 I hope that in the next 15 minutes or so, we will sort of start to see light at the end of the tunnel. We will get to understand what this wave function is all about, what can it tell us, what kind of information it can give us and what are the properties that it has to satisfy other than simply being a solution of a differential equation. What else, what is the physical significance and all that thanks to the work of this brilliant physicist Max Born. We are going to talk about Born Interpretation of this enigma of a wave function. So, Born Interpretation essentially says this. In classical wave equation remember Schrodinger equation started off as a classical wave equation of for de Broglie waves. In classical wave equation the mod square of psi denotes intensity of the wave. If you say psi can be imaginary also. So, if you multiply psi by its complex conjugate then you get mod psi square which is the intensity this was known. So, Born started thinking along this lines that we have this wave function which can be complex can be real all right. It is great that all possible information can be derived from psi but what is the meaning of intensity here? We are saying that this psi is the amplitude of matter wave. So, square of intensity of matter wave what does it mean? Intensity of matter wave Born realized would be the something to do with probability what does that mean? I look at some kind of a matter wave. In some region I see lots of intensity. In some region I see less intensity that essentially means that in the first region it is more probable to find the particle. In the other regions probability of finding the particle is less. So, intensity is related to probability and what I said was that this mod square of psi is really the probability density which is written as rho of x. What is the meaning of probability density? See it is at a point right psi is defined at some value of x, y, z or some other coordinate. So, you cannot really talk about probability at a point you have to talk about a region. So, if you are working in one direction we can say that we can talk about probability of finding the particle between x and x plus dx where dx is a very small increase increment in length. So, Born said that if you multiply mod psi square by dx then you get the probability of finding the particle between x and x plus dx. So, what is dx that is also important and you will see this is going to become most very strongly manifested when we talk about hydrogen atom wave functions later on. So, Born interpretation essentially says that mod psi square is the probability density and that is what makes life a lot simpler. So, now we realize that we do not have to worry about matter waves forget matter waves now we are worried about what we can call probability waves. And at this point let me stop for a moment and pay my gratitude and respect to our respected teacher Professor Balaichand Kundo who we unfortunately lost last year Professor Kundo was our teacher he was an inorganic chemist he taught us this portion when I was in college Presidency College Calcutta many many years ago and what I say now is merely a repetition of what he had told us I do not know whether it is written in as many words in any book. So, with my deepest respects to Professor Balaichand Kundo what we have now are probability waves right away the situation becomes much less disturbing I mean we can breathe easy we do not have to worry about why we are not seeing matter all around we are why we are not seeing matter wave all around us probability waves. Okay what about three dimensions in three dimensions probability of finding a particle at any point defined by xyz will be psi at xyz and t mod square of that multiplied by d tau where d tau is the volume of the small element that we get in Cartesian coordinates we get it by increasing x to x plus dx y to y plus dy and z to d plus dz. When we talk about spherical polar coordinates later we will see we get a little more interesting expression but the crux of the matter is psi tells us or rather psi psi star psi star psi is what theorists like to call it psi star psi is your probability density please do not forget the density part it is going to be very important later on. Now what you can do is you can find out things like average values because from classical treatment of mathematics the mean value is defined as integral when I say minus infinity to plus infinity I essentially mean the entire range of x when you integrate over the entire range of x then the function the average value of the function is whatever is the population distribution of probability distribution multiplied by the function you want to work with that gives us the average value of that function. So, average value of x is integral minus infinity to plus infinity x multiplied by p of x integrated over dx of course and average value of x square is x square multiplied by p of x where p of x is the distribution function how is the population distributed you can say or how is what is the probability distribution x square multiplied by the distribution function integrated over the entire range of x. So, that is exactly what we have written here what is the average value of a to know that first of all how do you find a you have to make a hat operate on psi. So, then in order to find the average you have to multiply by psi star integrate over all space. So, that is how we get the average value of any quantity that we get we are going to work out a at least one problem maybe in the next module while revising that is where this meaning of this average value and later on most probable value these will become much clearer I hope. So, this is how remember we calculate average value now what we have written here is the expression for well let