Question

Problem 1 Consider a quantum bit: Write down the most general vector representing its b. Normalize the states a. state. 310) i) where a, b are complex and ф is real. c. For each of the states in b, find the orthogonal states 1) d. Express the states |0) and |1) in terms of lvs) and |wi) Do everything using Dirac notation

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Answer 1

The most general state of vector representing state in terms of the & Bra notation, also the kit & bra notation is represents in matrix form In terms of the basis state {1 Phi_n}: n = 1 2 3 }an arbitrary state vector 1 Phi > can be written as 1 Phi > = Sigma_n 1 Phi_n > < Phi_n Vertical-bar Phi > \r\n Let us now write < Phi_n Vertical-bar Phi > = Phi_n Then we have by analogy with the position vector 1 Phi > = [Phi_1 Phi_2 Phi] This is a representation of 1 Phi > as a column vector with respect to the set of basis states { 1 Phi_n > = 1, 2 } & it is known as kit vector & the Bra vector is defined as < Phi Vertical-bar = [Phi^*_1 Phi^*_2] i e row vector & it is complex conjugant of kit vector b) 1 Phi_3 > = 3 Vertical-bar 0 > + i Vertical-bar 1 > then < Phi_3 Vertical-bar = 3 < 0 Vertical-bar = i < 1 Vertical-bar then < Phi_1 Vertical-bar Phi_1 > = g + = 10 Hence Normalize state is - r\n Vertical-bar Phi_1 >_N = 1/squareroot 10 (3 Vertical-bar 0 > + i Vertical-bar 1 >) Vertical-bar Phi_1 >_N = 3/squareroot 10 10 > + i/squareroot 10 Vertical-bar 1 >