent Schrodinger Equation - The Simplest Version!
2:45 - The One-Dimensional Particle in a Box + Energy Diagrams
4:44 - Substituting Our Values into the Schrodinger Equation
5:36 - The Second Derivative of the Wave Function
6:23 - 2nd Order Differential Equation
7:28 - Boundary Conditions (At The Walls)
8:52 - Quantization of Energy
11:29 - A Physical Understanding of our Mathematical Solutions

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