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?

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.

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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[ArvinAsh]

Excellent episode! Masterful explanation of a very difficult topic. Couldn't have done it better myself. I won't need to make a video on this topic, as I will just refer people to yours. Ha.

[michaelmccoll]

I think a combination of you and up and atom and 3 blue 1 brown doing a deep dive into quantum mechanics to possibly the best series ever created

[dhanashrikulkarni]

All my YouTube recommendations are videos made by Parth😁😁🤩

[genericallyentertaining]

You're brilliant. Clearest explanation of this I've seen. I love how you didn't really assume any prior knowledge and explained every step; this was just what I needed.

[v.gedace]

Studied Chemistry in the second half of the 80´s. Phyiscs and this topic was also part of it. We solved it for Hydrogen. It was really fun (no sarcasm). But still today (around + 35 years later) I could´nt get my head around the fact that time has to be eleminated / ignored in the first step. Seems to me that time does does not exists. Waiting for someone who figures out that the dimensions are curled one quantum level and that this is the reason why we experience time, the (non-force) effect of gravity, black matter and energy and strange things, like time bending, when "we go closer" to (massive) objects / black holes ... Pretty sure some one will find out that black holes are not singularities. Black holes are the oposite. Starting with the event of horiziont the dimension are so far "un-curled" (= straight) that (from our point of view) we could not "see" beyond our > 3 dimensions because at this point there are way more dimensions. So black holes aren´t holes they are the opposite, they are columns. lol . greek mythology came in my mind ... Didnt they say that the world is carried by Atlas and columns?

[farhannoor]

Phys Major here, really glad you posted this. just shows whats up ahead for me in junior year. The walk-through was good, really. Makes one excited for their 1st QM course.

[Priyanka_]

This is one of my favourite channels! Tysm😭❤️

[Lixmage]

You Sir are extremely talented. This is probably the clearest, most elegant description of solving the wave equation I have ever seen uploaded to YouTube. Congratulations!

[mcrrocks]

This is probably the most succinct and intuitive way I've heard the particle in a box explained :)

[ryanlyle]

I love how you explained the connection between quantization and the usage of radians in the mathematics. Professors and books tend to skip over those aspects assuming it's trivial. Thanks for the help!

[robertreynolds]

Hi Parth, I feel compelled to say what an absolutely BRILLIANT job you do in explaining each lesson. I have been a secondary teacher for some 40 or so years; I marvel are your style. You are a 'natural'. Whatever you do, don't change a thing.

[stevewhitt]

I have seen a lot of videos on the Schrodinger Equation, but this video brings them all full circle, 2 pi radians :) Lots of details, great job! A compliment from Mr Ash says it all.

[indrayudhdas]

I like how you don't just explain how great is Schrodingers equation and But you really give them a feel of how Schrodingers equation actually work.great job👍👍

[semmering]

Sir, you are getting better and better each lesson.. Thank you so much!

[bonelessbooks]

I’m in modern physics 2 and we deal with the Schrödinger equation all the time. This is the best and most concise explanation of the Schrödinger equation, boundary conditions, particle in a box, and quantization. Thank you!!

[fortytwo]

Beautiful explanation!! The step-by-step development of the theory is really awesome!

[lisakowalski]

You do such a great job with concepts. And I don't know anyone who communicates "the joy of physics" better. Kudos Parth!

[AnthonyDavid]

Loved your patient description of what happens "at the walls". It leads naturally to the periodic solutions.

[hedgehogchaser]

You just explained to me in 13 minutes what my prof couldn’t in two weeks. Please do a video just like this for the time-dependent Schrödinger equation.

[Agent_Foxtrot]

How the Schrodinger Equation is actually used is a concept that always escaped me until now. Thank you for a fantastic explanation as well as for not shying away from more advanced concepts like derivatives and radians. Subscribed.