homework7sol - ECE 407: Homework 7 Solutions (By Farhan...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
1 ECE 407: Homework 7 Solutions (By Farhan Rana) Problem 7.1 a) The answer follows from elementary vector calculus result that that the gradient of any function is perpendicular to the surface of constant value of the function. In the present case, the velocity is related to the gradient of the energy function, () () k E k v k c r h r r r = 1 And therefore the velocity must be perpendicular to the surfaces of constant energy. Problem 7.2 a) First note that: B k B k r r r r . || = Now take the dot product on both sides of the crystal momentum equation with the magnetic field to get: () [] 0 0 . 0 . . . || = = = × = dt k d dt k B d dt k d B B k v B e dt k d B c r r r r h r r r r r r h r b) ( ) ( ) () () ( ) () () () () [ ] [ ] 0 . . 1 = × = = B t k v e t k v dt t k d t k E dt t k dE c c c k c r r r r r r h r h r r c) The complete argument follows in two steps: i) Since the energy of the electron remains unchanged and equal to its initial value o E , the motion in k-space of the electron must be confined to a constant energy surface corresponding to the initial energy o E of the electron. ii) Since the component of its crystal momentum parallel to the magnetic field also remains unchanged and equal to its initial value || k r , the motion in k-space of the electron must also be confined to the plane in k-space on which the components of all crystal momenta in the direction of the magnetic field is || k r (convince yourself that this later condition defines a plane in k-space that is perpendicular to the magnetic field).
Background image of page 1

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

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 4

homework7sol - ECE 407: Homework 7 Solutions (By Farhan...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online