MATHEMATICAL PHYSICS PHYSICS 227, Instructor: Marcel den Nijs SOLUTIONS HOMEWORK SET 6: [10 points total] PROBLEM 1 [2 points] Consider the Pauli spin matrices ˆ S x = 0 1 1 0 ; ˆ S z = 0 - i i 0 ; ˆ S z = 1 0 0 - 1 ; ˆ 1 = 1 0 0 1 a. Give the general definition for unitary matrices, and explain why the Pauli spin matricesare unitary operators.Unitary operators are matrices where the adjoint is the same as the inverse,ˆA-1=ˆA†such thatˆA†ˆA= 1.Recall that Hermitian matrices have the property that they are self adjointˆA†=ˆA, which means that the square of unitary hermitian matrices is equalto one; see part (b).b. Derive that are their own inverse,c. Show that they anti-commute, as ind. Consider the linear combination ( ˆ S x ) 2 = ( ˆ S y ) 2 = ( ˆ S z ) 2 = 1 ( ˆ S x ) 2 = 0 1 1 0 0 1 1 0 = 1 0 0 1 and the same for the others ˆ S x ˆ S y + ˆ S y ˆ S x = 0 ˆ S x ˆ S y + ˆ S y ˆ S x = 0 1 1 0 0 - i i 0 + 0 - i i 0 0 1 1 0 = i 0 0 - i + - i 0 0 i = 0 and the same for the other combinations. Note that the commutators are nonzero ˆ S x ˆ S y - ˆ S y ˆ S x = i 0 0 - i - - i 0 0 i = 2 i ˆ S z ˆ H = - B x ˆ S x - B z ˆ S z 1
with ~ B = ( B x , 0 , B z ) the magnetic field (a vector with real numbers). Find the eigenvalues and eigenvectors of ˆ H and the rotation matrix that diagonalizes ˆ H. The eigen values of the matrix B z B x B x - B z ( B z - λ )( - B z - λ ) - B 2 x = 0 → λ = ± p B 2 z + B 2 x are equal to ± 1 times the magnitude of the magnetic field. ( B z - λ ) x = B x y → ~a ± = x y = B x B z ∓ p B 2 z + B 2 x The operator is Hermitian (symmetric in this case) such that the right and left eigenvectors coincide and the eigenvectors are perpendicular ~a + · ~a - = B x , B z - p B 2 z + B 2 x · B x B z + p B 2 z + B 2 x = 0 The rotation matrix that diagonalizes ˆ H is formed by the eigen vetors as rows: ˆ R = ~a + ~a - It has the same effect as rotating the 3D coordinate system in such a way that the magnetic field now points in the z -direction, ˆ H = | ~ B | S z (group theory needed to show how in detail; see Quantum Mechanics classes.) PROBLEM 2 [2 points] a. Study example 3 on page 165 in Boas, about three coupled springs strung between two walls, Fig 12.1. Describe in your own words the motion corresponding to the eigenmode with frequency ω 1 = p K/m and the one with frequency ω 2 = p 3 K/m . The eigenmode with ω 1 = p K/m represents the oscillatory motion where the two masses move in tandem swinging both to the left and the right at constant distance from each other. The eigenmode with ω 1 = p 3 k/m represents the oscillatory motion where the two masses pulsate in opposite direction while their center of mass does not move. We did this as a demo in class on Friday last week using the air track.