us let us take a check on that we will come back and say it later. Next thing that we want to discuss is normalization of a function. So, if psi star psi is probability density when you integrate over all the entire range of your coordinates what should you get the particle has to be somewhere or the other right. So, integral of psi star psi over dx dy dz if you are working in Cartesian space is equal to 1 this condition is called the normalization condition. So, obviously divergent functions they cannot be normalized look at the function here cannot be normalized naturally look at the function here cannot be normalized it is becoming infinity at least at one point. So, what we see is that some of the values of psi which may be absolutely valid solutions of the wave equation. They are not going to satisfy bond condition and if they do not satisfy bond condition then they are what we call unacceptable. Acceptable values are acceptable wave functions are those that satisfy bond condition all right we will come to the remaining bond conditions also first condition we have said is normalization you should be able to normalize it to 1. These are some examples of acceptable wave functions well let us say what it should be what are the conditions of acceptability first is one it should be normalized what if I give you a wave function that is not normalized if I give you an unnormalized wave function something like psi which is say not normalized we can always say that I will normalize it by multiplying it with n what do I mean I mean that I n has to be chosen in such a way that n can be actually complex quantity also. So, n star multiplied by n integral psi star multiplied by psi d tau that has to be equal to 1 this is really the normalization condition. So, if I give you a non normal unnormalized wave function no big deal you can multiply it by some constant and you can normalize it is something that is doable we are going to encounter several examples of normalization in subsequent classes. Second condition that we that is very very important is that it has to be continuous third condition is this del psi del q I will write the first derivative that also should be continuous. Now, these come from the requirement not so much from bond interpretation, but these come from the requirement of the differential equation per se right. So, there is a little bit of difference between these two and the first one. See we are working with something like del 2 psi del q 2 second derivative. So, these are the conditions for the second derivative to exist as you will see that this condition number 3 especially is often violated we are going to see some examples. And the last condition I want to talk about is that it must be single valued. What is the meaning of single valued if you have a function which has more than one value at any given x then what is the probability density at that point? You cannot have more than one values of probability density at a point right that is a problematic situation. So, it must be single valued in order to satisfy bond condition. So, this is a perfectly acceptable condition you can think is it actually not perfectly acceptable. What about this it is continuous single value everything is fine look at this point at this point there is a discontinuity of del psi del q, but as you will see this is taken as an acceptable function in some cases that we are going to discuss. So, so this del psi del q is a condition that is not so strange. What about this let me write the functional form of this one I think this is sin theta by theta at first look you might think that it is not acceptable why because what is the value of sin theta by theta for theta equal to 0 denominator is 0 numerator is also 0. So, 0 by 0 undefined however if you look at the function if you zoom in zoom in as much as you want what do you see at value very very small values of theta if you approach from the right the value is going to be nearly 1. If you approach from the left then also the value will be nearly 1 and the limit exists because from both sides you approach the value of 1. So, what we do is we set this value sin theta by theta and write like this at theta equal to 0 is equal to 1. What is 1? 1 is essentially the limiting value of sin theta by theta for theta tending to 0. So, the value of sin theta by theta is defined to be 1 and this is called a removable discontinuity. So, this is acceptable. Let us show you some examples of unacceptable wave functions this one is unacceptable why because it is not continuous. It is like sin theta by theta but the same value is not obtained if you approach from left and from right. So, it is unacceptable. What about this? This is also unacceptable because del psi del q is not continuous but given the system something like this might become acceptable as we will see. This is definitely not acceptable because it is multivalued it is not single valued this is what I was saying and this is not acceptable we have discussed this already because psi goes to infinity. So, what we see is that there are several restrictions on the wave function that arise out of born interpretation and these restrictions give us what are called boundary conditions and these boundary conditions bring in quantization as we are going to see. So, all these conditions that are there they bring in restrictions and these restrictions eventually bring about quantization. So, the origin of quantization lies in born interpretation. So, even though in Schrodinger equation we do not see any quantum number quantum numbers are going to arise the moment we try to apply these boundary conditions and this is something that comes out beautifully when we discuss a free particle and particle in a box that is what we are going to take up next.