How Do You Solve a Spin-One-Half Hamiltonian Problem?

[noblegas]

Homework Statement

A single spin-one-half system has Hamiltonian

H=\alpha*s_x+\beta*s_y, where \alpha and \beta are real numbers, and s_x and s_y are the x and y components of spin .

a) Using the representation of the spin components as Pauli spin matrices, find an expression for H^2 in termms of the above parameters.

b) used the result from part(a) to find the energy eigenvalues.

c) Find the eigenvectors of H in equation H=\alpha*s_x+\beta*s_y in the Pauli spin matrix representation.

d) Supposed that a t time t=0 the system is an eigenstate of s_z, with eigen value +\h-bar/2. Find the state vector as a function of time in the Pauli spin matrix representation.

e) Suppose the z-component of the spin in the state found in part d) is measured at time t>0 . Find probability that the result is +\hbar/2

Homework Equations

s=(\hbar)*(\sigma)/2
(\sigma_x),(\sigma_y), and (\sigma_z)

The Attempt at a Solution

a) Just multiply H twice right? but just need to insert matrix of x-component and y component for spin x and spin y

b) No idea what the energy eigenvalue is; Wouldn't it be H ? could they mean : U=exp(-i*H*t/(h-bar))?

c)Do they want me to just write the equation H out explicitly, i.e. with the matrix components of x and y ?

d) No idea what the state vector is; is it \phi=\varphi_x+? is \varphi_x+= \hbar/2?

e) I probably need to square the state vector which would be (\hbar^2)/4 if my state vector in d is correct.

What do you think of my approach?

 

[jdwood983]

b) No idea what the energy eigenvalue is; Wouldn't it be H ? could they mean : U=exp(-i*H*t/(h-bar))?

No. The energy eigenvalues are given by E, not H:

<br /> \hat{H}\psi=E\psi<br />

 

[noblegas]

No. The energy eigenvalues are given by E, not H:

<br /> \hat{H}\psi=E\psi<br />

So then , it reallly requires no effort to do this part of the problem right? Should I find the value for psi?

 

[jdwood983]

So then , it reallly requires no effort to do this part of the problem right? Should I find the value for psi?

No, you need to find E.

 

[noblegas]

No, you need to find E.

If E is equal to H and H is given in the problem, then why wouldn't I have my H.

 

[jdwood983]

If E is equal to H and H is given in the problem, then why wouldn't I have my H.

\hat{H} is not equal to E. \hat{H} is an operator while E is a number.

 

[noblegas]

\hat{H} is not equal to E. \hat{H} is an operator while E is a number.

E is an eigen value; Is E = T+V = exp(-i*E_n*t/(h-bar)) , which is what my eigenvlaues energies.

 

[jdwood983]

E is an eigen value; Is E = T+V = exp(-i*E_n*t/(h-bar)) , which is what my eigenvlaues energies.

E is an eigenvalue, but it is still just a number. That value of E you have written looks more like the unitary operator U for the time-evolution of the Hamiltonian, but it would not be a solution of the eigenvalues for the Hamiltonian in general. You may want to review your textbook for definitions of operators, eigenvalues, and wave functions for further help.

 

[noblegas]

E is an eigenvalue, but it is still just a number. That value of E you have written looks more like the unitary operator U for the time-evolution of the Hamiltonian, but it would not be a solution of the eigenvalues for the Hamiltonian in general. You may want to review your textbook for definitions of operators, eigenvalues, and wave functions for further help.

Well they say I need to use H^2 to find E. Assuming that H\varphi=E\varphi . Then H^2\varphi=H(H\varphi)=H(E\varphi)=E(H\varphi)=E^2(\varphi) Am I headed in the right direction?

 

[jdwood983]

Well they say I need to use H^2 to find E. Assuming that H\varphi=E\varphi . Then H^2\varphi=H(H\varphi)=H(E\varphi)=E(H\varphi)=E^2(\varphi) Am I headed in the right direction?

No. E is still just a number while H is an operator. Do you know what an operator is?

 

[jdwood983]

For part (a), I would assume that H^2=|H|^2=H^*H, that is you need to multiply the complex conjugate to the original matrix. You know what the spin matrices are, I'm assuming, so you multiply \alpha and \beta to each of those matrices and then add them to form one 2x2 matrix:

<br /> H=\frac{\hbar}{2}\left(\begin{array}{cc}0&amp;\alpha\\ \alpha&amp;0\end{array}\right)+\frac{\hbar}{2}\left(\begin{array}{cc}0&amp;-i\beta\\ i\beta&amp;0\end{array}\right)<br />


<br /> H^*=\frac{\hbar}{2}\left(\begin{array}{cc}0&amp;\alpha\\ \alpha&amp;0\end{array}\right)+\frac{\hbar}{2}\left(\begin{array}{cc}0&amp;i\beta\\ -i\beta&amp;0\end{array}\right)<br />

Add the matrices of each and then multiply the two matrices and you will have H^2. You can then use the characteristic equation, \det(H-\lambda\mathbb{I})=0 to find the eigenvalues, \lambda, in terms of \hbar,\,\alpha and \beta. (Though I'm not sure why your professor suggested using the solution to H^2 to find the eigenvalues and not the original matrix, H.)