Linear Vector Space and Matrix Mechanics Chapter 1 Lecture 1.3 Dr. Arvind Kumar Physics Department NIT Jalandhar e.mail: iitd.arvind@gmail.com https://sites.google.com/site/karvindk2013/

Dirac Notations: The physical state of the system is represented by the state vectors. These state vectors are independent of the basis. e.g. for ordinary vectors, the components of vector may take different values in different co-ordinate system but the meaning of the vector does not change from one co-ordinate system to other. The Dirac introduced a new formalism to make the state vectors independent of particular representation.

The state of dynamical system in quantum mechanics under Dirac notations are represented by ket vectors. We use the symbol to represent ket. For example state vector will be represented by just putting it in middle of above symbol i.e. Ket vectors belongs to the Hilbert space. Ket vectors can be multiplied by complex numbers and may be added together to give other ket vectors

Each state of a dynamical system is represented by a ket vector and if the state result from the superposition of other states, then the ket of that state can be expressed as linear superposition of other ket vectors. By superposing a state with itself we cannot form a new state but we get original state again

We now introduce the concept of Bra vectors. These are element of the dual space corresponding to the vector space. To understand, we consider a number α which is a function of a ket vector . i.e. Corresponding to each ket vector there corresponds one number α. We also suppose that the function is a linear One. The number may be looked upon as scalar product of that with some new vector. This new vectors are called Bra vectors.

The bra-ket is used to represent the scalar product (also known as inner product) of two functions Bra vector help in extracting the amplitude of something to happen when the system is in state .

Note that In wave mechanics one deal with wave function But in general formalism of quantum mechanics we deal with the abstract ket vectors We can express state vectors in terms of some particular representation e.g. Position representation Scalar product in co-ordinate representation is

Properties of Ket and Bra: Every Ket has corresponding bra

Properties of Scalar product: The scalar product is not same as . This is because the scalar product is a complex number and hence order matters a lot. We write We can write If both state vectors are real then

More properties of scalar product:

Norm is real and positive: Schwarz inequality: This relation is analogous to relation in real Euclidean space

Triangle inequality: Orthogonality:

Orthonormal states: Forbidden states: The products like or are forbidden if belongs to same Hilbert space. However two vectors belongs different spaces e.g. one belong to orbital space and other to spin space then we can define These are called tensor product.

If a ket is not normalized, then we can form a normalized Ket as | If a ket is not normalized, then we can form a normalized Ket as With property,

Example: Ket are represented by column matrices And Bras are represented by row matrices. Consider the following two kets Find bra corresponding to . Ans: Bra is the complex conjugate of transpose of ket.

Find the scalar product Products does not make sense Because columns are not multiplied with columns and rows are not multiplied by rows.