Inner Product Spaces

Inner Product Spaces • Physical properties of vectors aka length and angles in case of arrows • Lets use the dot product • Length of • Cosine of the angle between • How can we use the dot product to define length & angle when the dot product uses length and angle • Main features of the dot product are राघववर्मा

Pk Pj Linearity Explained Linearity requires that Pj + Pk = Pjk Why do we require Linearity? How does this condition ensure linearity? राघववर्मा

Inner Product Space for Quantum Mechanics • Inner product in our special space will be denoted by V|W • A vector space with an inner product is called an inner product space • No explicit rule for evaluating the scalar product • The first axiom is sensitive to the order of 2 factors. This is to make V|V real. • Second  Poistive Semidefiniteness- vanishing only if the vector does. It better be because from our generalization we are going to use it to define length • Linearity of inner product when a linear superposition of a|W + b|Z  |aW + bZ appears as a second vector in the scalar product राघववर्मा

Asymmetry of our space • What if the first factor in the product is a linear superposition • Expresses anti-linearity of the inner product with respect to the first factor in the inner product • The inner product of a linear superposition with another vector is the corresponding superposition of inner product if the superposition occurs in the second factor. • While the superposition with all its coefficients conjugated if the superposition occurs in the first factor. • This assymetry is going to stay with us…. राघववर्मा

Some more properties of Inner Product • Two vectors are orthogonal if the inner product vanishes • The norm of the vector in my vector space is • A set of basis vectors, pairwise orthogonal will be called an orthonormal basis • If we use orthonormal basis only the diagonal terms survive. • We have defined all the properties of inner product but have not specified how to compute it. • We said that the basis is orthonormal if राघववर्मा

Computing Inner Product • The double sum • Then collapses to • So now you can appreciate why we defined • If it was not defined this way then forget about the norm being positive definite it would not have been real • We have already said that the vector is uniquely defined by its components. Let me denote it by an ordered set of numbers and let me use a column to represent its components i.e. & राघववर्मा

Inner Product (contd..) • The inner product in our space has been defined using the dot product as reference. The inner product also defines the norm • We need a number • To get a number from a column matrix we need to multiply it with a row matrix • Hence if |V is a column matrix • Then if W| is a represented by a row matrix • The inner product of V|W is represented by Which is then राघववर्मा

Dual Space and the Dirac Notation • The column matrix associated with a vector in my space is unique • I can get a scalar from inner product if I multiply a column vector by a row vector (its transpose conjugate) • Since the column vector is unique Hence the row vector by definition also becomes unique • This row vector is called a bra in Dirac’s notation • Thus there are two vector spaces, the space of kets & a dual space of bras • There is a ket for every bra and vice-versa • Inner product is defined only between bras and kets  Hence from elements of two distinct but related spaces. राघववर्मा

Dual Space (contd…) • In my n dimensional vector space there exists basis vectors |i • Similarly in the dual of this vector space there exist basis vector i| • In an orthonormal basis |i has all zeros and a 1 at the ith row • A vector can now be expanded in my orthonormal basis • A vector in my space can then also be expressed as राघववर्मा

Dual Space (Contd..) • Adjoint Operation • If V| is the bra corresponding to the ket |V what bra corresponds to a |V where a is some scalar • Relation between bras and kets in linear equation • To take the adjoint of a linear equation relating kets (bras) replace every ket (bra) by its bra (ket) & complex conjugate all its coefficients राघववर्मा

Orthonormal Basis • Why do we need an orthonormal basis? • Our requirement for a basis in an n dimensional space is just n linearly independent vectors • Lets see in three dimensional space • If the unit vectors u v & w are orthonormal, the dot product reduces to three terms. • If the basis is only orthogonal then • Moral of the story is that we need orthonormal basis राघववर्मा

Example of Schmidt Orthonormalisation • Construct an orthonormal set of vectors from the vectors राघववर्मा

Subspaces • Given a vector space V a subset of its elements that form a vector space among themselves is called a subspace. • We will denote a particular subspace i of dimensionality ni by Vini • Given two subspaces Vini & Vjmj we define their sum • Vini Vjmj = Vkmk • This set contains all elements of Vjmj • All elements of Vini • All possible linear combinations of above राघववर्मा

Linear Operators • An operator  is an instruction for transforming any given vector |V into another vector |V • We have seen that one can multiply a vector by a scalar just stretches the vector • An action of an operator on a vector, in general, transforms the vector • Our Stern Gerlach Experiment transformed the original vector, the spin state of the silver atom which was oriented in all possible directions in the direction of the inhomogeneous magnetic field. • We will restrict ourselves to operators which will not take the ket on which they act upon outside the vector space under consideration • If |V is a vector in my Vector Space then so is  |V = |V राघववर्मा

|3 |2 + |3 R(|2 + |3) |2 |1 Linear Operators • Operators can act on bras as well • V| = V| • Linear Operators •  |Vi = |Vi • { |Vi + |Vj = |Vi + |Vj • Vi|  Vi| • (Vi| + Vj|)  Vi| + Vj| • Identity Operator I leaves the vector alone • Operator on V3® • R(/2 i)  Rotate all vectors by an angle /2 about the x axis राघववर्मा

Operator R राघववर्मा

Inner Product Spaces