In this video, we'll discuss uncertainty principle. So to observables, A and B are considered compatible if the corresponding operators commute with each other. So if you can say that these operators A and B representing the physical observables A and B commute, and therefore the commentator bracket is zero, then they are considered compatible if they don't commute, then they are considered incompatible. Now, commuting operators have two very important properties, one is the commuting operators share the same set of agenda eigenvectors. So let's suppose that [an] are the Eigen vectors of operator A satisfying this eigen vector equation here and the operators A and B commute, then we can write onto this Eigen value equation, we can apply operator B from the left and then from the associativity axiom. We can calculate this product first and using the Eigen value equation turn it into little [an] times an] but little an] is a scalar so, you can take it outside change the order however, this equation here, because A and B commute, you can switch the order and write it as A times an]. Now we look at this equation here and notice that this is another Eigen value equation for operator A in this case the Eigen vector is B times an] cat B times an] cat is an Eigen cat of A with the same Eigen value, an]. So what you can say is that B times an] is the same Eigen vector Eigen cat of A with the same Eigen value therefore it should be represented as a constant multiple scalar multiple of the Eigen vector [an] but then this is just an Eigen value equation for B and therefore an] Eigen cat is a simultaneous Eigen cat of operator A and operator B. Another important properties for commuting operators is that two operators that share the same complete set of Eigen vectors commute with each other. So let's consider two operators A and B share the same set of Eigen vectors, [an], so you can write down the Eigen value equation for A here and Eigen value is an] Eigen vectors. Eigen cats are also Eigen cats of B in this case the Eigen values are B, bn now then we consider A times B acting on a cat an because B be an is an eigen cat of B with the eigen value bn. So bn come out and then this simply using the Eigen value equation for A It gives you an times bn similarly, you can do B times A times an cat, you get the same results. Now, this doesn't prove that these two commute you have to show this A times B are acting on a cat it gives you the same results as B times A acting on a cat. For any arbitrary cat that only then you can say A times B is equal to B times A So let's do that. So consider on any arbitrary cat alpha you can express that as a linear combination of this complete eigen set of, an] and the caution cn is given by the inner product between alpha an. Right? Now, let's consider A times B operator acting on this arbitrary cat alpha then you can express alpha as a linear combination so you do that. And cn is just a scalar so you can change the order, so act A times B operator on this individual Eigen Cat an first and then later multiply these scalar and sum so, when you do that, then we already know that A times B operator acting on an eigen Cats give you simply an times bn Eigen values times an cat. Now Cn, an, bn are all scholars so scalar multiplication are commentated so you can change the order in whichever way you want so change the order bn times an and write it like this. But then that is equal to the bn times an operator and now that simply is equal to bn I'm sorry, B times A operator acting on a cat alpha so now we just have proved that A times B operator acting on a cat alpha. Any arbitrary cat alpha is equal to operator B times operator A acting on on arbitrary Cat alpha and that proves that these two operators A and B commute with each other. So why is this important while recall a measurement does not alter the quantum state If the system was initially in one of the eigen states of the operator representing the measured value, measured variable, or observable. So if we make two successive measurements of quantities whose operators commute, then the first measurement collapses the system into one of its eigenstates, but that eigenstate is also on eigenstate of the operator of the second measurement, and therefore the second measurements will not change the eigenstate. And what that means is that we know exactly what value we will measure with the second measurements, because the initial state is one of the eigenstates of that operator. On the other hand, if the two operators do not commute, first operator collapses the system into one of its eigenstate, but that is not an eigenstate of the second operator, and therefore the second measurements will collapse the quantum system into its own eigenstate which is distinct from the eigenstates of the first operator. And therefore in this case we have an inherent uncertainty we do not know exactly what value we will measure, we can only speak of probabilities of measuring certain values. So this leads to the well known uncertainty principle let's say that this is A bar here is the mean value of the quantity A which is represented by the formation operator A hat. So in the bracket notation A bar is the expectation value of Operator A, which is basically the matrix or the sandwich product of alpha, bra A operator and alpha cat, alpha here is the initial state the vector representing the initial state of the quantum system. Now, let's define this delta A operator as A operator minus the mean value now, A bar here is just a scalar a real number and therefore delta A is also a hermitian operator as long as operator A itself is permission. Now, to examine the variance of the quantity A that we will measure when we do repeated measurements, we calculate the expectation value of this operator delta A square actually. So to do this, we expand the initial state quantum state alpha in terms of the eigenstates of operator A so alpha here is now expressed as a linear combination of an cats, which are eigen cats of operator A. And this is a reasonably straightforward algebra so you should be able to follow it but here, basically this is the expansion of alpha bra in terms of an bras and this one is the expansion of alpha cat in terms of an cats. And we basically act this on these cats one by one twice and we turn these A operator minus A bar square into an Eigen value minus A bar, a number squared in this, which simply leads to this equation here. Now, because Cn squared absolutely square of this caution cn represents the probabilities that the system is found to be in one of the eigenstate, an the quantity here, an minus A bar square is the squared deviation of the value from the average value. This here actually represents the mean squared deviation or variants that we will find when we measure this quantity, in a repeating manner when the initial state was in alpha. Now we can similarly define mean and variance for another operator B so here is the mean or expectation value of B and this one here is the variance or the square deviation of B now, suppose these two permission operators A and B do not commute. And in general you can say that when the two operators don't commute, you can write down the commentator bracket and then express the remnant the non zero value as I times certain other operators C, now we consider this quantity here, G of lambda defined as this. This is a norm, this is a bra vector, this is a cat vector And if you take an inner product between the same bra and cat, that's the norm and it is always a positive definite, if these factors happen to be no vectors then it's zero, if they are not no letters, it will always give you a positive number. Now the left hand side this quantity here can be rearranged as this so, this is a bra corresponding to this operator so you can express this in terms of alpha bra and take the Hermitian Conjugate of this operator here and operate that from the right. Okay? And then here the dagger Hermitian A joint operation can be distributed into these two and you get this you just take the Hermitian I joined to B Hermitian I joint of A and you turn these I into plus I to take complex congregate. And then we do the straightforward algebra to rearrange re expressed this quantity G Of lambda in this form and, we can rewrite these three terms separately as shown here. And then this term can then be re expressed as this and if you remember this whole quantity is supposed to be Greater than or equal to zero now, this inequality has to be valid for any value of lambda so we choose lambda to be this particular thing that leads to this inequality. And this is the general form of uncertainty principle and it tells us that the minimum size of uncertainty, minimum amount of uncertainties in the measurements of two quantities, if these two Hermitian two operators represent the variables that we are measuring.
