PHYS 216 Exam 2, Fall 2016

Physics 216 Mathematical Physics Quiz 1, October 17, 2016, Time: 90 minutes Solve 8 problems out of 10 (12.5 points for each) 1. Let {−→e 1, −→e 2, −→e 3} be a basis of a three dimensional vector space V which are mapped by the operator T to the vectors {−→e ′ 1, −→e ′ 2, −→e ′ 3} defined byT−→e i = 3 � j=1 Tji−→e j such that −→e ′ 1 = −→e 1+−→e 3, −→e ′ 2 = 2−→e 1 + −→e 2, −→e ′ 3 = 3−→e 2 + −→e 3. Find the matrix representation of T with respect to the basis {−→e 1, −→e 2, −→e 3} and show that the set {−→e ′ 1, −→e ′ 2, −→e ′ 3} is also a basis. 2. Consider the two vectors −→u = (1, 3 − i, 2 + i) and −→v = (3, 1 + i, 2i) in the vector space C3. Find (−→u , −→v ) , (−→v , −→u ) , ||−→u || , ||−→v || , |(−→u , −→v )| . Verify that ∥(−→u , −→v )∥ ≤ ||−→u || ||−→v || . 3. Let A =   2 3 −4 0 −4 2 1 −1 5   . Find det A and list the cofactors of the 9 elements of A to deduce adj(A) the classical adjoint matrix of A and the inverse matrix A−1. 4. Consider the Hermitian matrix A = � 1 2 − i √ 2 2 + i √ 2 2 � . Find the eigenvalues and nor- malized eigenvectors of A. Construct the diagonalizing matrix P. 5. Consider the operators L1 = −i � x2 ∂ ∂x3 − x3 ∂ ∂x2 � , L2 = −i � x3 ∂ ∂x1 − x1 ∂ ∂x3 � , L1 = −i � x1 ∂ ∂x2 − x3 ∂ ∂x1 � . Show that the commutaor [L1, L2] f � x1, x2, x3� = iL3f � x1, x2, x3� . 6. Prove the identity−→ L = −i−→r × −→ ∇ = i � �θ 1 sin θ ∂ ∂ϕ − �ϕ ∂ ∂ϕ � , where −→ L is the angular momentum. Evaluate − i r2 −→r × −→ L to deduce that −→ ∇ = �r ∂ ∂r − i r2 −→r × −→ L . 7. Let −→ V = x √ x2−y2 −→i + y √ x2−y2 −→j , x > y. Show that −→ V is a conservative field and thus could be written as −→ ∇φ. Evaluate the line integral � C −→ V · −→ dr where C is a curve in the xy plane connecting the points (1, 0) and (x, y) . Deduce from this the function φ (x, y) . Hint: Use x = r cosh θ, y = r sinh θ. 8. Let −→ V = xy−→i + z2−→j + 2yz−→k . Evaluate the surface integral � S −→ V .d−→a where S is the surface of a cube 0 ≤ x, y, z ≤ 1. Verify Gauss’ law by evaluating the volume integral � V −→ ∇ ·−→ V dxdydz where V is the volume bounded by the surface S. 9. Let −→ V = y2−→i + xy−→j + xz−→k . Evaluate the surface integral � S �−→ ∇ × −→ V � · d−→a where S is the upper hemisphere with equation z = R2 − x2 − y2, z ≥ 0 bounded by the circle C defined by x2 + y2 = R2. Verify Stokes’ theorem by evaluating the line integral � C −→ V · d−→r . Hint: Write d−→a = R2 sin θdθdϕ�r in spherical coordinates. 10. Verify the vector relation −→ ∇ · �−→ A × −→ B � = −−→ A · �−→ ∇ × −→ B � + −→ B · �−→ ∇ × −→ A � . 1