We are continuing with our previous classes. We have now a new topic which is actually included in your operators in quantum mechanics, expectation values of an observable. Now what is observable? As you already learned it, observable which is term included in your classical mechanics, in quantum mechanics it is your operators. That means classical observable is nothing but the operators in your quantum mechanics. Let us come to the word expectation value. Actually expectation value denotes the avarice value of a basic quantity. So how we represent it, how we can represent this expectation value of an observable? Suppose I have an observable A. This is my observable. Then its expectation value in quantum mechanics is denoted as in this way. So within this symbol you have to write the required observable symbol. So mathematically how we can define the expectation value? When a system is in a state described by wave function psi, then the expectation value of an observable A of the system is given by a within this symbol integration of minus infinity to plus infinity psi starts. That means complex conjugate of the state function. In the operator psi d tau, I have taken the volume integration. A is what? A is your operator. And psi star is your complex conjugate of the wave function psi. Now here we considered it the wave function psi is a normalized wave function. If psi is not a normalized wave function, so in that case we have to write the expectation value of a in this way. A, if to write the normalization of psi is star d tau. Next integration minus infinity to plus infinity psi star a psi so d tau. So when we take in this way, when your wave function phi is not a normalized wave function. So for that case we have to normalize the wave function. So we have to divide this term by the normalization factor. Let us take an example. Suppose we have a particle which is bound to move along x axis. We are within the suppose within the limit 0 to L. So in the corresponding normalized Eigen function or you can say state is given a size you can root over 2 by L sin n by x by L. So in that case according to the rule, suppose we have to determine the expectation value of momentum. So in that case we have to write expectation value of momentum is px within this symbol. Integration 0 to L psi star and you know this is the operator from of your momentum. So in that case we have to determine the value of momentum of your linear momentum psi dx. Now our warp is maryz now. So I have written this equation in explicit form. So I put the values of psi star and psi. Now I am going to operate this term by d dx. So after doing all this thing you will get this equation. You will write in this equation in this way minus i is cross 2 by L multiplying this two term you will get 2 by L. And also n pi by L term after how it will come? It will come when you operate sin n pi x by L by d dx or an integration 0 to L. sin is already there. So sin n pi x by L cos n pi x by L. Now we have to solve this integral. We have to solve this integral. Now perhaps you know from your trigonometric knowledge that twice sin theta cos theta is again sin twice theta. So this formula we can use here when you insert this two within this integral sign. So applying this formula I can write the equation i is cross n pi x by L integration 0 to L. 0 to L sin twice n pi x by L dx. Now after integrating this term you will get zero result because you can see here when you integrate any function of sin theta when you integrate the sin function you normally get zero result. So your final expectation value of your momentum is what? Zero.