In this video, we will discuss operators in Dirac's bra-ket notation. An operator in linear algebra it's something that acts on the vector and convert it into another vector. In bra-ket notation, we can write operator as X, and just to make sure that X is an operator, not just a number or any other entity, we added a hat on top of this symbol X and operator X on a ket vector always from the left. X operator acting on a ket vector Alpha and it turns it into another ket vector Theta. Now, in quantum mechanics, operators are important because typically, the physical observables like momentum, energy, spin are represented by operators. There are also other important class of operators that are not observable, such as rotation and translation, and these transformation operators are also expressed or represented by operators. In any case, mathematically, two operators are said to be the same or equal to each other, X operator equal to Y operator if the X acting on ket Alpha results in the same ket as Y acting on the same ket Alpha. When these two are equal for all kets in the vector space, then X and Y are said to be the same or equal to each other. Operators can be added together and the addition is commutative and associative. The operator can act on a bra vector as well, and in the case of acting on bra vector, the operator acts on a bra vector always from the right, as shown here. It produces another bra vector, as shown here. In general, the ket vector resulting in from X acting on a ket Alpha that is not a dual to the bra vector resulting from X acting on a bra Alpha. Instead, in order to obtain a dual, we have to define an operator denoted as X dagger. This is a dagger symbol. We read it X dagger, and it is called hermitian adjoint operator of X. Hermitian adjoint of X acting on a bra Alpha is dual to the operator X acting on a ket vector Alpha. If the hermitian adjoint is equal to itself, then the operator is called hermitian. Hermitian operators form an important class of operators in quantum mechanics because physically observable variables are represented by hermitian operators in quantum mechanics. Operators can be multiplied as well. Multiplication, however, is not commutative. X times Y is not the same as Y times X. But they are associative. Associativity applies to multiplication with factors too. In other words, if you multiply two vectors, two operators together XY and act on a ket vector Alpha, that is the same as acting Y on a ket vector Alpha first and then applying X on it. That is the same. That results in the same ket vector that is. Or you can do the same with acting on a bra vector as shown here. Also, if you take a hermitian adjoint to a product operator XY, that is equal to the product of hermitian adjoint of individual operators, but you have to switch the order of operation. XY operator taken, you multiply to operator first and then take hermitian adjoint that is the same as taking hermitian adjoint of individual operator, multiply them together in the reverse order. Finally, we define outer product. Outer product is defined as a product between a ket vector and a bra vector in the reverse order as compared to the inner product. The outer product of two vectors results in an operator, and this is analogous to the outer product in matrices or vectors when you multiply a column vector to a row vector, it results in a matrix. So far we considered the following products, the inner product of two vectors, the bra vector and ket vector multiply together, resulting in a scalar. This is called the inner product. Product between operator and the vector, so you can have an operator acting on a ket from the left or operator acting on a bra from the right. We also define the product between two operators and we have defined finally the outer product between two vectors where ket is multiplied from the left to a bra vector on the right. There are illegal products, so you can't really have a ket to ket multiplication or bra to bra multiplication. You cannot have an operator acting on bra from the left or you can't have an operator acting on a ket from the right. Except for these illegal products, for all legal products, associativity always holds. This is called the associativity axiom. For example, if you have a triple product or double product of bra vector alpha and then operator x, and then ket vector data, you can multiply this first or you can multiply this first. It doesn't really matter. Incidentally, if you have an alpha bra and x operator and data ket, then it is equal to the data bra multiplied to x operator permission at joint and alpha ket. They are the same if the operator is permission. Here is a table that summarizes everything that we have discussed in the two previous videos. On the left here is the quantity vector, dual vector, inner product, outer product operators. It lists these quantities in the bracket notation. We represent those quantities using these notation. Here is the ket vector, bra vector, which is dual to this ket. Inner product between two vectors, bra from the left, multiplied from the left to a ket, outer product in the opposite direction, resulting in an operator, whereas the inner product results in a scalar. Operator is indicated by this hat on top and the operator vector products. You can operate the operator from the left on the ket or operator can be multiplied to a bra vector from the right. You could have the product of two operators, and then there is what we call a matrix element, which is a double product of bra operator and a ket. There is a perfect one-to-one analogy between these bracket notation and the matrix multiplication. A ket vector corresponds to a column vector and a dual bra vector corresponds to a row vector with a complex conjugate. Then of course, a dot product between a row vector and a column vector results in a scalar. That's a dot product, which is the inner product in the matrix vectors. Outer product is the product of column vector on the left and then the row vector on the right. It results in a matrix. Operator is expressed by a matrix or corresponds to a matrix and then the operator vector product is the matrix-vector product and then operator product is the matrix multiplication and the matrix element is equal to these row vector times matrix times column vector. Here in the last column we list the corresponding quantity in function representation. A ket vector is typically expressed by a function and a bra vector complex conjugated function and the inner product of two functions we have defined before. It's a complex conjugate times the other function integrated across the space over the entire volume. Operator is sum operator. It could be a differential operator, it could be an integral operator, it could be whatever other operator that turns one function into another function. Then finally, the matrix element is something similar to the inner product, except that the operator is sandwiched between these two function, one complex conjugated and the other unconjugated.
