Insights Blog
-- Browse All Articles --
Physics Articles
Physics Tutorials
Physics Guides
Physics FAQ
Math Articles
Math Tutorials
Math Guides
Math FAQ
Education Articles
Education Guides
Bio/Chem Articles
Technology Guides
Computer Science Tutorials
Forums
Intro Physics Homework Help
Advanced Physics Homework Help
Precalculus Homework Help
Calculus Homework Help
Bio/Chem Homework Help
Engineering Homework Help
Trending
Featured Threads
Log in
Register
What's new
Search
Search
Search titles only
By:
Intro Physics Homework Help
Advanced Physics Homework Help
Precalculus Homework Help
Calculus Homework Help
Bio/Chem Homework Help
Engineering Homework Help
Menu
Log in
Register
Navigation
More options
Contact us
Close Menu
JavaScript is disabled. For a better experience, please enable JavaScript in your browser before proceeding.
You are using an out of date browser. It may not display this or other websites correctly.
You should upgrade or use an
alternative browser
.
Forums
Homework Help
Advanced Physics Homework Help
Exploring Solutions to Spin Evolution Equations
Reply to thread
Message
[QUOTE="uxioq99, post: 6854744, member: 733136"] [USER=635497]@hutchphd[/USER] Sorry, I realized when I typed "\exp(-\omega/2t)" it produced ##\exp(-\omega/2t)## instead of ##\exp(-\frac{\omega}{2} t)## Wouldn't the units of ##\omega## cancel those of time given that it is an angular frequency? Would ## \begin{pmatrix} \Psi_1(x,t) \\ \Psi_2(x,t) \end{pmatrix} = \begin{pmatrix} \exp(\frac{\omega}{2} t) \\ \exp(-\frac{\omega}{2} t) \end{pmatrix} ## Be a valid solution of ##E\vec\Psi = H \vec \Psi## where ##H = -\omega S_y##? Wouldn't ## -i\hbar \begin{pmatrix} \frac{\partial \Psi_1}{\partial t} \\ \frac{\partial \Psi_2}{\partial t} \end{pmatrix} = -\frac{\hbar \omega}{2} \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} \begin{pmatrix} \Psi_1 (x,t) \\ \Psi_2 (x,t) \end{pmatrix} ## because ## -i\hbar \frac{\partial}{\partial t} \left(e^{\pm\frac{\omega}{2} t}\right) = \mp i \frac{\hbar \omega}{2} e^{\pm\frac{\omega}{2} t} ## and ## -\frac{\hbar \omega}{2} \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} \begin{pmatrix} e^{-\frac{\omega}{2} t} \\ e^{\frac{\omega}{2} t} \end{pmatrix} = i\frac{\hbar \omega}{2} \begin{pmatrix} -e^{\frac{\omega}{2} t} \\ e^{-\frac{\omega}{2} t} \end{pmatrix} ## I believe that a solution to the Schrodinger equation cannot always be real, so my selected form for ##\Psi(x,t)## does not appear to be correct. Additionally, the normalization factor I mentioned earlier ##\frac{1}{\sqrt{e^{\omega t} + e^{-\omega t}}}## is time dependent. I know that I neglected the appropriate derivatives in the energy operator, but I was unsure of what else to do. When I consider the similar problem for ##H = -\omega S_z##, I had that the Hamiltonian became ## H = -\frac{\hbar \omega}{2} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} ## that admitted a solution ## \begin{pmatrix} \Psi_1(x,t) \\ \Psi_2(x,t) \end{pmatrix} = \frac{1}{\sqrt{2}}\begin{pmatrix} e^{i\frac{\omega}{2} t} \\ e^{-i\frac{\omega}{2} t} \end{pmatrix} ## The normalization ##\frac{1}{\sqrt{2}}## remained constant in time because of the presence of these oscillators. I am confused by the presence of the imaginary unit ##i## in the ##S_y##. It must be present by definition, but it cancels the ##i## in the energy operator ##E = i \hbar \frac{\partial}{\partial t}##. Then, the solutions appear to be exponential instead of oscillatory. This fact conflicted my intuition because -- as said previously -- the wavefunction is real and does not appear to be easily normalizable. Also regarding my generalization to any 2 by 2 matrix. I am aware that the Pauli matrices span ##\mathbb{C}^{2\times 2}## and so the general solution could be written by decomposing the Hamiltonian into these constituent matrices. I had hoped that an alternate characterization in purely the language of introductory linear algebra could be fashioned by examining the equation as a system of ODEs and considering the spectrum. If ## \frac{d\vec x}{dt} = A \vec x(t) ## then ## \vec x (t) = S\exp(\Lambda t) S^{-1} \vec x_0 ## by considering ##\exp## as a matrix power series. [/QUOTE]
Insert quotes…
Post reply
Forums
Homework Help
Advanced Physics Homework Help
Exploring Solutions to Spin Evolution Equations
Back
Top