Go Back   Physics Help Forum > College/University Physics Help > Quantum Physics

Quantum Physics Quantum Physics Help Forum

Like Tree1Likes
  • 1 Post By topsquark
Reply
 
LinkBack Thread Tools Display Modes
Old Mar 23rd 2017, 10:46 AM   #1
Junior Member
 
Join Date: Mar 2017
Posts: 2
Hamilton operator affecting observable

'm working on this problem "Consider an experiment on a system that can be described using two basis functions. In this experiment, you begin in the ground state of Hamiltonian H0 at time t1. You have an apparatus that can change the Hamiltonian suddenly from H0 to H1. You turn this apparatus on at time t1. Then, at time tD > t1, you perform a measurement of an observable,D. In matrix notation, the Hamiltonians and the operator D are given below:

H0 = [0 -4; -4 6], H1 = [1 0; 0 3]"

b) If you perform many, many measurements, what will be the average observed value of D as a function of t1 and tD?
d)You perform the experiment, but suspect your apparatus is malfunctioning and
turning off at some systematic time t2 between t1 and tD. In other words, at some time t2, you suspect the Hamiltonian is reverting to H0. What qualitative effect would this have on your results from part b)? [Note: this only requires a qualitative description, not a full calculation.]"

So I found that <D> = (2/5)(exp(2i[tD-t1])+exp(-2i[tD-t1])+1) which would be the answer in b). I can't seem to figure out d). I was thinking that because H0 is the ground state, the energy would be lower than the one in b). Therefor D would be lower as well. But than again, the two eigenvalues of H0 is -2 an 8. Now -2 is lower than the eigenvalues of H1, but 8 is higher than both the eigenvalues of H1. How can I know which state we are in?
UiOStud is offline   Reply With Quote
Old Mar 23rd 2017, 04:40 PM   #2
Forum Admin
 
topsquark's Avatar
 
Join Date: Apr 2008
Location: On the dance floor, baby!
Posts: 2,450
Originally Posted by UiOStud View Post
'm working on this problem "Consider an experiment on a system that can be described using two basis functions. In this experiment, you begin in the ground state of Hamiltonian H0 at time t1. You have an apparatus that can change the Hamiltonian suddenly from H0 to H1. You turn this apparatus on at time t1. Then, at time tD > t1, you perform a measurement of an observable,D. In matrix notation, the Hamiltonians and the operator D are given below:

H0 = [0 -4; -4 6], H1 = [1 0; 0 3]"

b) If you perform many, many measurements, what will be the average observed value of D as a function of t1 and tD?
d)You perform the experiment, but suspect your apparatus is malfunctioning and
turning off at some systematic time t2 between t1 and tD. In other words, at some time t2, you suspect the Hamiltonian is reverting to H0. What qualitative effect would this have on your results from part b)? [Note: this only requires a qualitative description, not a full calculation.]"

So I found that <D> = (2/5)(exp(2i[tD-t1])+exp(-2i[tD-t1])+1) which would be the answer in b). I can't seem to figure out d). I was thinking that because H0 is the ground state, the energy would be lower than the one in b). Therefor D would be lower as well. But than again, the two eigenvalues of H0 is -2 an 8. Now -2 is lower than the eigenvalues of H1, but 8 is higher than both the eigenvalues of H1. How can I know which state we are in?
Two things. 1st it might help (though it probably doesn't matter much) if we had an inkling of what property D is measuring. The 2nd concerns the time average of D. As it turns out <D> is better noted to be
$\displaystyle <D> = \frac{4}{5} \left ( cos (2 [ t_D - t_1] \right ) + \left ( \frac{2}{5} \right )$.

Because this is real we know that D is Hermitian. This form should make it easier to consider what happens at $\displaystyle t_2$. I am also going to assume that the energy eigenvalues are also good quantum numbers.

So, armed with that and the fact that (I presume) the basis vectors for some $\displaystyle D_0$ is the ground state, and $\displaystyle D_1$ would be the larger eigenvalue. We can say that the wavefunction before $\displaystyle t_2$ is under time evolution. After $\displaystyle t_2$ the wavefunction is a mixed state of the $\displaystyle D_1$ state and the $\displaystyle D_0$ state. So what should happen in regard to the time evolution after $\displaystyle t_2$?

-Dan
UiOStud likes this.
__________________
Do not meddle in the affairs of dragons for you are crunchy and taste good with ketchup.

See the forum rules here.
topsquark is online now   Reply With Quote
Reply

  Physics Help Forum > College/University Physics Help > Quantum Physics

Tags
affecting, hamilton, observable, operator



Thread Tools
Display Modes


Similar Physics Forum Discussions
Thread Thread Starter Forum Replies Last Post
Lax operator astrogal Theoretical Physics 0 Oct 17th 2013 04:42 AM
Elastic properties affecting momentum transfer Ketman Kinematics and Dynamics 2 Jun 11th 2010 02:34 PM
Factors affecting E-field werehk Electricity and Magnetism 3 Jun 14th 2009 06:15 PM
Checking understanding on factors affecting equilibrium temperature qazxsw11111 Thermodynamics and Fluid Mechanics 0 Feb 25th 2009 02:19 PM


Facebook Twitter Google+ RSS Feed