In this video, we will introduce Dirac's bra–ket notation. In Dirac's bra-ket notation, a state vector specifying a quantum state is denoted by a "ket". The notation is here. So there is a vertical bar on the left and then an angled bracket on the right. Then in between there is some description about the quantum state. A ket is postulated to contain all information that we need to know about the system. Ket is a vector in a vector space in mathematical terms and the dimensionality of the vector is specified according to the nature of the physical problem that we're dealing with. For example, for the case of electron spin, as we have seen in Stern-Gerlach experiments, there are only two degrees of freedom and therefore the ket vector is two-dimensional. In another example, such as infinite potential well problem where there are infinitely many eigenstates. In those cases, the ket space is infinite dimensional and the ket vector is also infinite dimensional. Kets can be added together or multiply by a scalar constant, and an important physics postulate is that a ket Alpha and a constant multiple c of ket Alpha, they represent the same state as long as the constant is not 0 of course. If we make an analogy to a geometrical vector, what this means is that only the direction of the vector contains physically meaningful information, the amplitude or the magnitude of the vector does not. The multiplicative constant can always be determined at the very end by imposing the normalization condition. An observable, such as momentum or spin can be represented by an operator. This operator acts on a ket, and it turns a ket into another ket as shown here in this equation. Operator A acts on ket Alpha and it turns into another ket Beta. Now, in general Beta, the resulting ket is not a constant multiple of original ket Alpha, which means that performing a measurement on a quantum system turns the quantum state into another state. It alters the quantum state. There exist however, a special set up kets for a given operator which satisfy this equation here. You operate operator A on certain ket a prime here, it produces a constant multiple of the original ket. There can be many of them. How many of these kets exist depend on the operator A obviously, but it is possible that there exists certain set of kets that satisfy this equation. These kets are called the eigenkets of the operator a, and the multiplicative constant on the right-hand side are called the eigenvalues. What does this mean physically in quantum mechanics? It means that when a quantum system was originally in an eigenstate, making a measurement does not change the state. It preserves the original state. The quantum system is preserved. Now we introduce bra vectors denoted as this. This time you have an angled bracket on the left and a vertical bar on the right, and it is a dual to ket vectors. The perfect analogy would be row vectors versus column vectors. There is a one-to-one correspondence between a ket space and a bra space. Those vector space consisting of kets has a perfect one-to-one correspondence to another vector space composed of bra vectors. Further, let us define bra vector dual to a ket, c constant times alpha as not c times alpha bra, but c complex conjugate times the alpha bra. In the matrix analogy, what this means is that when we turn a ket vector into a bra vector, we are transposing to turn column vector into a row vector, and also taking a complex conjugate. Further define the inner product. Inner product is denoted as this symbol here. On the left, we have a bra vector, and on the right you have a ket vector. This is a product between these two vectors, and it results in a scalar, usually a complex number. The inner product has two important properties, so if you take the inner product between a bra vector beta, and a ket vector alpha, that is equal to the inner product of bra vector alpha, and then ket vector beta complex conjugated. Also, if you take an inner product with itself, a bra vector of alpha, and the ket vector of the same state alpha, then it is always positive definite. If this vector has zero amplitude, zero magnitude, then it is zero. Otherwise, you always have a positive number. Two kets are orthogonal when the inner product is zero, and the norm of a ket is defined as a square root of the inner product with itself. If the norm is zero, then we call that a null ket. Finally, a ket, which is not a null ket, can be normalized, and if you have a ket alpha here, which is not normalized, you evaluate the norm, and you divide the original ket by its norm. It's a scalar multiplication. You can always do that, and that results in a ket alpha tilde. This alpha tilde has a norm which is unity, and this type of ket is called normalized. The bra-ket notation is abstract and general. We made analogy two column vectors and row vectors and matrices. However, that's just analogy. Bra vector is not a row vector. Ket vector is not a column vector. Bra vector can be represented by a row vector and ket vector can be represented by a column vector, but that doesn't mean that they are the same. Bra and ket concepts are a lot more general and abstract, so if you want to specify the quantum states corresponding to the solutions of infinite potential of the problem, you can simply write it as this. Write down the psi sub n, the symbol we used to specify the eigenfunctions. You just write that down and then sandwich it with a vertical bar here, and then on angled bracket on the right or you can just spell it out; a state whose energy is given by this equation and surrounded with this vertical bar, and angled bracket on the right or you can write down whatever you want in between just to specify a state. That all works, and this all works in lieu of specifying the exact wave function, and therefore, you can imagine that these bracket notation is being abstract, is a lot more powerful, and a lot more convenient. For example, if you have a quantum state, that does not have any spatial dependence like spin. You cannot write down spatial wave functions because spin does not depend on spatial coordinate at all. How do you specify such quantum state? Bracket notation has no problem specifying such a state, whereas the wave function notation is simply cannot specify such a state. For mathematical manipulation, however, calculations, you have to turn these abstract things, bras, and kets into a concrete thing that you can actually add, multiply, and integrate, and so long. In those cases, we typically express these bras and kets as either matrixs, column and row vectors, or functions.
