852_2010hw4_Solutions

852_2010hw4_Solutions - PHYS852 Quantum Mechanics II,...

This preview shows pages 1–3. Sign up to view the full content.

PHYS852 Quantum Mechanics II, Spring 2010 HOMEWORK ASSIGNMENT 4: Solutions. Topics covered: rotation with spin, exchange symmetry 1. A vector pointing in the θ,φ direction, can be formed by starting with a vector pointing along ~e z , then applying an active rotation by θ about the y-axis, followed by a rotation by φ about the z-axis. (a) Verify this for an ordinary vector, by starting with the vector (0 , 0 , 1) T and using R y ( θ ) = cos θ 0 sin θ 0 1 0 - sin θ 0 cos θ ; R z ( φ ) = cos φ - sin φ 0 sin φ cos φ 0 0 0 1 (1) The vector (0 , 0 , 1) T is simply ~e z . Applying ﬁrst the y rotation, and then the z rotation gives: R z ( φ ) R y ( θ ) 0 0 1 = sin θ cos φ sin θ sin φ cos θ (2) we can recognize this as the unit vector ~e θφ that points in the θ,φ direction. (b) Thus for a spin-1/2 system, the spin-up state with respect to the θ,φ direction can be found in the basis of S z eigenstates, by starting with the spin-up state along ~e z , and applying unitary rotation operators, i.e. | ↑ θφ i = e - i ~ φS z e - i ~ θS y | ↑ z i . (3) In this way, ﬁnd the states | ↑ θφ i and | ↓ θφ i . ﬁrst we note that e - i ~ φS z = ± e - iφ/ 2 0 0 e iφ/ 2 ² (4) and e - i ~ θS y = cos( θ/ 2) I - i sin( θ/ 2) σ y = ± cos( θ/ 2) - sin( θ/ 2) sin( θ/ 2) cos( θ/ 2) ² (5) this gives us e - i ~ φS z e - i ~ θS y = ± e - iφ/ 2 0 0 e iφ/ 2 ²± cos( θ/ 2) - sin( θ/ 2) sin( θ/ 2) cos( θ/ 2) ² = ± cos( θ/ 2) e - iφ/ 2 - sin( θ/ 2) e - iφ/ 2 sin( θ/ 2) e iφ/ 2 cos( θ/ 2) e iφ/ 2 ² (6) Thus we have | ↑ θφ i = ± cos( θ/ 2) e - iφ/ 2 - sin( θ/ 2) e - iφ/ 2 sin( θ/ 2) e iφ/ 2 cos( θ/ 2) e iφ/ 2 ²± 1 0 ² = ± cos( θ/ 2) e - iφ/ 2 sin( θ/ 2) e iφ/ 2 ² (7) and | ↓ θφ i = ± cos( θ/ 2) e - iφ/ 2 - sin( θ/ 2) e - iφ/ 2 sin( θ/ 2) e iφ/ 2 cos( θ/ 2) e iφ/ 2 ²± 0 1 ² = ± - sin( θ/ 2) e - iφ/ 2 cos( θ/ 2) e iφ/ 2 ² (8) 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
(c) Compute the operator S θφ using unitary rotation operators to transform S z , and compare it to the result using the 3 × 3 rotation matrices. The operator S θφ is given by deﬁnition as S θφ = ~e θφ · ~ S , which gives S θφ = sin θ cos φS x + sin θ sin φS y + cos θS z = ~ 2 ± cos θ e - sin θ e sin θ - cos θ ² (9) Note that we have computed this by rotating the unit vectors. According to the lecture, there are two additional equivalent transformations. We can apply the inverse transformation to each component of ~ S via unitary operators, or we can apply the inverse transformation collectively to all three components via the 3
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 12/21/2011 for the course PHYS 852 taught by Professor Moore during the Spring '11 term at Michigan State University.

Page1 / 6

852_2010hw4_Solutions - PHYS852 Quantum Mechanics II,...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online