Question

When dealing with standard vectors (with purely real elements) we are used to finding the angle between the vector from But what happens when we are dealing with vectors that have complex elements. In quantum mechanics, in general, the inner product is a complex number, which does not define a real angle The Schwarz Inequality helps us in this regard However, according to it, the only thing we can know is that the absolute value of the inner product is no greater than 1 So one can define the angle in the general case as a) Consider two linearly independent vectors in a two dimensional vector space given by the plane containing the two vectors (any two vectors with a non-zero angle (except multiples of T) between them are linearly independent) Use the Gram-Schmidt procedure to obtain two orthogonal basis vectors. Take the second vector expressed in terms of the first (don't normalize the second one), observe the basic rules for scalar products in a Hilbert space, and proof the Schwarz inequality b) Find the angle between the vectors 2-2i

Answer 1