13 - Physics 6210/Spring 2007/Lecture 13 Lecture 13...

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

View Full Document Right Arrow Icon

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

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

Unformatted text preview: Physics 6210/Spring 2007/Lecture 13 Lecture 13 Relevant sections in text: § 2.1 Example: Spin 1/2 in a uniform magnetic field. Let us consider the dynamical evolution of an electronic spin in a (uniform) magnetic field. We ignore the translational degrees of freedom of the electron. The electron magnetic moment is an observable which we represent as μ =- e mc S , (Here e > 0 is the magnitude of the electron electric charge.) The Hamiltonian for a magnetic moment in a (uniform, static) magnetic field B is taken to be H =- μ · B = e mc S · B . Let us choose the z axis along B so that H = eB mc S z . Evidently, S z eigenvectors are energy eigenvectors. Thus the S z eigenvectors are stationary states. Let us consider the time evolution of a general state, | ψ ( t ) i = a | + i + b |-i , | a | 2 + | b | 2 = 1 . We have | ψ ( t ) i = e- i ¯ h eB mc S z | ψ ( t ) i = ae- iωt/ 2 | + i + be iωt/ 2 |-i , where ω = eB mc . From this formula you can easily see that if the initial state is an S z eigenvector, then it remains so for all time. To see dynamical evolution, we pick an initial state that is not an energy eigenvector. For example, suppose that at t = 0, a = b = 1 √ 2 , i.e., | ψ (0) i = | S x , + i . What is the probability for getting S x = ± ¯ h 2 at time t ? We have (exercise) P rob ( S x = ¯ h 2 ) = |h S x , + | ψ ( t ) i| 2 = cos 2 ( ωt 2 ) , P rob ( S x =- ¯ h 2 ) = |h S x ,-| ψ ( t ) i| 2 = sin 2 ( ωt 2 )...
View Full Document

This note was uploaded on 02/18/2012 for the course PHYSICS 6210 taught by Professor M during the Spring '07 term at AIU Online.

Page1 / 4

13 - Physics 6210/Spring 2007/Lecture 13 Lecture 13...

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