Parth G | The LAST STEP in QUANTUM MECHANICAL Wave Function Calculations | Normalization of the Wave Function @ParthGChannel | Uploaded 3 years ago | Updated 5 hours ago
A wave function is meaningless unless it is normalised (or normalized, for the US lot).
In my video discussing how to solve the Schrodinger Equation (found here: https://www.youtube.com/watch?v=sPZWtZ8vt1w) we saw that the wave function for a particular system could be found by solving the Schrodinger Equation for that system and its specific environment. But towards the end of the video, I just mentioned casually that our solution needed to be multiplied by some factor, in order for it to be correct. And I stated that the reason we multiplied by this factor, was due to something known as "normalization".
In this video, we will discuss what this means, and why it's important. To understand normalization, we first need to remember that a system's wave function, when squared, gives us the probability of getting certain experimental results with our system. For example, if our system is a single electron, then we can square its wave function (technically we can find the square modulus), and then the area underneath this squared wave function graph gives us the probability of finding the electron in different regions of space.
This is an important statement, because this means that the total area underneath any wave function squared graph must be equal to 1. The total probability of finding our electron somewhere must be 100%, or 1 - since we know the electron does indeed exist. And this area cannot be greater than 1 (which suggests a greater than 100% chance of finding our electron), nor can it be less than 1 (which suggests that there is some chance we won't find the electron at all). Neither of these possibilities make any physical sense.
Therefore, when we solve the Schrodinger equation and find the wave function of any system being studied, we first need to make sure that the total area underneath the square of that wave function is equal to 1. We do this by multiplying the wave function (squared) by a constant factor - either stretching or shrinking the wave function until it's the right size for the total area to be 1.
As it turns out, the stretched wave function (i.e. the wave function we originally calculated from the Schrodinger equation, multiplied by a constant factor) is also a solution of the Schrodinger equation, but this time it also accounts for the probability reasoning given above. And so, the normalized wave function is our real solution.
Normalizing the wave function is usually the final step in the mathematical process of solving the Schrodinger equation!
With all of that being said, if you enjoyed this video then please do consider subscribing for more fun physics content. Let me know what other areas you'd like me to cover!
Thank you so much for your support with this video, please check out my socials:
Instagram - @parthvlogs
Patreon - patreon.com/parthg
Some of you have asked about the equipment I use to make these videos, so here are my Amazon Affiliate Links - I get a small commission if you click through and purchase something on Amazon :)
My camera (Canon EOS M50): https://amzn.to/3lgq8FZ
My Lens (Canon EF-M 22mm): https://amzn.to/3qMBvqD
Microphone and Stand (Fifine): https://amzn.to/2OwyWvt
A wave function is meaningless unless it is normalised (or normalized, for the US lot).
In my video discussing how to solve the Schrodinger Equation (found here: https://www.youtube.com/watch?v=sPZWtZ8vt1w) we saw that the wave function for a particular system could be found by solving the Schrodinger Equation for that system and its specific environment. But towards the end of the video, I just mentioned casually that our solution needed to be multiplied by some factor, in order for it to be correct. And I stated that the reason we multiplied by this factor, was due to something known as "normalization".
In this video, we will discuss what this means, and why it's important. To understand normalization, we first need to remember that a system's wave function, when squared, gives us the probability of getting certain experimental results with our system. For example, if our system is a single electron, then we can square its wave function (technically we can find the square modulus), and then the area underneath this squared wave function graph gives us the probability of finding the electron in different regions of space.
This is an important statement, because this means that the total area underneath any wave function squared graph must be equal to 1. The total probability of finding our electron somewhere must be 100%, or 1 - since we know the electron does indeed exist. And this area cannot be greater than 1 (which suggests a greater than 100% chance of finding our electron), nor can it be less than 1 (which suggests that there is some chance we won't find the electron at all). Neither of these possibilities make any physical sense.
Therefore, when we solve the Schrodinger equation and find the wave function of any system being studied, we first need to make sure that the total area underneath the square of that wave function is equal to 1. We do this by multiplying the wave function (squared) by a constant factor - either stretching or shrinking the wave function until it's the right size for the total area to be 1.
As it turns out, the stretched wave function (i.e. the wave function we originally calculated from the Schrodinger equation, multiplied by a constant factor) is also a solution of the Schrodinger equation, but this time it also accounts for the probability reasoning given above. And so, the normalized wave function is our real solution.
Normalizing the wave function is usually the final step in the mathematical process of solving the Schrodinger equation!
With all of that being said, if you enjoyed this video then please do consider subscribing for more fun physics content. Let me know what other areas you'd like me to cover!
Thank you so much for your support with this video, please check out my socials:
Instagram - @parthvlogs
Patreon - patreon.com/parthg
Some of you have asked about the equipment I use to make these videos, so here are my Amazon Affiliate Links - I get a small commission if you click through and purchase something on Amazon :)
My camera (Canon EOS M50): https://amzn.to/3lgq8FZ
My Lens (Canon EF-M 22mm): https://amzn.to/3qMBvqD
Microphone and Stand (Fifine): https://amzn.to/2OwyWvt
