PHY 215A Quantum Mechanics Homework Set #1: Vector Spaces and Inner Products, Assignments of Quantum Mechanics

Download PHY 215A Quantum Mechanics Homework Set #1: Vector Spaces and Inner Products and more Assignments Quantum Mechanics in PDF only on Docsity! PHY 215A, Quantum Mechanics: Homework Set #1 Due: 10/10/07 1. Vector Spaces. 15 points. Consider the ordinary vectors in 3D, axî+ ay ĵ + azk̂ with complex coefficients ax, ay, az. (i) does the subset of all vectors with az = 0 form a vector field? Show why or why not. (ii) does the subset of all vectors with az = 1 form a vector field? Show why or why not. (iii) does the subset of all vectors with ax = ay = az form a vector field? Show why or why not. 2. A Different Vector Space. Subspaces. 15 points. Consider the set of all polynomials in the variable x of degree less than N. Considering the polynomial as the vector, answer the following: (i) does this set form a vector space? If so, suggest a convenient basis and give the dimension of the space. If not, which defining property does it lack? (ii) answer the same question if we require the polynomials to be even functions of x. (iii) answer the same question if we require the leading coefficient, i.e. the coefficient of xN−1, to be 1. (iv) answer the same question if we require the value of the polynomials in the set to have the value 0 at x = 1. 3. Gram-Schmidt. 15 points. (a) Given the vectors |U1 >= (1, 0, i), |U2 >= (i, i, 0), |U3 >= (1, 1, 1), perform Gram-Schmidt orthonormalization beginning with |U1 >. Show work, and identify your solution clearly. 4. Inner Product. 15 points. (a) Find the inner product < A|B > of the vectors |A >= (2 − i, i, i), |B >= (1,−2i, 1). (b) Next, given the 3×3 transformation matrix T whose only non-zero elements are T1,3 = T2,2 = T3,1 = i, find the inner product of |A ′ >= T |A > and |B′ >= T |B >. Compare the results to (a) and explain similarities and differences. (c) What kind of transformation is T ?